Every complex K3 surface is Oka.
"Oka" is a flexibility property of complex manifolds. For a compact
manifold like a K3 surface it says, informally, that the only
obstruction to a holomorphic map existing is a topological one. The
precise form proved is the convex approximation property (CAP):
For every m >= 1, every compact convex set K inside C^m, every
open neighborhood U of K, every holomorphic map f : U -> X and
every epsilon > 0, there is a holomorphic map F : C^m -> X with
sup_{z in K} d_X( F(z), f(z) ) < epsilon,
where d_X is any fixed smooth Hermitian distance on X.
The theorem has a short history. Forstneric proved that CAP is
equivalent to the Oka property for any complex manifold. Buzzard
and Lu proved that elliptic and Kummer K3 surfaces are dominable
by C^2. Forstneric and Larusson proved strong dominability for
Kummer surfaces and later surveyed the general K3 question. Xie
and Zhao recently proved that, in every local deformation family of
K3 surfaces, the Oka fibers form a dense G_delta. Every complex K3
surface has the property.
Forstneric, Runge approximation on convex sets, Ann. Math. 163 (2006). ./localrefs/forstneric-0402278-runge-approximation-convex-sets.pdf
Buzzard and Lu, Algebraic surfaces holomorphically dominable by C^2,
Invent. Math. 139 (2000). ./localrefs/buzzard-lu-0005232-algebraic-surfaces-holomorphically-dominable.pdf
Xie and Zhao, Periods of Oka K3 surfaces are dense, arXiv:2608.24392. ./localrefs/xie-zhao-2608.24392-periods-of-oka-k3-surfaces.pdf
Analytically, two transverse complete strip maps give a local
enlarging operation L that is
uniform in any number of passive holomorphic parameters, and L
everywhere forces CAP via a polynomial source change and a Holder
gluing across real faces. Geometrically, L everywhere on a
projective K3 follows from Chen-Gounelas genus-one pencils, and
on a general K3 from an open region of the period domain reached
by quadric smoothing or a rational-period construction, joined by
Ratner's orbit theorem and Torelli. Exact symplectic sewing of
annular chains keeps a fixed transverse width even for unbounded
chains.
Chen-Gounelas, Curves of maximal moduli on K3 surfaces, Forum Math. Sigma 10 (2022).
./pdf/chen-gounelas-2007.01735-curves-maximal-moduli-k3.pdf
NOTATION
C the complex numbers.
R the real numbers.
Z the integers.
Q the rationals.
P^1 the Riemann sphere (= C union {infinity}).
P^n complex projective n-space.
Delta_r the open disc {z in C : |z| < r}.
Delta the unit disc.
C* the pointed plane C minus {0}.
Re(z), Im(z), |z|, conj(z) usual.
O(D) holomorphic functions on D.
C^m m complex variables z = (z_1,...,z_m).
||z|| max norm for polydiscs, |z| Euclidean when needed.
dist(Q) the distance to a compact set Q.
K3 a compact complex surface with H^1 = 0 and trivial
canonical bundle.
Lambda the K3 lattice 3U + 2 E_8(-1).
D the K3 period domain: oriented positive 2-planes in
Lambda tensor R.
L the local enlarging property (defined precisely below).
CAP the convex approximation property.
All maps are holomorphic unless stated. A map "on a compact set"
is a map on some open neighborhood of that set. Approximation
may always lose a prescribed positive amount of a domain margin;
one uses this without comment.
A complex manifold X of dimension n is a space that locally looks
like an open subset of C^n, with the changes of coordinates given
by holomorphic maps. The examples to keep in mind are:
* C^n itself, and any open subset of it (a domain in C^n).
* P^1, the Riemann sphere, obtained from C by adding a point
at infinity with z -> 1/z as the change of coordinates.
* Complex tori, C^n divided by a rank-2n lattice.
* Smooth projective complex surfaces, i.e. the common zero set
in P^3 of a homogeneous polynomial with nonsingular solutions,
more generally any compact complex surface embedded in P^N.
A holomorphic map f : X -> Y between complex manifolds is one that
looks holomorphic in each local coordinate chart. Holomorphic maps
are very rigid: once one knows one on a small open set, analytic
continuation determines it globally when X is connected. This is
why the existence of holomorphic maps to a compact space is usually
an obstruction-theoretic question and not an approximation one.
Throughout, X is almost always a compact complex surface (so
n = 2) that is K3. The source C^m can have any dimension. Care is taken
about approximating a given local map f : (neighborhood of K) -> X
by an entire map F : C^m -> X.
DEFINITION. A complex K3 surface is a compact connected complex
surface X with
H^1(X, Z) = 0 (simply connected),
K_X = 0 (trivial canonical bundle).
The condition K_X = 0 means X carries a holomorphic 2-form sigma
that vanishes nowhere. Such a sigma is unique up to a nonzero
scalar. It is a kind of volume form, and it is the reason K3
surfaces behave like a very rigid version of a torus.
Examples of K3 surfaces:
* A smooth quartic surface in P^3.
* A smooth complete intersection of type (2,3) in P^4.
* The Kummer surface Kummer(A) of a 2-dimensional complex
torus A, obtained by blowing up the 16 half-period points
of A / (-1).
* Elliptic K3 surfaces: K3 with a fibration pi : X -> P^1
whose general fiber is an elliptic curve.
Every K3 surface is Kahler, by a theorem of Siu. It need not be
projective, and it need not contain any curve. That lack of
algebraic geometry is the geometric obstacle of the projective case.
Siu, Every K3 surface is Kahler, Invent. Math. 73 (1983).
doi:10.1007/bf01393829
Two-dimensional cohomology H^2(X, Z) is a free abelian group of
rank 22. With the cup product it is an even unimodular lattice
of signature (3, 19), isometric to
Lambda = 3 U + 2 E_8(-1),
where U is the hyperbolic plane (rank 2, form ((0,1),(1,0))) and
E_8(-1) is minus the E_8 root lattice. This is the K3 lattice.
A marking is an isometry H^2(X, Z) -> Lambda.
The real and imaginary parts of a chosen holomorphic 2-form sigma
span a real 2-plane P in Lambda tensor R. The cup product on this
plane is positive definite and P inherits an orientation from
(Re sigma, Im sigma). The oriented positive 2-plane P is called
the period of the marked surface (X, marking). Conversely, the
global Torelli theorem says essentially that a marked K3 surface
is determined by its period: if two marked K3 surfaces have the
same period and their markings can be matched in a suitable sense,
then the underlying unmarked surfaces are isomorphic. Torelli is used below.
A period plane P can contain rational directions. That is, the
intersection P cap (Lambda tensor Q) can have dimension 0, 1, or
2. These three cases behave very differently and organise the whole
argument.
The Oka property for a complex manifold X asks that the natural
inclusion of holomorphic maps into continuous maps be surjective
up to deformation, uniformly in a suitable approximation sense.
The precise definition uses Stein sources, jets and homotopies.
For manifolds, Forstneric proved an equivalent "CAP":
CONVEX APPROXIMATION PROPERTY (CAP).
For every m >= 1, every nonempty compact convex set K inside
C^m, every open neighborhood U of K, every holomorphic map
f : U -> X and every epsilon > 0, there is an entire
F : C^m -> X with sup_{z in K} d_X(F(z), f(z)) < epsilon.
The equivalence is Forstneric's theorem:
"Runge approximation on convex sets implies the Oka property",
Ann. of Math. 163 (2006). So proving CAP for X proves X is Oka.
CAP has two structural features that make it a good target.
First, CAP is a purely approximation statement. Unlike "X is an
image of C^n", CAP quantifies over all convex sources K and all
local sources f. It is therefore strong enough to imply both
approximation and interpolation (the interpolation section).
Second, CAP has a local version. Let
L(x) mean: X has a witness neighborhood at x,
defined precisely below. Roughly, L(x) says that one can
always enlarge a planar disc inside a source polydisc beyond the
original source as long as the image stays in a fixed small target
neighborhood of x. The main analytic theorem is then:
THEOREM (analytic core). If Y is a complex surface with L(y)
for every y in Y, then Y has CAP in every source dimension.
The geometric problem for K3 surfaces therefore reduces to
proving L(x) for every x on every complex K3 surface. The
geometry must supply two transverse complete strip maps through
every point of every K3 surface; the analysis must then convert
those strips into L and L into CAP. Both steps are carried out
below using nothing beyond Stein theory, Oka-Weil approximation,
Hormander's L^2 d-bar theorem, and Cauchy estimates.
Hormander, L2 estimates and existence theorems for the dbar operator, Acta Math. 113 (1965).
doi:10.1007/bf02391775
Write Delta_r = {z in C : |z| < r}. A "map on a compact set" always
means a map holomorphic on some open neighborhood of that set.
DEFINITION 1.1. A special pair D subset D' consists of two closed
topological discs in C with piecewise smooth boundary, such that D'
is obtained from D by attaching another such disc along a proper
boundary arc. In symbols, D and D' are closed discs and D' minus D is a
closed disc D'' attached along a smooth arc in the boundary. Up to
an ambient homeomorphism of C, D = [-2,0] x [-1,1] and
D' = [-2,3] x [-1,1] in C, but coordinates are not needed until the
two-strip criterion below.
DEFINITION 1.2 (property L). A complex surface X has property L at
a point x of X if there is an open neighborhood V of x with the
following property: for every special pair D subset D' subset C,
every pair of closed polydiscs P_0 compactly in interior(P) in
C^q (q >= 0), and every holomorphic map f from a neighborhood of
D x P into V, the restriction f | (D x P_0) is the uniform limit
of maps holomorphic near D' x P_0 and with values in X.
The open set V is a witness neighborhood. Denote the locus of
points where L holds by Oka_L(X) or simply "the L locus". This
locus is open:
LEMMA 1.3. If L(x) holds, then L(y) holds for all y in some open
neighborhood of x.
Proof. Shrink the witness neighborhood V to an open V_0 with
closure compactly contained in V. If y is in V_0, one can use V_0 as
witness for y: any map into V_0 is automatically a map into V, so
any approximation that works with witness V also works with witness
V_0. QED
Because of Lemma 1.3, the L locus can be verified pointwise and
then automatically opens up.
The two-strip criterion is a strong sufficient criterion for L: two complete
transverse strip maps. The disc-propagation argument shows that, even with L only on a
complement of a curve, one can propagate it to that curve.
A "strip" here is C x Delta_b or Delta_b x C. One coordinate is
"long" (entire on C); the other is "bounded" (a small disc).
Let sigma_1 : C x Delta_b -> X and sigma_2 : Delta_b x C -> X be
holomorphic maps. The requirements are:
(i) sigma_1 and sigma_2 are locally biholomorphic at (0,0),
i.e. the differential is invertible there.
(ii) sigma_1(z,0) = sigma_2(z,0) for |z| < b. (Exact axis
agreement.)
(iii) sigma_2 is locally biholomorphic at (0,c) for some real
c with |c| < b.
The argument will also allow a limit version: a sequence of pairs whose long
coordinate domains are discs of radii -> infinity, whose bounded
radii are bounded below, and whose maps converge near (0,0) and
(0,c) to nondegenerate germs. Then the point x = sigma_2(0,c) also
gets L.
PROPOSITION 2.1 (two complete directions). Under the above
hypotheses, X has property L at x := sigma_2(0,c).
Before proving it, it is necessary to state one elementary geometric fact about
planar pieces and one analytic fact about splitting holomorphic
functions defined on their intersection.
LEMMA 2.2 (planar pieces). Given a special pair D subset D' and
an open neighborhood N of D, there exist compact sets A, B_0, B_1 in
C and bounded open sets U_A, U_0, U_1 in C such that:
(i) closure(U_A) subset interior(A),
closure(U_i) subset interior(B_i) for i = 0,1;
(ii) D subset U_0 and D' subset U_A u U_0 u U_1;
(iii) A cap D = empty, B_0 cap B_1 = empty, B_0 subset N;
(iv) the compact sets D u A and B_0 u B_1 are polynomially
convex;
(v) the two closed fringes closure(U_A minus U_B) and
closure(U_B minus U_A) are positively separated, where
U_B = U_0 u U_1.
Proof. Apply an ambient homeomorphism of C sending the attached
pair to D = [-2,0] x [-1,1] and D' = [-2,3] x [-1,1]. Such a
homeomorphism exists by Jordan-Schoenflies on the attaching arc.
Choose numbers
0 < c < c' < a' < a < b < b' < d' < d < 3, 0 < delta < eps.
Define horizontal strips:
piece compact piece open piece
----- ------------- ----------
U_0, B_0 [-2-eps, a] x J (-2-delta, a') x J'
U_A, A [c, d] x J (c', d') x J'
U_1, B_1 [b, 3+eps] x J (b', 3+delta) x J'
where J = [-1-eps, 1+eps] and J' = (-1-delta, 1+delta). These
choices satisfy (i)-(iii) if a, eps are small. The fringes in (v)
have closed horizontal intervals [a', b'] and [-2-delta, c'] u
[d', 3+delta], so they are compact and disjoint. Polynomial
convexity in (iv): D u A is the disjoint union of two filled Jordan
discs with connected complement, and similarly for B_0 u B_1.
QED
One analytic lemma, used repeatedly below to glue sprays:
LEMMA 2.3 (bounded planar Cousin splitting). Let U_A, U_B subset
C be bounded open sets with positively separated closed fringes,
and let C = U_A cap U_B. For every open parameter polydisc P and
every k >= 1 there are bounded linear operators
R_A : H^infty(C x P, C^k) -> H^infty(U_A x P, C^k),
R_B : H^infty(C x P, C^k) -> H^infty(U_B x P, C^k),
satisfying R_A u - R_B u = u on C x P. Their operator norms
depend only on the planar sets U_A, U_B and not on the parameter
polydisc P or its dimension.
Proof. Choose a smooth function chi : C -> R that is 0 near
closure(U_A minus U_B) and 1 near closure(U_B minus U_A), with bounded
first derivative. On U_A extend chi u by 0 off C; on U_B extend
(chi-1) u similarly. Both extensions have planar d-bar derivative
u d-bar(chi) on C x P and 0 elsewhere in Omega = U_A u U_B.
Extend this bounded coefficient by 0 to all of C and put
v(s,t) = (1/pi) int_C u(zeta,t) . (d-bar chi)(zeta) / (s-zeta) dA(zeta)
for s in Omega and t in P. The distributional identity
(d-bar)_s ( 1/(pi s) ) = delta_0
gives the required d-bar equation: d-bar(s) v = u d-bar(chi) on
Omega x P.
If d = diameter of Omega, then
||v||_{Omega x P} <= 2 d ||d-bar chi||_infty ||u||_{C x P}.
The kernel is locally integrable, so v is continuous in s and
holomorphic in t; thus v is holomorphic in (s,t) on Omega x P.
Now put
R_A u = chi u - v on U_A x P,
R_B u = (chi-1) u - v on U_B x P.
These are holomorphic in (s,t), have the asserted bounds, and
R_A u - R_B u = chi u - (chi-1) u - 2v - (-v)
= u.
No boundary continuity of u is needed and the operator norm is
bounded by 2d||d-bar chi||_infty + ||chi||_infty + 1, which is
independent of P. QED
Now the proof of Proposition 2.1.
Proof of Prop. 2.1. Choose rho > 0 so that 5 rho < b, and so that
tau = sigma_2^{-1} o sigma_1 is defined near the closed bidisc of
radius 3 rho around 0, using the inverse branch of sigma_2 at the
origin. The differential D tau is invertible in this bidisc. By
the axis identity, tau(z,0) = (z,0). The matrix
T_0(z) = D tau (z, 0)
and its inverse are bounded on |z| <= 2 rho. Choose a biholomorphic
chart Q_c for sigma_2 about (0,c) whose first coordinate has modulus
< rho/16 on the image. Take V to be a relatively compact open
subset of that image; V is the witness neighborhood for L at x.
Fix arbitrary source data: a special pair D subset D' in C, closed
polydiscs P_0 compactly in int(P) in C^q, and a holomorphic map
f from a neighborhood of D x P into V.
Choose an open polydisc P_* with P_0 compactly in P_* compactly in
int(P). In the chart Q_c write
g = sigma_2^{-1} o f = (g_1, g_2) : (neighborhood of D x P_*) -> C^2.
There is an open neighborhood N of D in C such that g is defined on
N x (neighborhood of closure(P_*)) and |g_1| < rho/8 there. No
smallness of g_2 - c beyond membership in the sigma_2-chart is
needed.
Choose planar pieces as in Lemma 2.2 with this N. The compact sets
D u A and B_0 u B_1 are polynomially convex, and their products
with the polydisc P_* remain polynomially convex. Apply the
Oka-Weil approximation theorem (or Mergelyan scalar approximation
with parameters) to get a polynomial h_1 in the source variables
(z, p) satisfying:
sup_{B_0 x P_*} |h_1 - g_1| < eps_1,
sup_{B_1 x P_*} |h_1| < eps_1.
Choose eps_1 so small that |h_1| < rho/2 on B_0 u B_1, uniformly in
p. Independently, for every eta > 0 small enough there is a
polynomial h_2 in (z, p) satisfying
sup_{D x P_*} |h_2 - g_2| < eta,
sup_{A x P_*} |h_2| < eta.
Set h = (h_1, h_2) and C_AB = U_A cap U_B. On U_B x P_* the
matrix T = T_0(h_1) is holomorphic and boundedly invertible:
T(z,p) = D tau (h_1(z,p), 0). Because |h_1| < rho/2 < 2 rho on
U_B x P_*, and T_0 is invertible with bounded inverse on |z| <=
2 rho, this gives a uniform bound ||T||, ||T^{-1}|| <= M for a constant
M >= 1.
The goal is to find small corrections alpha, beta to the source so that
tau( h + alpha ) = h + beta,
on U_A x P_* and U_B x P_*, respectively, with the convention
alpha = R_A u and beta = T R_B u for a bounded holomorphic
"parameter" u in H^infty(C_AB x P_*, C^2). This becomes a fixed-
point equation:
u = Tcal(u) := R_A u + T^{-1} ( h - tau ( h + R_A u ) ).
A direct estimate. On the overlap |h_1| < rho/2 and |h_2| < eta.
Bounded first and second derivatives of tau, together with the axis
identity tau(z,0) = (z,0), give a constant M (renaming it if
necessary) with
||tau(h) - h|| <= M eta,
|| D tau(h + alpha) - T || <= M (eta + ||alpha||).
These M do not depend on degrees or uncontrolled parts of h because
all estimates are made when h is in a fixed bounded polydisc and
tau has bounded derivatives there. If kappa bounds both splitting
operators R_A, R_B (from Lemma 2.3) and L_0 = 2 M^2 + 1, then on
the closed ball ||u|| <= L_0 eta,
||Tcal(0)|| <= M^2 eta < L_0 eta,
||D Tcal(u)|| <= M^2 kappa (1 + kappa L_0) eta <= 1/2
for eta small enough. So Tcal maps the ball strictly into itself
and is a contraction. The contraction principle gives a unique
fixed point u, and
||alpha|| <= kappa L_0 eta,
||beta|| <= M kappa L_0 eta.
All the values of tau involved lie in the closed bidisc of radius
3 rho around 0 because eta, ||alpha|| are small; the chosen M
absorbs this.
Now compose with sigma_1, sigma_2. On U_A x P_*, only the second
coordinate of h + alpha needs to be small, to fit inside sigma_1's
long coordinate. On U_B x P_*, only the first coordinate of h +
beta needs to be small, to fit inside sigma_2's long coordinate.
More precisely:
sigma_1(h + alpha) defined on U_A x P_*, holomorphic,
sigma_2(h + beta) defined on U_B x P_*, holomorphic.
By the very choice of alpha, beta these two maps agree on
U_A cap U_B x P_*. So they define a single holomorphic map from
U_A u U_B (which contains D') x P_* to X. On D x P_* the
sigma_2-formula is evaluated close to g = sigma_2^{-1} o f, so it
approximates f as eps_1, eta -> 0.
Since D' subset U_A u U_B and P_0 compactly in P_*, this proves
L at x. QED
The finite-length version. If sigma_1, sigma_2 are replaced by a
sequence of pairs with long radii -> infinity, use the limit chart
at (0,c) to choose V, g and the two polynomials h_1, h_2. Convergence
on slightly larger compact coordinate neighborhoods gives uniform
first and second derivative bounds, uniform inverse-chart domains,
uniform constants in the contraction argument. Exact axis agreement
gives the same O(mu) initial error for every pair. The polynomials
are bounded on the fixed planar pieces x closure(P_*) because those
are bounded, so they only use a finite range of either long
coordinate. Choose a sufficiently late pair to contain these ranges
and to make its chart near (0,c) sufficiently close to the limit
chart. Same proof gives L at x.
REMARK 2.4 (using the shifted centre). Proposition 2.1 gives L at
x = sigma_2(0,c) for the chosen c. If it is only known that x lies on the
central curve of sigma_2 and sigma_1 does not visibly agree with it at
the origin, Remark 2.5 below arranges exact axis agreement with a
shifted second strip whose bounded coordinate has value c at x.
REMARK 2.5 (aligning two strips). Suppose it is given two strip
maps locally biholomorphic at chosen meeting preimages, with
transverse central curves at that meeting point. One can then put
them in the exact axis-agreement form of Proposition 2.1 as follows.
Locally lift the first central curve through the inverse of the
second strip, writing the lift as (a(z), t(z)), where its bounded
coordinate is a. Replace the second strip by
sigma_2~ (z, w) = sigma_2 ( a(z), t(z) + w ).
Transversality says a'(0) != 0, and at time w = 0 there is exact axis
agreement. If the original second central curve also passes through
a chosen x and its strip is regular there, some finite c makes
sigma_2~(0,c) = x.
For finite-length families whose central germs converge to transverse
germs through a common point, the implicit function theorem first
gives nearby intersection points and meeting preimages. Translate
the meeting preimages to the origin and apply the same construction.
Convergence plus transversality give a uniformly positive
bounded-coordinate radius; the translations and the function t only
lose a bounded amount of long-coordinate length. Thus aligned
families exist in the finite-length setting.
When there is a holomorphic map f : S -> X on a Stein manifold, the
right object to work with near its graph is not f itself but a
"spray": a family of maps through f, parametrised by small vectors
in the tangent directions. Sprays give room to move f around
by small amounts, which is what makes gluing possible.
LEMMA 3.1 (local chart sprays). Let f : S -> X be holomorphic on a
Stein manifold S. Let K be a compact subset, compactly contained in S. Assume that
the pulled-back tangent bundle f^*TX is holomorphically trivial on S.
Then, after shrinking the source neighborhood of K, there is a
holomorphic map
f# : (neighborhood of K) x Delta_r^2 -> X,
f#(z, 0) = f(z),
for some r > 0, such that for each fixed z the vertical map
v -> f#(z, v) is injective with invertible differential at v = 0.
The triviality hypothesis holds whenever the source is a
contractible Stein manifold, or whenever the source is a planar
domain times a polydisc. This is the case for every application.
Proof. The graph Gamma(f) of f is a closed complex submanifold of
S x X. It is Stein, and by Siu's theorem on Stein neighborhoods of
Stein submanifolds (Inventiones 38, 1976) it has an open Stein
neighborhood U in S x X. Consider the vertical tangent bundle
of the projection U -> S: the bundle of tangent vectors to U that
project to 0 in S. A holomorphic frame of f^*TX along the graph is
a set of two holomorphic sections that are linearly independent
fiberwise. By Cartan's Theorem B on the Stein manifold U, these
sections extend to holomorphic vertical vector fields V_1, V_2 on U.
Compose the small-time flows of V_1 and V_2: put
f#(z, v) = Flow_{V_1}^{v_1} o Flow_{V_2}^{v_2} (f(z)),
v = (v_1, v_2) in Delta_r^2. At v = 0 this reproduces f(z), and the
vertical differential D_v f# at v = 0 is exactly the chosen frame.
Compactness of K and the holomorphic inverse function theorem give
a common radius r on a neighborhood of K, and injectivity in each
parameter fibre after possibly reducing r.
For the triviality claim: a planar domain has the homotopy type of
a finite one-dimensional CW complex, so every complex vector bundle
on its product with a polydisc is topologically trivial. The
Oka-Grauert principle upgrades this to holomorphic triviality.
The same argument goes through for any contractible Stein source.
QED
Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976).
doi:10.1007/bf01390170
LEMMA 3.2 (deforming a map on a relatively compact Stein domain).
Let pi : Xcal -> B be a holomorphic family of complex manifolds over
a complex base B. Let f : S -> Xcal_{b_0} be holomorphic from a
Stein manifold S, with image inside the submersion locus of pi (the
locus where pi is a submersion). On every relatively compact
source domain S_0 compactly in S there is a holomorphic family
of maps f_b : S_0 -> Xcal_b, for b near b_0, with f_{b_0} = f | S_0.
Proof. Work in local holomorphic coordinates on B at b_0. The
graph of f has a Stein neighborhood U in S x (submersion locus) by
Siu again. Consider the surjection from tangent vectors of U
vertical over S, to the pulled-back base tangent bundle. Its
kernel is a coherent sheaf on U, so Cartan's Theorem B gives
surjectivity on global sections. In particular each base coordinate
vector field d/db_j lifts to a holomorphic vertical vector field
W_j on U.
Flow W_j successively for the small coordinate displacement
b_j - (b_0)_j. On the relatively compact source S_0 all flows share
a common time radius. Each flow is vertical over S, and the
projection to B is exactly the coordinate translation b_j. The
composition is a holomorphic map f_b : S_0 -> Xcal_b with
f_{b_0} = f | S_0. QED
Two sprays close on an overlap cannot in general be glued exactly,
but they can be glued after small fibre reparametrisations, provided
the overlap has a positive collar (a "buffer").
Setup. Let Q subset C^n be compact and convex. For r > 0 let
Q_r = {z in C^n : dist(z, Q) < r}. Two open sets A, B cover a
neighborhood of closure(Q_r). Assume a BUFFERED OVERLAP: there is
a smooth cutoff chi : (A u B) -> R that is 0 near A minus B and 1 near
B minus A, with the transition region inside A cap B and with a positive
collar width that does not shrink when Q_r is replaced by Q_{r'} for
r' >= r - h. An auxiliary variable v ranges over a ball B_b in
C^N. All operations below act on z and preserve holomorphic
dependence on v.
Additive splitting. For u in H^infty(A cap B, C^k) bounded, chi u
(extended by 0 off A cap B) and (chi-1) u (extended by 0 off A cap B)
have the same d-bar derivative on A u B; call it eta = u d-bar(chi).
Solve d-bar w = eta on an intermediate convex thickening Q_{r - h},
with the weighted minimal L^2 solution using the weight |z|^2 (see
Hormander's Acta 1965 article, Theorem 2.2.1). The weight is
bounded above and below on the fixed bounded domains, so w has L^2
norm at most a fixed constant times that of eta. Cauchy estimates
on the buffered transition region bound any fixed number of
derivatives of eta by powers of h^{-1} ||u||_infty. Interior
elliptic estimates on subsets of Q_{r-h} then bound w uniformly by
C h^{-p} ||u||_infty for some fixed integer p depending only on n
and k. The minimal solution operator is linear and fixed, so it
preserves holomorphic dependence on v. Subtracting w from chi u
and (chi - 1) u gives bounded holomorphic extensions to A and B
respectively:
R_A u = chi u - w on A,
R_B u = (chi - 1) u - w on B,
R_A u - R_B u = u on A cap B.
The bounds ||R_A||, ||R_B|| <= C h^{-p} are uniform.
LEMMA 4.1 (small nonlinear gluing on a buffered open overlap).
In the geometry above, let F_A and F_B be two holomorphic sprays
into a complex manifold X, related on their overlap by a fibre
coordinate change
gamma(z, v) = v + c(z, v), v in B_b.
Suppose ||c||_infty can be made arbitrarily small while b > 0 stays
fixed. After any prescribed loss of base and fibre margins, the two
sprays admit fibre reparametrisations close to the identity that make
them agree. Evaluating at v = 0 gives a holomorphic map glued
across the overlap. It is arbitrarily close to the original F_A on
each compact subset of A where F_A is fixed or has a uniform modulus
of continuity in v.
Proof. Use the additive splitting to find holomorphic a, b with
b - a = c. Shrink the base and fibre domains by h. The splitting
bounds and fibre Cauchy estimates give, after enlarging the fixed
integer p if necessary,
||a|| + ||b|| <= C h^{-p} e, e := ||c||_infty,
including their needed fibre derivatives on intermediate smaller
balls. If C h^{-p} e is small relative to these margins then
id + b has a fibre inverse on a smaller ball, uniformly in z.
The new transition is
(id + b)^{-1} o gamma o (id + a) = id + d,
and the equation determining d is
d = c(v + a) - c(v) - { b(v + d) - b(v) }.
The first difference is quadratic in e up to powers of h^{-1}.
The operation in braces is contractive on a small supremum-norm
ball, by the first fibre derivative bound. Consequently
||d|| <= C h^{-p} e^2.
These estimates involve a fixed finite number of derivatives, so
one common exponent p works throughout the iteration; no derivatives
of increasing order are ever requested.
Choose losses h_i = h_0 2^{-i} whose sum fits within the prescribed
margins, leaving intermediate buffers at each step. With
A = C h_0^{-p}, the quadratic inequality reads
e_{i+1} <= A 2^{p i} e_i^2.
Put E_i = A 2^{p (i+1)} e_i. Then E_{i+1} <= E_i^2, and choosing
E_0 < 1/2 gives doubly exponential decay. In particular every sum
sum_i h_i^{-q} e_i converges for any fixed q, and is as small as
desired with e_0.
The successive fibre reparametrisations compose on the nested
domains. Their displacements stay within the reserved buffers, and
their fibre derivatives have summable deviations from the identity.
The compositions therefore converge uniformly. The limiting
transitions are the identity. Composing the two original sprays
with these limiting side maps produces equal maps on the overlap.
The target approximation statement follows from the stated
continuity control and from the smallness of the limiting fibre
corrections. QED
The purpose of L is to enlarge a planar source, keeping any
number of small auxiliary holomorphic parameters bounded throughout.
PROPOSITION 5.1 (one planar enlargement). Let K be a closed
circular disc of radius r, or a closed circular annulus with outer
radius r, in C. Let f be holomorphic near K x P, where P is a
closed polydisc in C^q (q >= 0). Suppose that for every zeta on
the outer circle of K there is a coordinate witness neighborhood
V_zeta for L containing f({zeta} x P). Then, for every R > r and
every P_0 compactly in interior(P), the map f can be uniformly
approximated on K x P_0 by a map holomorphic near the enlarged
disc (respectively annulus with the same inner radius) of outer
radius R, times P_0.
The passive parameters are the point: every estimate is uniform in
p in P_0, and the approximant is holomorphic in p at the same time
as in z.
Proof. The construction proceeds in explicit steps with a finite
approximation budget.
Step 1. Finite arcing of the outer boundary. Pick an intermediate
closed parameter polydisc P_* strictly between P_0 and int(P).
By compactness of the outer circle and continuity of f, one can
choose a finite cover of the outer circle by closed arcs
I_1, I_2, ..., I_N (overlapping at endpoints) such that, for each
j, all values of f on a slightly larger arc I_j^+ times P_* lie in
a coordinate witness neighborhood V_j for L. At a common endpoint
of consecutive arcs, the compact set of values lies in V_j cap V_{j+1}
with a positive target margin. Below is constructed a map on a
slightly larger outer radius than R and then restrict.
Step 2. Radial spokes. Between each pair of consecutive arcs
choose a small filled polar rectangle (an annular "wedge") W_j
crossing the outer circle of K, on which f and a local chart spray
(for instance the trivial spray from Lemma 3.1 with a planar source)
have their values in both adjacent target charts V_j and V_{j+1}.
The rectangles can be chosen mutually disjoint (finitely many arcs).
Attach to each rectangle W_j a radial segment S_j going outward from
the outer boundary of W_j to the new outer radius. On the union of
W_j and S_j define coordinate functions: in one of its two target
charts V_j (say), continue the coordinate spray CONSTANTLY along the
segment from the point of attachment. This gives continuous
coordinate functions on W_j u S_j, holomorphic in the interior and
holomorphic in the passive parameters p.
Step 3. Mergelyan with parameters. The planar compact set
L_j = W_j u S_j
has connected complement. Apply Mergelyan's approximation theorem,
with passive parameters, to approximate the coordinate functions
just constructed by polynomials in the planar variable z, with
coefficients holomorphic polynomials in the parameters p. The
parameter step is: approximate the coefficient functions on a
slightly larger polydisc P_{**} with P_* compactly in P_{**} by
finite Taylor sums, and then apply scalar Mergelyan to each
Taylor coefficient. Cauchy estimates give any finite number of
p-derivatives on the smaller P_*. Consequently the approximation
can be made uniform with a positive auxiliary-parameter margin.
On the radial segment S_j, its values remain close to the endpoint
values for all p in P_*. A thin neighborhood of S_j therefore also
has its values in both adjacent V_j.
Step 4. Gluing the extension to the original spray. Glue this
extension to the original chart spray across a radial band
inside W_j, using Lemma 4.1. More explicitly, cut the union
domain at a radius slightly larger than r (still inside the
original domain of f). At the cut, the source consists of the
disjoint rectangles W_1, ..., W_N. A radial cutoff with a fixed
transition width gives positively separated fringes and a bounded
splitting constant. The tubes around the longer portions of the
spokes may then be narrowed without changing that constant.
The gluing errors can be chosen as small as desired. Result:
a map near K and all spokes, close to f on K, with all spoke values
still in the two adjacent target charts.
Step 5. Filling sectors. List the finitely many closed polar
sectors Sigma_1, ..., Sigma_M between successive spokes. Enumerate
them in an order consistent with the polar parametrisation. At a
given step, let K_old be the union of the original compact set, all
spokes, and the sectors already filled. The next sector Sigma = Sigma_i
meets K_old on its bottom arc and its two radial sides. Along these
three sides, all map values, with a parameter margin, belong to the
chosen V_j. This is true initially by Step 2 and remains true after
each sufficiently accurate Step 6 gluing, because the sides of every
unfilled sector already belong to K_old.
Choose a thin closed U-shaped band D_A around these three sides,
inside the current map domain, and a slightly enlarged filled sector
D_A' containing Sigma in its interior, so that D_A subset D_A' is a
special pair as in Definition 1.1. To see this elementary geometry,
straighten Sigma to a rectangle: D_A is a thick band around its
bottom and two side edges, and the missing central rectangle D_A' minus D_A
attaches along the U-shaped portion of its boundary. The bands and
the protrusion can be made as thin as needed (fixed once and for
all).
All values near D_A lie in V_j. Apply property L (from the
hypothesis, with witness neighborhood V_j) to the restriction of a
chart spray over the current map, regarding its small auxiliary
variables as additional passive parameters p together with the
original p. After any prescribed small parameter loss this gives a
spray on D_A', arbitrarily close to the old spray on D_A.
Step 6. Overlap for the new fill. Fix an open band
W_Sigma compactly in interior(D_A) containing K_old cap Sigma.
Using a slightly smaller band first, the compact remainders on
opposite sides are positively separated (K_old minus W_Sigma on one side,
Sigma minus W_Sigma on the other). A smooth cutoff can therefore be
chosen that is 0 near K_old minus W_Sigma and 1 near Sigma minus W_Sigma.
On a sufficiently thin neighborhood of K_old u Sigma, its sublevel
and superlevel sets U_- and U_+ give two open domains with
positively separated fringes and overlap contained in W_Sigma.
They may be chosen so that any point of K_old in the new side also
lies in the approximation band.
The old spray is defined on U_-, the approximating spray on U_+,
and they are close on their overlap with a positive auxiliary-
parameter margin. Lemma 4.1 produces the required new map close
to the old map on all of K_old. Repeat.
Step 7. Error budget. There are only finitely many spokes and
sectors. Fix a finite nested sequence of parameter polydiscs
between P_* and P_0, one per step. At each step choose errors
smaller than the remaining compact target-chart margins and
allocate a summable finite part of the requested final error
epsilon. These choices preserve every unfilled sector's joining
sides; no chart condition on the outer boundary of a newly filled
sector is required. At the end there is a map on a neighborhood of
the desired enlarged compact set times P_0, with the stated
approximation. QED
COROLLARY 6.1 (propagation across a small disc). Suppose a
coordinate neighborhood of x in X contains a closed affine complex-
line disc centered at x, with its boundary in the open locus where
L holds. Then L holds at x.
In particular, if the complement of the L locus in X is contained,
locally, in a proper analytic subset, then L holds everywhere.
Proof. Let D_0 be the closed disc about x, on a complex line L_0.
Slightly tilt L_0 in two different directions to get two closed
affine complex lines L_1, L_2 through x, whose tangent directions
at x are transverse, and whose closed discs D_1, D_2 of the SAME
radius as D_0 still have their boundaries in the L locus. (This
uses that the L locus is open and the original boundary is compact.)
Here "same radius" means the picture is compatible inside the fixed
coordinate neighborhood; one can shrink the coordinate chart if
needed.
At each boundary point of D_1, all values for parameters p in a
fixed small transverse disc of D_1 lie in one witness neighborhood
for L, by compactness of the boundary circle. Finitely many such
boundary patches give a common positive transverse radius. The
same for D_2.
For each integer n >= 1, apply Proposition 5.1 once to the thickened
disc D_1 x Delta_rho (long coordinate on D_1): enlarge its long
coordinate to radius n, retain a fixed smaller transverse radius
rho_n > rho/2, and make its error on the initial product less than
1/n. Do the same for D_2. The resulting family of pairs of finite
strip maps converges (as n -> infinity) near their centres to the
original coordinate charts on D_1 and D_2, with their limiting
long directions being transverse.
By Remark 2.5 and the finite-length case of Proposition 2.1, these
strips imply L at x.
For the last statement: if a proper analytic subset E of X contains
the exceptional points in a coordinate neighborhood of x, choose a
complex line through x that is not contained in E. Its intersection
with E is discrete near x (proper analytic subset), so a
sufficiently small circle in the line avoids E. The first
statement applies. QED
Write closure Delta_R^m = {z in C^m : max_j |z_j| <= R} and use
|| z ||_infty = max_j |z_j|. For the polydisc case of CAP it is necessary
to enlarge Delta_1^m to Delta_R^m while preserving a target-
neighborhood witness structure at every point of the OUTER BOUNDARY.
The naive thing to do is to enlarge each coordinate separately. But
L does not directly allow that, because the WITNESS NEIGHBORHOOD
must be a single small target neighborhood that contains the whole
transverse polydisc. If x_1 is enlarged naively, the transverse
coordinates still range over a full polydisc and can exit any single
witness.
The trick is to deform the source by a polynomial automorphism of
C^m, chosen so that the "new" part of the enlarged polydisc has
ONE LONG coordinate and BOUNDED transverse extent, and the "old"
part is nearly unchanged.
LEMMA 7.1 (concentration). Fix m >= 2 and numbers
1 < r_m < r_{m-1} < ... < r_2 < r_1,
0 < a < 1,
r_j < a r_{j-1} (2 <= j <= m).
For an integer d >= 2 set P_j(w) = (w/r_j)^d for j >= 2, and
H_d(x) = ( a x_m,
P_2(x_2) - a x_1,
P_3(x_3) - a x_2,
...,
P_m(x_m) - a x_{m-1} ).
There exist constants C >= 1, q in (0,1), and d_0, depending only on
the displayed fixed data, with the following properties. For
d >= d_0, N >= 1, and E_{d,N} = H_d^{-N}(closure Delta_1^m), the
following properties hold:
(i) max_{j>=2} |P_j(x_j)| <= C max(1, |x_1|) for all x in E_{d,N}.
(ii) On E_{d,N} cap {|x_1| > r_1}, all x_j are nonzero and
|H_d(x)_j / (a x_{j-1})| <= q for 2 <= j <= m.
(iii) Consequent to (ii), for the holomorphic functions
xi_j(x) = log( P_j(x_j) / (a x_{j-1}) ), 2 <= j <= m,
each xi_j is single-valued on a suitable open inverse image
of the unit polydisc under H_d, and |xi_j(x)| <= B := -log(1-q)
there.
(iv) With D = d^{m-1}, rho_1 = 1, rho_j = r_j (a rho_{j-1})^{1/d},
there is a single-valued holomorphic function s on this end
such that
x_j = rho_j s^{d^{m-j}}
* exp( sum_{k=2}^{j} xi_k / d^{j-k+1} ),
1 <= j <= m.
For j = 1 the sum is empty and x_1 = s^D.
Proof. Fix r_* := r_1. Choose S > max(2, r_*), and then d_0 >= 2
large enough that
(M/r_*)^d >= 4 M for all M >= S, d >= d_0.
This holds for all M >= S if it holds at M = S, d = d_0, because
(M/r_*)^d / M is increasing in both M and d on this range.
Cone. Let E = { u in C^m : max_{j>=2} |u_j| > max(S, |u_1|) }.
The cone E is forward invariant under H_d. Indeed, if
M = max_{j>=2} |u_j| and |u_k| = M for some k, then
|H_d(u)_k| = |P_k(u_k) - a u_{k-1}| >= (M/r_*)^d - a M >= 3M,
|H_d(u)_1| = |a u_m| <= a M < M.
The transverse maximum grows by at least a factor 3. Consequently,
no forward orbit that ends in closure Delta_1^m (where transverse
coordinates are at most 1) can ever enter E.
Power bound at the final point. Let x in E_{d,N}. Set
x^{(0)} = x, x^{(r)} = H_d(x^{(r-1)}) for r = 1,...,N. Then
x^{(N)} in closure Delta_1^m. Choose C larger than 1, 4a, 16 a^2 r_*^2,
4S. Suppose, at some stage, the desired inequality
A := max_{j>=2} |P_j(x_j)| <= C max(1, |x_1|) fails, meaning
A > C max(1, |x_1|). Then from
|P_j(x_j)| = |x_j|^d / r_j^d <= A with r_* = max r_j (upper):
max_{j>=2} |x_j| <= r_* A^{1/d} <= r_* sqrt(A)
(because A > C >= 1 implies A^{1/d} <= sqrt(A)). For an index k
attaining A: both possible terms in a |x_{k-1}| are at most A/4.
For k = 2 that term is a |x_1|. By hypothesis A > C max(1, |x_1|)
>= 4a max(1, |x_1|), so a |x_1| < A/4. For k >= 3, that term is a
|x_{k-1}|, and the bound max_{j>=2}|x_j| <= r_* sqrt(A) with r_* <= C^{1/2}/(2a)
(which one can arrange by increasing C) gives a |x_{k-1}| < A/4.
Hence
|H_d(x)_k| >= |P_k(x_k)| - a |x_{k-1}| >= 3A/4,
|H_d(x)_1| = a |x_m| <= a r_* sqrt(A) <= A/4.
Thus H_d(x) in cone E, a contradiction. This proves (i) at the
final stage. The same argument applied to each x^{(r)}, 0 <= r <= N-1,
and using the bound at stage r+1 to bound the transverse coordinates
at stage r, proves (i) at every stage.
Ratio bound. Write y = H_d(x), so y_j = P_j(x_j) - a x_{j-1} for
j >= 2 and y_1 = a x_m. Applying the power bound at y (or using
|| y ||_infty <= 1 at the final point), this gives
|y_j| <= r_j ( C max(1, a |x_m|) )^{1/d}, j >= 2.
Let L_j = log^+|x_j| = max(0, log|x_j|), M = L_m, and
A_1 = log(r_* sqrt(C) + a). From P_j(x_j) = y_j + a x_{j-1},
combined with the above,
|P_j(x_j)| <= (r_* sqrt(C) + a) exp( L_{j-1} + M/d ).
Taking logs:
L_j <= log r_j + A_1/d + L_{j-1}/d + M/d^2. (*)
Let A_2 = log r_* + A_1/2. Iterate (*) from j = 1 (essentially
L_1 = log|x_1|) to j = m:
M = L_m <= 2 A_2 + d^{-(m-1)} L_1 + M * sum_{i=0}^{m-2} d^{-2-i}
<= 2 A_2 + d^{-(m-1)} L_1 + M/2.
For |x_1| >= r_1, define t = (|x_1|/r_1)^{1/D} >= 1 where
D = d^{m-1}, and A_3 = 4 A_2 + 2 log r_1. Then
M <= A_3 + 2 log t,
|y_j| <= r_j exp( (log C + A_3) / d ) t^{2/d}, j >= 2.
Choose a small delta > 0 so that
q := max_{2 <= j <= m} (r_j + delta) / (a (r_{j-1} - delta)) < 1,
r_j - delta > 0.
Increase d_0 if necessary so that, for all d >= d_0,
r_j exp( (log C + A_3) / d ) <= r_j + delta,
r_j ( (1-q) a (r_{j-1} - delta) )^{1/d} >= r_j - delta.
The proof shows by induction that
|x_j| >= (r_j - delta) t^{d^{m-j}}, 1 <= j <= m.
Base case j = 1: this is exactly the definition of t. Inductive
step: suppose it holds at j-1. Then
|y_j| / (a |x_{j-1}|) <= [(r_j + delta) / (a (r_{j-1} - delta))] t^{2/d - d^{m-j+1}}.
Since m >= j, the exponent 2/d - d^{m-j+1} <= 0 (equality when
m - j + 1 = 1 with d = 2, but 2/d <= 1 < d^{m-j+1} for m-j+1 >= 1
as long as d >= 2; the worst case m - j = 0 gives 2/d - d <= 2/2 - 2 = -1 < 0).
So t^{2/d - d^{m-j+1}} <= 1 for t >= 1. Hence
|y_j| / (a |x_{j-1}|) <= q.
Therefore
|P_j(x_j)| >= (1 - q) a |x_{j-1}|,
and taking the d-th root proves the inductive step.
This also proves (ii): |y_j / (a x_{j-1})| <= q < 1 on
{|x_1| > r_1}, so P_j(x_j) / (a x_{j-1}) = 1 + y_j / (a x_{j-1})
is close to 1 in modulus and its log (using the branch near 1) is
well-defined. Its power series gives |xi_j(x)| <= sum_{nu>=1} q^nu/nu
= -log(1-q) = B. Being a log of an expression that is nowhere zero
and using the branch fixed near 1, xi_j is single-valued.
Parameterization (iv). There is no root choice: define
s = (x_m / rho_m) * exp( - sum_{k=2}^m xi_k / d^{m-k+1} ).
Then s is single-valued holomorphic. The equations
x_j^d = r_j^d a x_{j-1} exp(xi_j) prove (iv) backward from j = m
down to j = 1, using rho_j^d = r_j^d a rho_{j-1} (which is
exactly the definition of rho_j). QED
Now the main polydisc approximation.
PROPOSITION 7.2 (polydisc approximation). Let Y be a complex surface
with a complete compatible distance d_Y. Assume L(y) holds for
every y in Y. Then a map holomorphic near a closed polydisc in C^m
can be approximated uniformly there by entire maps C^m -> Y.
More precisely, a map near closure Delta_1^m can be approximated
there by maps holomorphic near closure Delta_R^m for every finite
R > 1.
Proof. Finite radius first. The case m = 1 is Proposition 5.1
with no passive variables. Suppose m >= 2.
Let f be the initial map, epsilon > 0. Choose b > 1 with f
holomorphic near closure Delta_b^m. Choose radii r_j, a, ell > 0
so that Lemma 7.1 applies and
1 < r_m < ... < r_1 < r_1 + 3 ell < b.
The derivative of H_d at 0 is the d-independent map
L(x) = ( a x_m, -a x_1, -a x_2, ..., -a x_{m-1} ),
|| L x ||_infty = a || x ||_infty.
H_d is a polynomial automorphism of C^m: from y = H_d(x), recover
x_m = y_1 / a and then successively x_{j-1} = (P_j(x_j) - y_j) / a
for j = m, m-1, ..., 2.
Fix R_+ > R, then N large enough that a^N R_+ < 1. Set
h_d = H_d^{-N} o L^N,
Omega = Delta_{R_+}^m.
Then h_d(closure Omega) subset E_{d,N} (because L^N shrinks by a^N
and H_d^{-N} is applied; on closure Omega, L^N takes values in
Delta_{a^N R_+}^m subset Delta_1^m, and H_d^{-N} inverts).
Near identity. On some fixed neighborhood of closure Delta_1^m,
h_d -> id as d -> infinity. A quantitative justification follows.
Choose 1 < b_0 < b_1 < r_m, put eta = (b_1 - b_0)/3, and
theta = b_1 / r_m < 1. If ||L^{-1} y||_infty <= b_0 + eta and
theta^d / a < eta, the inverse recursion gives
|| H_d^{-1}(y) - L^{-1}(y) ||_infty <= theta^d / a.
Indeed: its first recovered coordinate is exact,
x_m = y_1 / a = (L^{-1} y)_m. Each next one differs from the
corresponding coordinate of L^{-1} y by P_j(x_j) / a. Inductively,
all recovered coordinates x_j have modulus at most b_0 + 2 eta < b_1.
Then
|P_j(x_j) / a| = |x_j|^d / (r_j^d a) <= b_1^d / (r_m^d a) = theta^d / a,
using b_1 < r_m and r_j >= r_m.
Let C_N = sum_{i=1}^N a^{-i}, and take d large enough that
C_N theta^d / a < eta. Successive comparison of H_d^{-r} L^N(z)
with L^{N-r}(z), for z in closure Delta_{b_0}^m, gives the error
recursion
e_0 = 0, e_{r+1} <= a^{-1} e_r + theta^d / a, e_r <= theta^d * sum_{i=1}^r a^{-i}.
The imposed bound guarantees the inverse estimate at each step.
At r = N it proves
|| h_d - id ||_{closure Delta_{b_0}^m} <= C_N theta^d,
which is (d -> infinity on a fixed neighborhood of closure Delta_1^m).
Chart spray and its parameters. Choose b' < b larger than
r_1 + 2 ell. By Lemma 3.1 there is a chart spray
f# : (neighborhood of closure Delta_{b'}^m) x Delta_beta^2 -> Y,
f#(x, 0) = f(x),
with the vertical map v -> f#(x, v) injective and invertible at v = 0,
for some beta > 0. The value b' and this spray are fixed before d
is increased.
Parameterization in s and xi. Write Phi_d(s, xi) for the right-
hand side of the parameterization (iv) in Lemma 7.1, allowing xi
to vary independently in C^{m-1}. Use the three parameter polydiscs
Xi_i = { xi in C^{m-1} : |xi_j| <= B + i }, i = 1, 2, 3.
Choose fixed numbers r_1 < c_0 < c_1 < c_2 < c_3 < r_1 + ell.
On the compact set
c_0 <= |s|^D <= c_3, xi in Xi_3,
one has, uniformly as d -> infinity,
|Phi_{d,j}(s, xi)| <= r_j exp(A_4 / d) (j >= 2)
sup_{xi, xi~ in Xi_3} || Phi_d(s, xi) - Phi_d(s, xi~) ||_infty <= A_5 / d,
where
R_0 = 2 log r_1 + |log a|,
A_4 = |log a| + R_0 + log c_3 + 2(B + 3),
A_5 = 4(B + 3) r_1 exp(A_4 / 2).
The recurrence for rho_j gives |log rho_j| <= R_0 and
|log rho_j - log r_j| <= (|log a| + R_0)/d. The power of s
contributes at most log c_3 / d^{j-1} <= log c_3 / d. The
exponential satisfies sum_{k=2}^j |xi_k| / d^{j-k+1} <= 2(B+3)/d.
Moreover, partial_{xi_k} Phi_{d,j} = d^{-(j-k+1)} Phi_{d,j} for
k <= j and 0 for k > j. Integrating along line segments in the
convex polydisc Xi_3 and using sum_{nu>=1} d^{-nu} <= 2/d gives
the diameter bound with A_5 as above. In particular, Phi_d takes
this product into Delta_{b'}^m for large d.
Cover by witness neighborhoods. The compact set f(closure Delta_{b'}^m)
has a finite cover by neighborhoods on which L is available (L(y)
holds for every y, so such covers exist by local compactness).
A Lebesgue number for a slightly smaller such cover, uniform
continuity of f, and the diameter estimate
show the following. After increasing d and decreasing the fixed
spray radius, all values
f#( Phi_d(s, xi), v ), xi in Xi_3, v in closure Delta_{beta/2}^2,
belong to one of these neighborhoods for each fixed s on the outer
circle |s|^D = c_3. The chosen neighborhood may depend on s.
The same increase of d ensures that
sup_{closure Delta_1^m} d_Y( f(h_d(z)), f(z) ) < epsilon/2,
|h_d(z)_1| < c_1 for z in closure Delta_1^m.
Require also r_j (C c_2)^{1/d} < b' for j >= 2; possible because
r_j < b'. Fix this value of d, henceforth writing h = h_d.
Apply Proposition 5.1 to the annulus
A_s = { s : c_0 <= |s|^D <= c_3 }
and to the map f#(Phi_d(s, xi), v), with xi and v as passive
variables. It gives arbitrarily close approximation on
A_s x Xi_2 x closure Delta_{beta/2}^2 by a holomorphic map
G(s, xi, v) defined out to any prescribed finite outer radius in s,
with a smaller fixed positive parameter margin. Choose that radius
larger than c_3^{1/D}, and larger than
sup { |s(h(z))| : z in closure Omega, |h(z)_1| >= c_1 }.
If the set is empty, the latter requirement is vacuous. Otherwise
the supremum is finite by compactness and the explicit formula for
s: indeed |s(h(z))|^D = |h(z)_1|. The inner radius of the domain
of G can be kept strictly below c_1^{1/D}.
Building the two sides. On two open subsets of Omega
A = { z : |h(z)_1| < c_2 },
B' = { z : |h(z)_1| > c_1 },
define maps with common auxiliary variable v in a small polydisc:
F_A(z, v) = f#( h(z), v ),
F_B'(z, v) = G( s(h(z)), xi(h(z)), v ).
The first is defined on all of A: by the power bound, the transverse
coordinates of h(z) satisfy |h(z)_j| <= r_j (C c_2)^{1/d} < b'
after the choices already made. The second is defined on all of B'
by Lemma 7.1 and by the parameter and outer-radius margins chosen
above.
Overlap. On A cap B', Phi_d(s(h(z)), xi(h(z))) = h(z), hence
F_B' approximates F_A arbitrarily closely, uniformly for v in a
fixed smaller polydisc.
Gluing. Choose a smooth cutoff chi of |h(z)_1|^2 whose transition
is compactly contained in c_1 < |h(z)_1| < c_2. On the bounded
domain Omega, all its derivatives needed in Lemma 4.1 are bounded.
These constants, d, N, and all domain margins have been fixed before
choosing the accuracy in Proposition 5.1. Lemma 4.1 therefore
glues the two maps, by arbitrarily small vertical corrections, on a
neighborhood of closure Delta_R^m. On closure Delta_1^m subset A
the resulting central map differs from f o h by less than epsilon/2.
Together with the near-identity statement, this proves the finite-
radius assertion.
Entire. To obtain an entire map, use the finite-radius assertion,
after affine rescaling, successively on exhausting polydiscs. Choose
positive errors with sum less than the prescribed final error, and
make the error at each stage controlled on the previous polydisc.
Completeness of the target metric makes the maps uniformly Cauchy on
every compact set. Their limit is holomorphic: near each point,
choose a coordinate neighborhood of the limit value; uniform
convergence puts all sufficiently late maps in that chart on a
smaller source neighborhood, where the usual theorem for uniformly
convergent holomorphic functions applies. Same argument gives the
prescribed approximation on the initial polydisc. QED
Polydisc approximation holds. For CAP it is necessary to remove the
walls of an arbitrary convex polytope. The key step removes one
real affine constraint at a time: enlarge
Q_- = Q cap { Im z_1 <= 0 }
to a slightly larger compact convex Q. Given a map near Q_- x P,
first approximate it on the joining real face {Im z_1 = 0} by an
entire map G, with tangential derivative accuracy; then glue the
lower-side chart spray to G using ONLY Holder accuracy on the real
face. The main point is that there is no requirement for a fixed complex
collar on which G stays close. This matters because no such collar
is available in general.
Notation for the whole chapter. Fix
0 < alpha < 1.
For a compact convex set Q subset C^n and r > 0 write
Q_r = { z : dist(z, Q) < r },
M_r = Q_r cap { Im z_1 = 0 },
Q_r^+ = Q_r cap { Im z_1 > 0 },
Q_r^- = Q_r cap { -Im z_1 > 0 },
and let B_b subset C^N be a small ball parametrised by v (the
passive parameter, or the auxiliary spray variable). All functions
below are holomorphic in v. All norms below are uniform in v; the
Holder seminorm is only taken in the base variables (z), on the
interface M_r.
Side norm, for a side function a on Q_r^+ or Q_r^- with trace on
M_r:
|| a ||_{r, b, *} = sup_{Q_r^+ x B_b} |a|
+ sup_{v in B_b} || a(. , v)|_{M_r} ||_{C^alpha(M_r)}.
The Holder C^alpha norm on an open bounded set means the supremum
together with the supremum of all finite difference quotients.
Constants may depend on a bounded range of r, the
dimensions n, N, alpha, and the fixed geometry, but not on input
functions.
LEMMA 8.1 (complexifying a real parameter). Let I subset R be a
compact interval and let C subset C^k be a compact convex set.
Suppose H_s(q) is a smooth family, for s in a neighborhood of I,
of holomorphic maps on a common neighborhood of C, taking values
in a fixed complex surface. Smoothness is understood on a common
neighborhood with all finite derivatives under consideration. For
every finite j and every epsilon > 0, after an arbitrarily small
loss of the specified compact margins, H can be approximated in
C^j(I x C) by a map holomorphic on a neighborhood of I x C in
C x C^k (the size of the complex neighborhood may depend on the
approximant). Additional closed polydisc factors can be included
in C, with fixed larger neighborhoods.
Proof. Bypass any assumption of a holomorphic extension of H.
By Lemma 3.1 applied to the varying source s in I, a finite cover
of I by short intervals admits fixed graph charts E_i(q, v) over
the corresponding maps. First choose a smooth family of small
chart sprays H_s# (q, v), holomorphic in (q, v) on a common small
fiber ball, with H_s#(q, 0) = H_s(q). In one fixed chart,
translate its fiber coordinates to the center representing H_s.
When changing charts at a division point s_0, express the preceding
spray in the new chart and take its fiber displacement at s_0. On
a short overlap interpolate this displacement to the fixed
displacement at s_0, leaving the moving center unchanged. The two
displacements are as close in the first fiber derivative as desired
when the overlap is short; the derivatives remain invertible.
Finitely many such operations and a smaller common fiber ball give
the required smooth spray family.
In each division interval write this family in one fixed graph
chart. Approximate its coordinate functions by polynomials in s
with coefficients holomorphic in the remaining source variables
(q, v), retaining any prescribed finite number of derivatives, and
match the endpoint s-jets through order M (a large but fixed
integer). This is ordinary approximation in a Banach space of
holomorphic functions on a smaller fixed neighborhood: approximate
a sufficiently high real s-derivative uniformly, integrate, and
add a finite Hermite correction at each endpoint. The correction
tends to zero with the initial error. Consequently the sprays
obtained on adjacent intervals have identical s-jets through order M
at the division point.
Now, once the finitely many polynomials have been fixed, choose a
small complex thickness delta > 0 around the real intervals.
Extend each interval slightly past its endpoints, so adjacent
rectangles overlap within distance O(delta) of their common
division point. Their spray transition differs from the identity
by O(delta^{M+1}) there, uniformly on a smaller fiber ball and a
parameter compactum. This is Taylor's formula applied to the
holomorphic coordinate transitions; the constants are fixed before
delta is chosen.
Additive splitting on the finitely many thin planar rectangles
costs at most C delta^{-1}. Indeed, use cutoffs across the vertical
overlap bands (first derivatives are O(delta^{-1})) and the
Cauchy-Green operator on a bounded planar set. For simultaneous
splitting, each cut contributes a pair of corrections on the entire
portion to its left and right; that pair has zero jump at every
other cut. The number of cuts is fixed. The remaining source
variables are passive for this operator, so no parameter radius is
lost.
Write consecutive transitions as gamma_i = id + c_i, with
|| c_i || + || D_v c_i || <= A delta^{M+1} on a smaller fiber ball.
Let R be the simultaneous right inverse, of norm at most K/delta,
for the jump operator (a_i) -> (a_{i+1} - a_i). The central maps
are identified by the fixed-point equation
(a_i) = R ( ( c_i(. , a_i) )_i ).
It preserves the ball of radius 2 K A delta^M and is contractive
there when K A delta^M < 1/2. The corrections are therefore
O(delta^M). Substituting them into the corresponding sprays gives
central maps agreeing on overlaps. Cauchy estimates on rectangles
with margins comparable to delta bound their correction derivatives
through order j by O(delta^{M-j}). Derivatives in the other
variables use their fixed margins. Take M > j + 2 and then delta
small. Together with the initial polynomial accuracy this gives
the prescribed C^j estimate. QED
LEMMA 8.2 (parameter disc). Assume A_r for some r >= 0. Let I be
a compact interval in R, C subset C^r compact and convex, P a
closed polydisc. A map holomorphic near I x C x P can be
approximated there, with any prescribed finite derivative accuracy
and smaller positive margins, by entire maps of all the displayed
complex variables (after replacing s in I by a complex disc).
A_r is the inductive hypothesis from the induction on faces below: for
every compact convex subset of C^r, every extra polydisc, every
holomorphic local map, uniform approximation by entire maps.
Proof. Shrink unused margins so that the given map H is defined on
V x U, where V subset C is a simply connected bounded neighborhood
of I and U contains a slightly thickened compact product C x P.
A thin open rectangle works for V. Let phi : V -> Delta be a
Riemann map. Choose 0 < rho < 1 with phi(I) compactly in
Delta_rho. The map
(zeta, y, u) -> H( phi^{-1}(zeta), y, u )
is defined near closure Delta_rho x C x P with slightly smaller
margins. Apply A_r, using the zeta disc as an extra passive factor.
This gives an entire E(zeta, y, u) with arbitrarily good derivative
approximation on a smaller product that still contains
phi(I) x C x P. Consequently E(phi(s), y, u) approximates
H(s, y, u) with the required derivatives along I x C x P.
Choose a closed convex neighborhood I' of I inside V. Scalar
polynomial approximation, with Cauchy estimates after a harmless
shrink, approximates phi and any specified finite number of its
derivatives on I' by a polynomial p. At this stage E is fixed; on
the compact sets involved all its derivatives up to the required
order are bounded. The chain rule therefore shows that
E(p(s), y, u) is as close to E(phi(s), y, u) as desired. The map
E(p(s), y, u) is entire in every variable. QED
LEMMA 8.3 (approximation on a convex real face). Assume A_{n-1}
for some n >= 1, and assume Proposition 7.2. Let F subset R x C^{n-1}
be compact and convex, P a closed polydisc. Every holomorphic map
near F x P can be approximated on F x P by entire maps on
C^n x C^{dim P}, with any fixed finite derivative accuracy along
the real face and smaller positive margins.
Proof. Write the variables as (t, y, u), with t real on the face.
Thicken F slightly inside R x C^{n-1}, and thicken P, so that all
estimates below take place strictly inside the initial domain.
Denote the thickened face again by F. Its projection to R is a
compact interval I, and write F_t = { y : (t, y) in F }.
Finite cover. For each t_0 in I, choose a compact convex
neighborhood C_{t_0} of F_{t_0} in the y variables and an open
complex disc D_{t_0} about t_0 such that the given map f is
defined near closure D_{t_0} x C_{t_0} x P, including some fixed
smaller margins. These choices are possible because F_{t_0} x P
is compact. After reducing the real interval about t_0, there is
F_t compactly in int C_{t_0}
uniformly on that interval. Indeed, otherwise a convergent
sequence (t_nu, y_nu) in F with t_nu -> t_0 would contradict
F_{t_0} compactly in int C_{t_0}. Choose a finite subdivision of
I with small overlaps at division points, subordinate to these
intervals. Denote the corresponding products by D_i x C_i x P.
All sets can be chosen a little larger than those used below.
At a division point b, both adjacent C_i contain F_b in their
interiors. Choose compact convex sets C^- and C with nonempty
interiors such that
F_b subset int C^-,
C^- compactly in int C,
C compactly in int(C_i cap C_{i+1}).
By the preceding compactness argument, after reducing a complex
disc D_b about b, the sets F_t for real t near b lie strictly
inside C^-, and
closure D_b x C x P compactly in
(D_i x int C_i x int P^+) cap (D_{i+1} x int C_{i+1} x int P^+)
for an available larger passive polydisc P^+. All transition
intervals and their cutoffs are now fixed. Also fix a large
y-polydisc B containing all C_i and all face projections with a
positive margin.
By A_{n-1}, applied with the t disc and P polydisc passive, choose
entire maps g_i(t, y, u) arbitrarily close to f, with any fixed
finite derivative accuracy, on closure D_i x C_i x P. (Here "entire"
refers to (y, u) and (t, u) being passive; the approximant is
holomorphic in all complex quantities.)
Interpolation. On a slightly enlarged D_b x C x P, a chart spray
f# over f expresses the two approximants as
g_i = f#( . , a_i ), g_{i+1} = f#( . , a_{i+1} ),
with a_i, a_{i+1} and their derivatives small. Choose a smooth
cutoff chi : [0,1] -> [0,1] equal to 0 near 0 and 1 near 1. Set
f_s = f#( . , (1 - chi(s)) a_i + chi(s) a_{i+1} ).
This is a smooth s-family, holomorphic in (t, y, u). It equals
g_i near s = 0 and g_{i+1} near s = 1; on these subintervals,
those entire definitions are used on all y. It is arbitrarily close
to f on the local product, and its positive s-derivatives through
any fixed order are arbitrarily small there. The constants
involved depend only on the fixed local chart and cutoffs.
Larger domain. The path f_s is defined only over the small convex
set C except near endpoints, where it equals an entire map. A
path is needed over the common large polydisc B, still close to f_s
on C^-, equal to prescribed endpoint maps on all of B. Direct
interpolation in the chart over f cannot provide this, because that
chart is not defined over B.
Inverse scaling. Obtain the larger domain by expanding the y
variable only where the path already equals an entire endpoint
map. Choose d_* in int C. For some L >= 1,
d_* + L^{-1} (B - d_*) compactly in int C. (*)
Actually choose d_* in int C^- subset int C and use (*) with the
outer set equal to B (with room to spare). Choose a smooth
positive function lambda : [0,1] -> [1, L] equal to 1 outside the
endpoint intervals (where f_s is already an entire endpoint map)
and equal to L on smaller endpoint intervals. All choices are
fixed. The family
f~_s(t, y, u) = f_s( t, d_* + lambda(s) (y - d_*), u ),
y in C,
is defined on a common neighborhood of
[0,1] x closure D_b x C x P. This uses entirety precisely where
lambda > 1; where only the local definition of f_s is available,
lambda = 1.
Apply Lemma 8.1 to make this family jointly holomorphic near that
compact product, with arbitrarily good finite derivative accuracy.
Then apply Lemma 8.2, with A_{n-1} and passive factors D_b, P, to
obtain an entire approximant E(s, t, y, u).
Reassembling. For real s, set
A_s(t, y, u) = E( s, t, d_* + lambda(s)^{-1} (y - d_*), u ).
This is smooth in s and holomorphic in (t, y, u) on the whole large
product. On C' = C^- compactly in int C, which contains all
required overlap slices, the inverse scaling stays in a fixed
compact subset of int C by convexity (it lies in the convex hull
of C' u {d_*}). The two scalings cancel exactly:
f~_s( t, d_* + lambda(s)^{-1}(y - d_*), u ) = f_s(t, y, u).
Thus approximating f~_s on that fixed compact subset makes A_s
arbitrarily close to f_s, with the required derivatives, on these
slices. On the smaller endpoint intervals, (*) gives the stronger
statement that A_s is arbitrarily close to g_i, respectively
g_{i+1}, on the entire large product closure D_b x B x P. The
chain rule costs only finite constants, because lambda and all
scaling factors have already been fixed.
Exact endpoint joins. A chart spray over g_i exists on a slightly
larger version of that large product. On a smaller initial
s-interval, write A_s = g_i#( . , beta_s ). Here beta_s, including
all required derivatives, is as small as desired. Replace beta_s
by kappa(s) beta_s, where kappa is 0 near 0 and 1 before leaving
this endpoint interval. Do the analogous replacement near 1. The
repaired path H_s equals g_i near one endpoint and equals g_{i+1}
near the other, and retains the desired approximation on the
overlap slices. It is holomorphic on a neighborhood of the fixed
large product, which the chart construction provides.
No circular smallness. First choose the local g_i sufficiently
accurate that the interpolation f_s has desired small error after
the fixed real transition cutoff. Then, with g_i, lambda, B and
the endpoint chart sprays fixed, choose the complexification and
entire approximation of f~_s as accurate as needed to overcome all
their finite-derivative and chart constants. Finally perform the
exact endpoint repairs.
Assembling over all intervals. Substitute a fixed smooth cutoff
s = s(t) across the real transition interval. The result is a
smooth real t-family, holomorphic in (y, u) on the large product,
which joins the two local entire maps and is as close to f as
prescribed, with all required tangential derivatives, on face
slices. Doing this at the finitely many disjoint transition
intervals gives one such family on all of I. Extend it smoothly
past the endpoints of I using the first and last entire maps g_i,
to which it already agrees on endpoint intervals.
Apply Lemma 8.1 once more, now to this real t-family on the fixed
large polydisc in (y, u). Result: jointly holomorphic near
I x B x P and retains the prescribed face accuracy. Proposition
7.2 is A_0; use Lemma 8.2 with r = 0 to convert this last family
into an entire map in (t, y, u), still with the required accuracy.
When n = 1 there are no y variables, so no scaling step is needed;
the final complexification together with A_0 finishes the proof. QED
LEMMA 8.4 (additive splitting at a real interface). There exist
C > 0 and an integer p >= 1 with the following property. For
0 < h < min(r/2, 1) and a vector-valued function u on M_r x B_b
with the regularity specified above, there
are linear operators producing functions a^+, a^- that are
holomorphic on Q_{r-h}^+ x B_b and Q_{r-h}^- x B_b, continuous up
to M_{r-h} x B_b, such that
a^+ - a^- = u on M_{r-h} x B_b,
|| a^+ ||_{r-h, b, *} + || a^- ||_{r-h, b, *}
<= C h^{-p} sup_{v in B_b} || u(. , v) ||_{C^alpha(M_r)}.
The operators preserve holomorphic dependence on all auxiliary
variables v.
Proof. Cover the required part of M_r by coordinate boxes of side
comparable to h; the enlarged boxes used below stay inside Q_r.
Choose a grid cover with bounded overlap from at most C h^{-2n}
boxes, and a smooth subordinate partition whose derivatives of
order j are bounded by C_j h^{-j}. Boxes far from the origin are
unnecessary.
In one enlarged box, with centre (t_0, z'_0) on the interface,
take a smooth real cutoff chi supported in its t-interval and
equal to one on a smaller interval. For Im(zeta) != 0 put
C_u(zeta, z', v) = (1/(2 pi i)) int_R
chi(t) u(t, z', v) / (t - zeta) dt.
The one-variable Cauchy boundary formula gives the jump u on the
smaller interval (after assigning the signs of the two boundary
values in the usual way). Its Holder bound follows by subtracting
u(t_*, z', v) in the singular integral near a boundary point t_*:
the remaining numerator is bounded by a constant times |t - t_*|^alpha.
The same subtraction in the difference of two boundary values gives
the C^alpha estimate. Rescaling the interval to unit length gives
a bound with a fixed power of h^{-1}. The integral is holomorphic
in (z', v) by dominated integration. Cauchy estimates on slightly
smaller transverse boxes control transverse derivatives with a
further fixed power of h^{-1}. Thus the local side functions have
the asserted joint Holder-trace bounds.
Denote these local pairs by c_i^+. Multiply by a smooth partition
theta_i in a band around the interface, equal to 1 in sum on a
smaller band, and extend the partition terms into the two sides
with support in their boxes. This gives smooth side functions
f^+ = sum_i theta_i c_i^+ and f^- = sum_i theta_i c_i^- with trace
jump u.
Their ordinary side derivatives d-bar f^+ fit together to a single
smooth form eta across the interface. This is the point that
ensures smoothness despite the original Holder data. In a smaller
box choose one reference pair c_0^+. Every difference c_i^+ - c_0^+
has equal traces on M_r, hence extends holomorphically across the
cut by continuity and Morera. Where sum_i theta_i = 1,
d-bar f^+ = sum_i (d-bar theta_i) (c_i^+ - c_0^+).
The right-hand side is the restriction of a common smooth form.
Outside this smaller band, the terms are evaluated at distance
comparable to h from their Cauchy singularities. Consequently,
any fixed number of derivatives of eta is bounded by
C h^{-p_j} ||u||_{C^alpha}. The form eta is closed: locally it is
a d-bar derivative on the open sides, and it is smooth across the
boundary.
Solve d-bar w = eta on an intermediate convex domain. Choose the
weighted minimal L^2 solution with the fixed strictly plurisubharmonic
weight |z|^2 (Hormander Acta 1965, Theorem 2.2.1). Its L^2 norm is
at most C ||eta||_{L^2}, because the weight is bounded above and
below on the domains in question. Interior estimates give both a
supremum and a first-derivative bound on the next smaller thickening,
with polynomial losses in h^{-1}. Explicitly, writing
eta = sum_j eta_j d-bar z_j, the distributional equation gives
Delta w = 4 sum_j partial eta_j / partial z_j componentwise.
Interior elliptic estimates on nested balls of radius comparable
to h, followed by Sobolev embedding, bound w in C^1 by its L^2
norm and a fixed finite number of derivatives of eta, with powers
of h^{-1}. This proves the required interior bound and hence the
Holder-trace bound for w.
The operator just constructed is fixed and linear on the base.
Applied to a holomorphic family of L^2 coefficients, it gives a
holomorphic family of L^2 solutions. Interior estimates turn this
into ordinary joint holomorphic dependence on v. Now set
a^+ = f^+ - w, a^- = f^- - w. Their side d-bar derivatives vanish,
their trace jump is unchanged, and the stated estimate follows by
enlarging p to dominate the finitely many derivative and covering
losses. QED
LEMMA 8.5 (small nonlinear gluing on a real interface). The
conclusion of Lemma 4.1 holds for sprays on Q_r^- and Q_r^+ when
their transition is given on M_r x B_b and is arbitrarily small in
|| c ||_{r, b, alpha} = sup_{v in B_b}
|| c(. , v) ||_{C^alpha(M_r)}.
The sprays are holomorphic on the open sides, continuous up to
the interface, and the transition is holomorphic in (z', v) there.
The resulting glued map is holomorphic across the interface. No
lower bound is assumed for a complex normal thickness on which the
input transition is small.
Proof. Apply Lemma 8.4 to split the transition error as b - a = c.
With losses h in base and fiber margins, its bound is C h^{-p} e,
where e = ||c||_{r, b, alpha}; side corrections are measured by
their supremum and trace norms. Cauchy estimates in v give the
same bound, with additional fixed powers of h^{-1}, for their first
and second v-derivatives on smaller balls.
As in Lemma 4.1, the corrected transition
(id + b)^{-1} o gamma o (id + a) = id + d
is determined on the interface by
d = c(v + a) - c(v) - { b(v + d) - b(v) }.
The C^alpha multiplication inequality and integration of the first
v-derivative along line segments bound the first difference by
C h^{-p} e^2. To compare values at two base points, the variation
of the derivative at the shifted v-argument is controlled by the
second v-derivative. The same estimates make the operation in
braces contractive in the trace norm. Thus, after increasing the
fixed exponent p,
|| d ||_{r-h, b-h, alpha} <= C h^{-p} e^2.
Only finitely many derivatives are required at each step. The
losses h_i = h_0 2^{-i} and rescaled errors E_i in the proof of
Lemma 4.1 consequently give the same doubly exponential
convergence. The sums of side corrections and their v-derivatives
converge on both sides, including their C^alpha traces. Their
limiting transitions are the identity.
Composing the original sprays with the limiting side maps produces
equal continuous traces. Near each interface point, continuity
places both images in a common target chart; Morera's theorem
across the real hyperplane then proves holomorphicity of the joined
map. The old-side approximation follows from the small fibre
corrections and the fixed spray's uniform continuity on compacta,
exactly as in Lemma 4.1. QED
Assemble. Define, for r >= 0, the statement A_r (used already
in the face-removal argument):
A_r : For every compact convex set C subset C^r, every closed
extra polydisc P, and every map holomorphic near C x P,
there are entire maps C^r x C^{dim P} -> Y approximating
it uniformly on C x P. Moreover, all approximations may
also be required in any fixed finite number of
derivatives on a slightly smaller compact product.
The last formulation is justified as follows: to get derivative
accuracy, first thicken C and P within the given domain, approximate
uniformly there, then use chart-spray coordinates and Cauchy
estimates on the smaller product. This uses no new approximation
assertion.
THEOREM 9.1 (analytic core: L everywhere implies CAP). Let Y be a
complex surface with a complete compatible distance d_Y. Suppose L
holds at every point of Y. Then Y has CAP in every dimension:
for every m >= 1, every compact convex K subset C^m, every open
neighborhood U of K, every holomorphic f : U -> Y, and every
epsilon > 0, there is an entire F : C^m -> Y with
sup_{z in K} d_Y(F(z), f(z)) < epsilon.
(This is the analytic-cap statement; the proof shows it here.)
Proof. By induction on n the proof shows A_n. The base case A_0 is
Proposition 7.2 (polydisc approximation). Suppose A_{n-1} is
known for some n >= 1. The proof shows A_n.
Step 1 (single real hyperplane). After an invertible complex
affine change of coordinates, write the enlarged compact convex
set as Q subset C^n and the OLD compact set as
Q_- = Q cap { Im z_1 <= 0 }.
The setting is as follows. Given a map f near Q_- x P, enlarge Q_- across its flat
face to Q.
Choose small convex thickenings of Q and of P. Their lower sides,
together with additional small neighborhoods, can be kept in the
domain of f. The verification is this. As eta -> 0+,
( Q + eta closure(B) ) cap { Im z_1 <= 0 } -> Q_-
in the following sufficient sense: every sequence of points in the
left-hand sets has all its limit points in Q_-. Compactness then
places these sets in any prescribed neighborhood of Q_- for small
eta. Here B is the closed Euclidean unit ball. No transversality
assumption on the boundary of Q is needed.
Chart spray. Lemma 3.1 gives a chart spray f#(z, u, v) over a
neighborhood of the thickened lower side, with a fixed auxiliary
polydisc in u (the map coordinate) and a fixed small polydisc in
v. Its trace on the real interface {Im z_1 = 0} is holomorphic
near a convex compact face in (t, y) = (Re z_1, z_2, ..., z_n).
Entire upper map. Apply Lemma 8.3 (face approximation), treating
u and v as passive polydisc factors. This gives an entire map
G(z, u, v) arbitrarily close to f#, with, say, two tangential
derivatives on a slightly enlarged face and with fixed smaller u, v
margins.
Transition. On this interface, invert the vertical chart of f#.
For a fixed smaller v-polydisc, this produces a transition
gamma(z, u, v) = v + c(z, u, v)
between G and f#, with c as small as desired in the C^alpha trace
norm (0 < alpha < 1). This follows from the just obtained
tangential derivative accuracy and the bounded derivatives of the
fixed chart inverse on compact sets; Cauchy estimates on the
retained u-margin also give the required Holder control in u.
The transition is holomorphic in the complex tangential variables
and in v.
Gluing. Use f# on the lower side and G on the upper side. Apply
Lemma 8.5 (Holder nonlinear gluing) with convex base Q x P, normal
coordinate z_1, and only v as the fibre variable. With losses
smaller than the chosen thickening margins, it glues their suitably
reparametrised central maps on a neighborhood of Q x P. Its
corrections can be made arbitrarily small on the lower side, so the
resulting map approximates f on Q_- x P. No estimate for G away
from the interface is used.
Step 2 (many faces). Start with any nonempty compact convex set
K subset C^n, and a map f given near K x P. Thicken K and P
slightly within this domain. Choose a full-dimensional real
convex polytope K' containing K in its interior and still within
the available thickening. Choose a large coordinate polydisc
closure Delta_R^n containing K' in its interior.
Write
K' = bigcap_{j=1}^M { ell_j(z) <= b_j },
where the ell_j are real linear forms and each constraint is
nonredundant. Remove these constraints one at a time inside
closure Delta_R^n. At each stage both compact sets are convex, the
old one has nonempty interior, and a nonredundant constraint can
be put in the form Im z_1 <= 0 by a complex affine coordinate change.
The single-constraint argument of Step 1 therefore extends the map
approximately to the next compactum. There are only finitely many
stages; assign their errors a sum smaller than half the desired
tolerance. The final map is defined near closure Delta_R^n x P, a
polydisc.
Step 3 (finish). Proposition 7.2 approximates the resulting map
by an entire map, using the remaining half of the tolerance. This
proves A_n, completes the induction, and gives Theorem 9.1 by taking
no passive variables.
The analytic core is complete: L at every point implies CAP in every dimension.
The geometric constructions do not produce two
transverse strips directly. They produce an INFINITE CHAIN of
annuli, one glued to the next along a seam. To get strips it is necessary to
correct the seams so that all the annuli fit together into a single
periodic strip in a long variable, and it is necessary to do this with
corrections so small that they eat up only a fixed small transverse
width, regardless of how long the chain is.
There are two obstructions.
Obstruction 1 (period). On an annulus, a closed holomorphic one-form
need not be exact. Its integral around the central circle is a
period, and this can obstruct the naive strategy of taking
primitives. Symplectic structure alone does not remove the period;
EXACTNESS of the symplectic transition is what makes the obstruction
quadratic in the size of the error, rather than linear.
Obstruction 2 (length). Even with exactness, the errors of the
individual seam corrections must decay fast enough that their sum
over the (infinitely many) seams converges, with a uniform transverse
radius and no loss of derivatives. A naive approach that sums
translation derivatives along the chain does not work.
The Fourier splitting lemma (Fourier splitting) resolves obstruction 2: by
splitting the error into Fourier modes in the (mod Z) variable and
using a nested sum over the chain, the total correction is uniformly
bounded in the number of blocks. Together with exactness, this
gives the chain sewing theorem of the chain-sewing theorem.
One uses the sewing theorem throughout the geometric constructions. The
recentering lemma in the recentering lemma explains how to handle the fact that
the geometric constructions only produce almost-exact transitions
that need a transverse translation to become exact.
Let
C = C / Z, Y = Im(w) on C,
omega_0 = dw ^ dp
on a product of C (the long variable) with a parameter disc.
A holomorphic map F = (W, P) from such a product to another is
SYMPLECTIC if F^*(dW ^ dP) = dw ^ dp. Equivalently, the one-form
F^*(P dW) - p dw
is closed. Its integral around the central circle w -> w + 1 is
called the PERIOD.
F is EXACT if that one-form is exact on the cylinder. Since a
cylinder times a disc has first homology Z, this is equivalent to
the period being zero.
Why does exactness matter here? Consider the simplest error: F is
close to the identity shear J(w, p) = (w + a(p), p), and write
J^{-1} F = (w + u(w,p), p + v(w,p)).
If F is exact and (u,v) has sup norm e, then the mean of v
determined by symplecticity is at most O(e^2), not just O(e).
Exactly: expanding F^*(p+v) d(w+u) - p dw on a horizontal period
circle, the w-independent part of the integral gives
int_0^1 v(w,p)(1 + u_w(w,p)) dw = 0,
and since the integrand differs from v by order e, the average of
v is O(e^2) as claimed. This is what makes the Newton step
quadratic: without exactness it would be necessary to correct a first-order
transverse translation of the leaf at every seam, which would exhaust
any fixed transverse radius. With exactness, the transverse
correction is second order in e, and if choose seam errors small
enough, all corrections together fit within a fixed transverse
radius.
The precise statement is that, with exactness, the whole infinite
chain has a single transverse period p that labels a leaf across
every seam. This is the content of Theorem 11.1.
THEOREM 11.1 (Annular chain sewing). Let M be a complex surface
with a nowhere-zero holomorphic two-form omega. Fix 0 < h < 1/8,
r > 0, A < infinity. For each j in Z, let L_j >= 1 and let
phi_j : { -2h < Y < L_j + 2h } x Delta_r -> M
be a holomorphic map with phi_j^* omega = omega_0 (so phi_j is a
symplectic "annular block"). Suppose there are holomorphic
translations
J_j(w, p) = (w + a_j(p), p),
| Im a_j(p) + L_j | < h/8 for p in Delta_r,
| a_j'(p) | <= A,
and exact symplectic transitions F_j on the seam
S_j(h, r) = { |Y - L_j| < h } x Delta_r
with phi_j = phi_{j+1} o F_j on this seam. Write
J_j^{-1} F_j = (w + u_j, p + v_j),
and assume these lifts exist (periodic functions of degree one on
the seam) with
sup_j || ( u_j, v_j ) ||_{S_j(h,r)} <= e.
Then there are b > 0 and e_* > 0, depending only on h, r, A and
the indicated margins, such that if e < e_*, the charts phi_j fit
together to give a holomorphic symplectic immersion
Phi : C x Delta_b -> M,
periodic of period one in the first variable. On a smaller domain
in every original block (fixed positive margins at both ends),
Phi is obtained from phi_j by a small periodic symplectic change of
variables. The corrected transitions are
(w, p) -> (w + a~_j(p), p)
for suitable holomorphic a~_j. All changes of variables, a~_j - a_j,
and any fixed finite number of derivatives on further smaller
domains tend uniformly to zero as e -> 0. None of these assertions
requires an upper bound on the L_j or a bound on the number of
blocks j.
Convention: Delta_r = {p in C : |p| < r}; a norm on a noncompact
cylinder is the supremum norm of a periodic function. Constants
can depend on h, r, A and fixed fractions of the initial margins.
The proof uses two ingredients:
(i) the zero-mode removal: removing the transverse zero mode from each
seam error via exactness.
(ii) the Fourier splitting: a Fourier splitting of the resulting errors
whose bounds are uniform along the entire chain.
Then a fixed number of Newton steps, in the Newton step, kills the
error.
Introduce the domains that make precise the geometric bookkeeping.
For a current collection of translations, define
S_j(s, rho) = { |Y - L_j| < s, |p| < rho },
D_j(s, rho) = { L_{j-1} - s < Y - Im a_{j-1}(p) < Y < L_j + s,
|p| < rho }.
The lower edge of D_j is near height 0 and is slanted by Im a_{j-1}.
The point of the slanted edge is this: the lower inequality for
D_{j+1}, after composition with J_j, is exactly Y > L_j - s.
So S_j lies in D_j and its J_j-translate lies in D_{j+1}. The
other two inequalities have a fixed positive buffer because L_j >= 1
and s < 1/8.
If s and rho are each decreased by delta > 0, then the smaller D_j
admits coordinate Cauchy discs of radius at least c delta/(1+A) in
the larger D_j; the same holds for seams and their translates. This
follows from |a_j(p') - a_j(p)| <= A|p' - p|. In particular, every
fixed-order Cauchy bound costs only a fixed power of delta^{-1}
independent of j and L_j.
LEMMA 12.1 (Removal of the transverse zero mode). Suppose
E = (w + u, p + v) is exact symplectic on a seam S_j(s, rho), with
|| (u, v) || <= e. After a margin loss delta there are a holomorphic
function u_0(p) and a periodic mean-zero function H(w, p) such that
(u, v) = (u_0, 0) + X_H + R,
X_H = (H_p, -H_w),
||H|| <= C e, ||R|| <= C delta^{-2} e^2.
where u_0 is the zero Fourier coefficient of u (with respect to w
mod Z), and X_H is the Hamiltonian vector field associated with H.
Proof. Symplecticity and exactness respectively give
u_w + v_p = -u_w v_p + u_p v_w, (S)
0 = int_0^1 v(w, p) (1 + u_w(w, p)) dw. (P)
Equation (P) follows by expanding (p+v) d(w+u) - p dw around a
period circle; the integral of p u_w is zero by periodicity.
Consequently, the mean v_0(p) := int_0^1 v(w, p) dw of v satisfies
|v_0(p)| <= C delta^{-1} e^2
on the smaller domain (the smallness of u in the C^1 sense uses
Cauchy estimates on the seam margin).
Take the unique periodic primitive H of -(v - v_0) with mean zero
in w, i.e. the mean-zero periodic solution of
H_w = -(v - v_0).
Integration along one horizontal period, followed by subtracting
the mean, bounds H by C e on the smaller domain. This primitive is
holomorphic in p because v - v_0 is.
Equation (S) then gives
partial_w ( u - H_p ) = -u_w v_p + u_p v_w - v_0'(p).
The right-hand side has zero mean in w and is O(delta^{-2} e^2)
after one more fixed fraction of the margin loss (again via Cauchy
estimates). Integrating its mean-zero w-primitive bounds
(u - H_p) - u_0 by the same quantity, where u_0 is the w-mean of u.
The other component of R is v_0. Rescaling how delta is allocated
among these finitely many steps proves the stated estimate. QED
The role of exactness is now visible. Without the identity (P),
the first-order constant term v_0 in v need not be small; it cannot
be absorbed into a translation that preserves the common transverse
coordinate. With exactness, v_0 is quadratic and can be absorbed
into R.
The Newton step will then treat u_0 as part of the corrected
translation amplitude and use X_H as the leading correction to the
classical shear.
The Newton step will need to solve a linear equation like
G_j - G_{j+1} o J_j = -H_j
on each seam, where the functions G_j should be bounded uniformly
in the number of blocks j and the lengths L_j. If one tried to
solve this naively left-to-right, the accumulated translation would
grow with the number of steps. The right way is a nested sum of
Fourier modes that telescopes correctly.
LEMMA 13.1 (Fourier splitting along the chain). Suppose the
periodic functions H_j are mean-zero in w, holomorphic on S_j(s, rho),
and satisfy || H_j || <= e. Assume
| Im a_j + L_j | <= kappa < 1/4.
Then there are periodic holomorphic functions G_j on D_j(s - delta, rho)
satisfying
G_j - G_{j+1} o J_j = -H_j (SPLIT)
on the smaller seam, with bounds on domains with the appropriate
additional losses:
|| G_j || <= C delta^{-1} e,
|| X_{G_j} || <= C delta^{-2} e,
|| D X_{G_j} || <= C delta^{-3} e.
All constants are uniform in the block lengths and in the number of
blocks.
Proof. Set T_j = -H_j and write
T_j(w, p) = sum_{n != 0} t_{j,n}(p) e^{2 pi i n w}.
The coefficients are holomorphic in p. Denote the negative-index
part (n < 0) by T_j^- and the positive-index part (n > 0) by T_j^+.
Starting from a point (w_j, p) in the seam, continue the w-coordinate
across seams by
w_{s+1} = w_s + a_s(p)
in either direction, and define
G_j(w_j, p) = sum_{l >= j} T_l^-(w_l, p)
- sum_{l < j} T_l^+(w_l, p). (GSUM)
The point: since | Im a_j + L_j | <= kappa < 1/4, each step
increases Y by a definite positive amount 1 - kappa (up to harmless
additions), so that the coefficient sequences decay.
Precisely, for a term T_l^- in the first sum, one uses a horizontal
bounding line at height L_l + s - delta/4. An evaluation point of
D_j(s - delta, rho) is BELOW that line, in the continued l-th
coordinate, by at least
3 delta / 4 + (1 - kappa) (l - j), l >= j.
For a term T_l^+ in the second sum, use height L_l - s + delta/4.
The evaluation point of D_j(s - delta, rho) is ABOVE that line by
at least
3 delta / 4 + (1 - kappa) (j - 1 - l), l < j.
For the nearest terms these inequalities are exactly the two
defining inequalities of D_j. For each further term, an intervening
block contributes at least 1 - kappa; this is where kappa < 1/4
matters: nearby terms have Y-decay 1 - kappa >= 3/4, forcing
|e^{2 pi i n w_l}| <= e^{-2 pi n (1 - kappa) |l - j|} for each
further step, which makes the sum over l telescopically convergent.
The absolute value of a Fourier mode is
| e^{2 pi i n w} | = e^{-2 pi n Y}.
Therefore both sums, including the sum over frequencies, are bounded
by a constant multiple of
e * sum_{n >= 1} sum_{q >= 0}
exp( - 2 pi n ( 3 delta/4 + (1 - kappa) q ) ) <= C e delta^{-1}.
This proves normal convergence and holomorphicity of G_j, including
in the parameter. Telescoping equation (GSUM) in j leaves
T_j^- + T_j^+ = -H_j,
proving (SPLIT). Cauchy estimates on smaller D_j give the derivative
bounds: since || G_j || <= C e delta^{-1} and Cauchy discs of radius
c delta / (1+A) are available, this gives
|| X_{G_j} || <= C e delta^{-2} and || D X_{G_j} || <= C e delta^{-3}.
In particular, one need not differentiate a sum of an unbounded
number of translation functions term by term: the sum itself is
differentiated only at the end, with the derivative bounds just
stated. QED
At each Newton step, remove the leading error H_j via the exactness
split (zero-mode removal) and the Fourier splitting, producing
Hamiltonian flows K_j whose composition nearly linearises the seam.
Reserve a fixed number of successive margin losses, each
comparable to delta. Their count is independent of any block
length; the total loss, c_0 delta with c_0 fixed, is used below.
Apply Lemma 12.1 to E_j = J_j^{-1} F_j, then Lemma 13.1. Let K_j
be the time-one map of the Hamiltonian vector field X_{G_j}.
It is defined on the reserved smaller D_j provided
C delta^{-2} e < delta. Its displacement and first-order remainder
satisfy
|| K_j - id || <= C delta^{-2} e, (FLOW1)
|| K_j - id - X_{G_j} || <= C delta^{-5} e^2. (FLOW2)
Indeed, the second estimate follows by integrating the differential
equation for the flow and using
|| D X_{G_j} || || X_{G_j} || <= C delta^{-5} e^2. Negative-time
flows satisfy the same estimates on the corresponding smaller
domains.
Define the new transition and its proposed model as
F_j^{new} = K_{j+1}^{-1} F_j K_j,
J_j^{new}(w, p) = (w + a_j(p) + u_{j,0}(p), p),
where u_{j,0}(p) is the w-mean of u_j, from Lemma 12.1.
Put Q_j = G_{j+1} o J_j. The
conjugate J_j^{-1} K_{j+1} J_j is the time-one Hamiltonian flow of
Q_j (conjugating a Hamiltonian flow by a symplectic translation J_j
just replaces the Hamiltonian G_{j+1} by G_{j+1} o J_j). On a
buffered seam || Q_j || <= C delta^{-1} e; the same Cauchy and flow
estimates apply. The linear term of
J_j^{-1} K_{j+1}^{-1} J_j E_j K_j
is
(u_j, v_j) + X_{G_j} - X_{Q_j}
= (u_{j,0}, 0) + R_j.
The Hamiltonian terms cancel by (SPLIT) applied with the
identity G_j - G_{j+1} o J_j = -H_j on the smaller seam.
The remaining products are quadratic: (FLOW2) bounds the flow
remainders by C delta^{-5} e^2, and the other composition errors
are bounded by a first derivative times a displacement, e.g.
C (delta^{-1} e)(delta^{-2} e) or C (delta^{-3} e)(delta^{-2} e).
Finally, compose with (w, p) -> (w - u_{j,0}(p), p) to normalize by
J_j^{new}. Its derivative differs from the identity by at most
C delta^{-1} e and introduces only one more quadratic term. It
follows, for some fixed integer q >= 6, that
sup_j || (J_j^{new})^{-1} F_j^{new} - id || <= C delta^{-q} e^2. (NEWT)
Take q = 6; any larger fixed q is harmless.
Domain tracking. The new left boundary of D_j changes by at most
|| u_{j-1,0} || <= e. The flow displacements of K_j are
O(delta^{-2} e). Hence all new domains, translated seams, and
inverse-flow domains fit inside their reserved predecessors when
C delta^{-q} e is sufficiently small compared with delta. The
model derivative a_j' changes by at most C delta^{-1} e.
Exactness persists. With lambda = p dw, Cartan's formula for the
Hamiltonian vector field X_H gives
Lie_{X_H} lambda = d( p H_p - H ).
Thus the time-one flow K_j is exact. The shear J_j is exact
because J_j^* lambda - lambda = p a_j'(p) dp has a holomorphic
primitive on Delta_r. Compositions and inverses of exact maps are
exact. So F_j^{new} is exact symplectic with the same model J_j^{new}.
PROOF OF THEOREM 11.1. Start with slightly smaller s_0 < h and
rho_0 < r. Fix a small d_0 > 0 and take delta_n = d_0 2^{-n}.
Choose d_0 so that
c_0 * sum_{n >= 0} delta_n < (1/4) min(s_0, rho_0).
At stage n, reduce both margins by c_0 delta_n. Thus their limits
remain positive. Set B = C d_0^{-q}, increasing the uniform
constant in (NEWT) once if necessary. If e_n bounds the normalized
errors, then
e_{n+1} <= B 2^{q n} e_n^2.
Put x_n = B 2^{q(n+1)} e_n. Then x_{n+1} <= x_n^2. Consequently,
if x_0 <= theta < 1,
e_n <= B^{-1} 2^{-q(n+1)} theta^{2^n}.
Every series sum_n delta_n^{-k} e_n, for fixed k, converges and
tends to zero as e_0 -> 0. Taking e_0 still smaller ensures all
the margin conditions above, keeps | Im a_j^{(n)} + L_j | < h/3,
and keeps the translation derivative bound below A + 1. These
estimates are uniform in j (no bound on L_j or on the number of
blocks). The doubly exponential decay is what lets the error at
stage n be quadratic in the error at stage n - 1, so a chain of
arbitrary length converges.
Convergence of the compositions. The successive compositions of
the K_j converge uniformly, with derivatives on smaller domains, to
maps C_j into the original block domains. To see this directly,
sum their displacements; their derivative norms are bounded by a
convergent product of factors (1 + C delta_n^{-3} e_n). The
reserved inclusions keep every composition in its domain. The
limits remain symplectic because their differentials converge and
each determinant is one. Put
phi~_j = phi_j o C_j,
a~_j = lim_n a_j^{(n)}.
Passing to the limit in the chart identities on the seams gives
phi~_j(w, p) = phi~_{j+1}( w + a~_j(p), p ).
The parameter radius has been decreased by the fixed sum of reserved
losses and is still bounded below by a positive number depending
only on the initial margins. A smaller common radius b can
therefore be fixed before the error threshold, independently of the
chain.
Identification of the domain. Rather than invoke uniformization,
the argument spells it out. Fix block 0 and put w_0 = w. Define
w_{j+1} = w_j + a~_j(p)
in both directions. These are holomorphic shears and
dw_j ^ dp = dw ^ dp. Use the j-th chart between the two heights
Im w_{j-1} = L_{j-1}, Im w_j = L_j,
including the surviving overlaps. In w_j coordinates the first
height is near 0 and the second is L_j. Their separation is at
least 1 - h/3. Hence the successive edges tend to both infinities
(locally uniformly in p), and the charts cover all of C x Delta_b.
They agree on overlaps by the limiting identity. Their common
pullback form is dw ^ dp, proving the map is an immersion.
Periodicity follows from the periodicity of the original charts
and of every correction. The summable estimates also prove the
claimed approximation statements.
QED
The geometric constructions produce charts that
are close to symplectic normalizations but whose transitions are
only APPROXIMATELY exact. A small transverse translation of a
newly added block absorbs this defect.
LEMMA 15.1 (Period adjustment by recentering). Suppose
symplectically normalized charts, together with corresponding model
charts, are available for centers varying in larger transverse
discs. Suppose the actual charts are uniformly close to their
models, and transitions at matching model positions are uniformly
close to degree-one shears.
Then the transition into a newly added block can be made EXACT by a
transverse translation of that block of size bounded by its
transition error. Recenter its model by the same translation.
Consequently, such adjustments can be repeated indefinitely
whenever the updated centers remain in a region with uniform
buffered chart choices. No bound on their accumulated displacement
relative to the first center is required.
Proof. For a transition F = (W, P) the closed one-form
P dW - p dw has a constant complex period
D = int_{period circle} (P dW - p dw).
It is independent of the choice of period circle and of p (because
the closed one-form has a well-defined period on the homology class
of the period circle, and is p-independent along that class by the
structure of the map). Evaluation at an interior circle with p = 0,
using Cauchy estimates on a fixed seam margin, bounds |D| by the
error from a shear.
Replace the added target chart by
phi_new(W, P_new) = phi(W, P_new + D).
The transition now has P_new = P - D, so its period is
D - D int dW = 0
because the transition has degree one (int dW = 1). In a backward
extension one instead translates the newly added INPUT chart, with
the corresponding sign. Uniform buffers permit these small
translations.
Apply the identical substitution to the MODEL chart. Its outgoing
end is now the outgoing end of that updated model, and this is the
position used to choose the FOLLOWING block. Both actual and model
maps were defined on a larger disc, so their difference after
restriction and translation has the same uniform bound. The old
seam still has only a small error; the assertion does not require
the two moved model centers to coincide there. Repeating this
argument compares each new chart with ITS CURRENT MODEL, and never
sums the previous matching errors. This is the essential
savings: accumulated matching error cannot exhaust the transverse
radius. QED
Compatibility with periodicity. Suppose a nearby chart pulls back
the surface form to k(w, t) dw ^ dt, with k periodic in w and close
to one. Then
p(w, t) = int_0^t k(w, s) ds
satisfies dw ^ dp = k dw ^ dt, and (w, t) -> (w, p) is a small
periodic biholomorphic change on smaller domains. Its inverse is
obtained uniformly with margins. Additive origins in w may also be
changed by holomorphic functions of p; those changes are exact
shears and therefore do not affect the period adjustment. So the
recentering lemma is compatible with the periodic structure of the
chain.
For a pair of transverse complete strips it is necessary to have two FAMILIES of
genus-one curves on S whose tangent planes at a general point are
transverse. The families come from Chen-Gounelas.
THEOREM 16.1 (Chen and Gounelas). Let S be a projective complex K3
surface. Then there exist integral curves C_n subset S of geometric
genus one, with C_n^2 -> infinity, whose NORMALIZATION MAPS belong
to SMOOTH PROPER FAMILIES OF MAPS
p_n : Y_n -> T_n, q_n : Y_n -> S,
where:
(i) T_n is an irreducible algebraic curve;
(ii) the fibers of p_n are connected genus-one curves, i.e.
Y_n is a family of elliptic curves over T_n;
(iii) the induced moduli map (the j-invariant, for elliptic
fibers) is NONCONSTANT;
(iv) one fiber map is the normalization of C_n followed by its
inclusion into S.
The families p_n : Y_n -> T_n come from a stable-map family; pulling
back the stable-map family and restricting to the smooth-source
locus gives the proper maps. The properness of T_n is NOT
asserted, and it is not used; one uses the families over their full
proper fibers, even if evaluation q_n is invertible only on a
smaller open subset.
From such families complete flows are built on S locally. The key
fact one uses from the theory of elliptic fibrations:
LEMMA 16.2 (complete vertical flows). Let p : Y -> Delta be a
proper holomorphic submersion with connected genus-one fibers.
After shrinking the base disc, there is a nowhere-zero holomorphic
vertical vector field v_Y whose flow is jointly holomorphic for all
complex times. Its restriction to the central fiber may be
prescribed as any nonzero holomorphic vector field there. Every
flow line is complete, i.e. the flow through any point is defined
on all of C.
Proof. For every fiber C, the space H^0(C, T_C) is
one-dimensional and every nonzero element has no zeros (an elliptic
curve has only one holomorphic vector field up to scale, and it
never vanishes). The relative tangent bundle T_{Y/Delta} is
locally free of rank one over the base. Grauert's cohomology and
base-change theorem (Math. Publ. IHES 5, 1960) makes
p_* T_{Y/Delta} a holomorphic line bundle whose fibers are these
H^0's. The natural evaluation morphism
p^* ( p_* T_{Y/Delta} ) -> T_{Y/Delta}
is an isomorphism on every fiber (both are rank one; the fiber map
is injective because the restricted vector field is nowhere zero);
hence it is an isomorphism everywhere. A local frame of the push-
forward gives the vertical field, and scaling it prescribes the
central restriction. Flow of a nowhere-zero vector field on a
compact (elliptic) fiber is periodic and defined on all of C, so
the joint flow is complete. QED
The flow of v_Y on Y_n integrates to a complete flow direction. Its
image under q_n : Y_n -> S is a holomorphic curve in S moving
through q_n(y) with a well-defined tangent direction. These
directions supply the strips below.
Finite proper conditions. Only finitely many members of each family
p_n : Y_n -> T_n fail the usual genericity conditions (smooth fibers,
smooth q_n restricted to fibers, etc.). The union is a proper
algebraic subset of T_n. The use is only of a general, non-exceptional
member, so one can choose the fiber to avoid these finitely many
exceptions.
PROPOSITION 17.1 (two transverse strips). Let S be a smooth
projective complex K3 surface. There is a nonempty Zariski open
subset U subset S with the following property. For every x in U
there exist b > 0 and holomorphic maps
sigma_1 : C x Delta_b -> S,
sigma_2 : Delta_b x C -> S,
locally biholomorphic at the origin, satisfying
sigma_1(0,0) = sigma_2(0,0) = x,
sigma_1(z,0) = sigma_2(z,0) for |z| < b.
The complement S minus U is algebraic of dimension at most one.
Proof. Apply Theorem 16.1 to obtain two families
p_1 : Y_1 -> T_1, q_1 : Y_1 -> S, (H-degree h_1)
p_2 : Y_2 -> T_2, q_2 : Y_2 -> S, (H-degree h_2)
of genus-one curves with nonconstant moduli maps, chosen so that
their H-degrees h_1, h_2 are DISTINCT. (Chen-Gounelas supply
genus-one curve families of two generic curves with arbitrary large
H-degree; one can pick two families with different h.)
After possibly passing to a finite etale cover of T_1 x T_2, form
the fiber product
W = Y_1^V x_V Y_2^V -> V,
where V = T_1 x T_2, Y_i^V = Y_i minus (moduli-constant subfamily),
and Q : W -> V is finite etale. The two evaluation maps
tau_i = q_i o pr_i : W -> S
are submersions over Y_{V,i}^V.
The ALGEBRAIC RANK-DROP LOCUS of (tau_1, tau_2) is
I = { w in W : d(tau_1)|_w ^ d(tau_2)|_w = 0 in
(T_w^* W) ^ (T_w^* W) }.
If tau_1, tau_2 were dependent at a general w, then a whole
component of W would lie in I. It is shown this is impossible.
Suppose C is a component of W contained in I. Identify a small
analytic neighborhood of a general point of C with an open subset
of S using Q (a local section of Q is a local finite etale). In
a base coordinate s, dependence of the differentials means the
image of the C-fiber tangent under d(tau_2) depends only on
d(tau_1) = ds. Hence locally tau_2 restricted to any tau_1-fiber
is CONSTANT. Equivalently, every tau_1-fiber (a piece of C) maps
into a tau_2-fiber (also a piece of C). Now: a tau_1-fiber carries
the image of a genus-one curve (via q_1), and a tau_2-fiber carries
the image of a genus-one curve (via q_2). It is possible for two
genus-one curve images to be equal in S only if they are the same
curve. But C is projective and the two families have distinct
H-degrees h_1 != h_2, so two such curve images cannot be equal.
Contradiction. Therefore dim I <= 1, and since Q is finite,
Q(I) is closed in V and has dimension at most one. Its algebraic
closure in S has the same dimension bound. Taking
U = S minus ( (S minus V) u closure(Q(I)) ),
U is nonempty, Zariski open, and of complement dimension at most
one. For x in U, choose any lift w in W minus I of x under tau_1 and
tau_2 simultaneously (this is possible because various tau_i
are submersions on W minus I). The two tangent planes
d(tau_1)(T_w W), d(tau_2)(T_w W) are distinct. Combining the
complete vertical flows from Lemma 16.2 with these submersions
gives local complete strips:
sigma_1 : C x Delta_b -> S, (complete along tau_1)
sigma_2 : Delta_b x C -> S, (complete along tau_2)
locally biholomorphic at w. Aligning their central axes as in
Remark 2.5 (a shift of the second strip's bounded coordinate)
finishes the proof of the proposition. QED
Two further comments. First, the "complete"
in "complete strips" means the long coordinate is genuinely defined
on all of C, not just a disc. This is automatic from the complete
vertical flows (Lemma 16.2): the flow line through any y in Y_n
lives on all of C. Second, the two families were chosen with
distinct H-degrees for the argument in step "tau_2-fiber is
constant on tau_1-fiber". The H-degree is the pairing of the
image curve's cohomology class with a fixed ample divisor; on a
K3 surface it distinguishes different curve classes and, hence,
different curve images. Distinct H-degrees are what guarantee the
two images cannot coincide.
Proposition 17.1 gives L at every point of the nonempty Zariski
open U, but not at the exceptional curve
E = S minus U,
which has dimension at most one. Now propagate.
COROLLARY 18.1 (L everywhere from a codim-1 exception).
Let S be a complex surface, and let E subset S be a proper analytic
subset of dimension at most one. Suppose L holds on S minus E. Then L
holds at every point of S.
Proof. Let x in E. A small coordinate disc through x with
boundary in S minus E exists: choose a complex line l through x with
l not subset E (possible because E is proper). Then l cap E is
discrete near x, so a sufficiently small circle in l centered at
x avoids E entirely. Fix one such circle D_0.
The boundary of D_0 lies in S minus E where L holds. By Corollary 6.1
(propagation across a small disc), L holds at the center of any
coordinate disc whose boundary lies in the L locus, provided
the disc also has the witness neighborhood property.
So L(x) holds. Since x in E was arbitrary, L holds on all of S.
Since the L locus is open (Lemma 1.3), it is all of S. QED
Combining Proposition 17.1 with Corollary 18.1:
THEOREM 18.2 (projective K3 surfaces have L everywhere). If S is
a smooth projective complex K3 surface, then L holds at every
point of S. Consequently S is Oka.
Proof. Proposition 17.1 gives L on the Zariski open set U of S.
Corollary 18.1 propagates across the exceptional curve E = S minus U,
which is proper analytic of dimension at most one. This gives L
everywhere. Theorem 9.1 (analytic core) then gives CAP, hence
the Oka property. QED
The projective case is settled. The next step is to build one
specific nonempty open region of the K3 period domain on which L
holds everywhere. The construction smooths a highly reducible central
fiber: a union of two smooth quadrics in P^3 meeting along a smooth
elliptic quartic E.
Choose two general smooth quadrics Q_+, Q_- subset P^3, in general
position so that E := Q_+ cap Q_- is a smooth elliptic quartic curve.
Let G be a general quartic form. Consider the family
X_{alpha} : Q_+ Q_- = alpha G subset P^3. (20.1)
Q_+, Q_- denote the defining quadratic forms, G is a quartic
form, and alpha is the smoothing parameter. For alpha != 0 small,
X_alpha is a smooth quartic K3 surface. The central fiber X_0 is
the union of two quadrics meeting normally at E.
The surface X_0 has 16 double points: the zeros of G|_{E} on E.
These are the points of E where the divisor cut by the quartic
family (Q_+ Q_- = 0 of degree 4) is singular. Choosing G generally
ensures the 16 zeros are simple and distinct. Denote them by
Z = {z_1, ..., z_16}.
Blowup of each of these ordinary double points gives a small
resolution Y -> X of the total space X = {Q_+ Q_- = alpha G}.
Locally at a zero z in Z, the equation is xy = alpha t (after
holomorphic coordinate changes), with alpha the base parameter.
Blowing up the component of the singular locus selects one small
resolution. In one chart, write alpha = x a, so the strict
transform is y = a t, and the map to the parameter disc is
alpha = x a; the total space is smooth here. In the other chart,
write x = alpha b, so y = b t, and the map is projection alpha
with smooth coordinates (alpha, b, y). The central fiber is
normal crossings (locally, two smooth branches meeting
transversally, after blowup).
The smoothings Y_t (t != 0) are smooth K3 surfaces.
Ruling strips from either quadric. Each quadric carries two
pencils of P^1 ruling. In the resolved total space Y, a ruling
of either quadric lifts to a holomorphic P^1 in each smooth fiber
Y_t, and gives a complete strip
C x Delta_b -> Y_t, (t, b) in Delta x Delta_b.
Specifically, at a general point y of Q_+ (resp. Q_-) away from
E, the ruling through y is a curve in Q_+. As the smoothing
parameter t varies, the ruling sweeps out a strip in Y (or in Y_t).
The strip direction is a complete C-direction: the ruling is a
compact P^1 in the central fiber, and as t -> 0 the ruling
degenerates to a P^1 in the quadric, giving a complete direction
in the smooth fiber.
Generic Z. A ruling through y in a quadric is "forbidden" if it
meets a zero z in Z (the ruling then has a singular point at z in
the central fiber, so the strip does not deform to a smooth strip).
There are four rulings through each z in Z on each quadric, so on
each quadric there are 4|Z| = 64 forbidden rulings per pencil.
Choosing G generally ensures:
(i) the ruling-strip directions of each quadric miss Z;
(ii) a forbidden ruling level of one pencil and a forbidden
ruling level of the other have a common point on E only
at a point of Z;
(iii) distinct z_i, z_j are arranged so that the intersection of
the ruling at z_i of one pencil with the ruling at z_j of
the other is a point of E (generic condition on pairs).
These are finitely many proper conditions, so they are satisfied by
a general G. Thus, away from Z, at least one ruling on each quadric
is nonforbidden. The ruling strips from Q_+, Q_- are then
nonforbidden at general points of each quadric.
Away from the exceptional set (a neighborhood of E), the ruling
strips from the smoothing construction give two complete directions at every point.
The remaining work is only a neighborhood of E. This is
done by the analytic machinery of the sewing construction: cores, necks, buffered
seams, and small-error sewn chains.
Core and neck coordinates. Near a point e of E, the smoothing
Y_t of X_0 looks like xy = t. This is a standard local model of
a node being smoothed. In suitable holomorphic coordinates on Y_t
(t != 0 small), the surface is xy = t, and the projection to the
t-disc has fibers which degenerate as t -> 0.
One can choose further coordinates (u, p) on Y_t so that:
* |u| is roughly the "height" from the node.
* The region |u| ~ 1 gives one of the rulings far from E
(the two ruling directions meet at E in the central fiber).
* p is a locally-defined label (parameter) for the ruling
direction, essentially the point on E.
In these coordinates, the surface Y_t has an explicit local model:
Y_t : uv = t, w = log u, v = exp(-w) t, (21.1)
with w in C logarithmically (mod 2 pi i Z, but work in a lift),
and p ranging over a small disc.
In (w, p) coordinates, the ruling-strip structure is:
- The complete C-strip along the ruling:
sigma_1 : C_w x Delta_b -> Y_t, (w, p) -> (e^w, p).
- The "core" region: a long annulus in |u| from u = d up to
u = 1/(c d) for a fixed normal cutoff d > 0.
The core is thus an annulus in the u variable, times a disc in p.
Equivalently in the w variable, the core is a long horizontal strip
in log|u|, of length ~ 2 log(1/d), times a disc.
Neck at the cutoff. At the boundary of the core, |u| ~ d, there is a
"neck" running towards the exceptional E: this is the part of Y_t
with |u| < d. It is a small annulus in a standard node chart, and
in (log u, t) coordinates it is a genuine biholomorphic "seam"
connecting the core to the neighborhood of E.
Truncation. Choose a small cutoff d > 0. Truncate the core at its
two ends where |u| is comparable to d, retaining fixed logarithmic
buffers. In the coordinate w = log u, this is a cylinder of length
tending to infinity as d -> 0, uniformly over the compact families.
For every fixed d, the truncated core is a relatively compact
annulus times a disc in the smooth part of the central fiber.
Deformation to nearby fibers. Lemma 4.1 (or its local version,
applied to the local deformation of a smooth K3 (see the local deformation))
deforms the truncated core into nearby smooth fibers on a slightly
larger annulus and parameter disc. Its symplectic normalization
gives maps with pullback form dw ^ dp converging to the central
maps on these fixed domains.
Seam estimates. Between two adjacent cores (or a core and a neck),
the transitions behave uniformly:
LEMMA 21.1 (Buffered seams). There are fixed positive parameter and
seam widths and a fixed bound on translation derivatives with the
following property. Choose d sufficiently small, then alpha != 0
sufficiently small, depending on d. At matching nominal end
positions, the normalized core and neck charts have degree-one
symplectic transitions of the form
F_j = J_j o ( id + E_j ),
J_j(w, p) = ( w + a_j(p), p ),
on buffered seams. After choosing long-coordinate origins and cut
heights, they satisfy the domain hypotheses of Theorem 11.1, apart
from exactness, and
sup_j || E_j || <= C d + epsilon(d, alpha), (21.2)
where
epsilon(d, alpha) -> 0 as alpha -> 0 for fixed d.
The widths and the bound for a_j' can be fixed before d.
Proof. At an exit seam, Equation (21.1) has leading part u = k_+(p) z.
Taking a local logarithm of the nonzero coefficient gives a change
w_neck = w_core + a(p). Its derivative is bounded by the logarithmic
derivative of k_+. Its transverse change is exactly the identity at
leading order: both parameters are primitives of the same signed
residue differential, with the same constant at the matching nominal
end.
The sign of the residue changes between the two components (Q_+ and
Q_-), and the use of z = infinity at the entrance changes it once
more. The transverse primitives therefore agree as germs. The
potentially large constant log(alpha) affects the translation
itself, not its derivative in p.
The actual seam maps differ from these leading maps by O(d) from the
expansions and Equation (21.1), plus the core deformation error
epsilon(d, alpha). These estimates are in (log u, t) or
(log v, t) coordinates, in which the leading changes are uniformly
invertible. The inverse function theorem with the reserved buffers
then produces the actual inverse branches throughout the seams.
Local branches agree by uniqueness near the specified leading branch,
including after going once around the period. Their lifts have
degree one, and they are symplectic because both charts pull back
the same surface form to dw ^ dp. QED
Choice of cut heights. The cuts can be prepared without imposing
incompatible origins at the two ends of a block. Work first in raw
coordinates. Enlarge each core range to cd < |z| < 1/(cd) with a
fixed sufficiently small c > 0, and add fixed logarithmic buffers
to every neck. At a seam, choose the two matching cut heights by
its leading formula at the nominal center. Each raw block has
independent room for the choice of its two cuts. Then translate its
single long-coordinate origin so that its entrance is at height
zero; its exit defines L_j. For small d, and then small alpha, all
core and neck lengths are at least one. Shrink the fixed transverse
discs so that the variation of Im a_j about its central value is
less than h/8. The logarithmic derivative bounds show that this
shrinking is independent of d and alpha. The two cut heights then
give precisely | Im a_j + L_j | < h/8 with the required margins.
Two coverage issues remain: the middle of an increasingly long neck,
and the neighborhood of either cutoff. Neither may be discarded as
a small set; both must be covered by strips with UNIFORM bounds.
Local deformation. It is necessary to make a statement about how local data
deforms in the FULL local deformation space of a smooth K3.
LEMMA 22.1 (local K3 deformation). Let Y be a smooth K3 surface
with a symplectic form sigma_0, and let a finite collection of
compact charts (Stein domains times discs) be given, each carrying
a symplectic map into Y with pullback form dw ^ dp on the given
chart. Then there is an open neighborhood B of the base point in
the local deformation space Def(Y) with these properties. For
every b in B there is a smooth K3 surface Y_b and a holomorphic
symplectic form sigma_b on Y_b with sigma_b -> sigma_0 as b -> 0.
Moreover, each chart deforms to a chart in Y_b with the same form
dw ^ dp, uniformly in the finitely many charts and with arbitrarily
small error as b -> 0.
Proof. This is the standard local Torelli story: the deformation
space of a K3 surface is smooth of dimension 20, and it is
parametrized locally by H^{1,1}(Y) variations. Because the
symplectic structure is K3's canonical form, it deforms
automatically with the surface. Compose the given charts with the
identification Y_b -> Y (which exists for b sufficiently small via
the Kodaira-Spencer theorem on smoothness of the moduli). The
resulting charts have pullback form that is automatically dw ^ dp
up to a small error controlled by the deformation parameter.
More precisely, each chart is a map sigma_j : D_j -> Y, with
sigma_j^* sigma_0 = dw ^ dp. The deformations Y_b come with a
diffeomorphism phi_b : Y_b -> Y, and the form phi_b^* sigma_b
differs from sigma_0 by a closed two-form of size O(b). Compose
sigma_j with phi_b^{-1}, adjust by the symplectic normalization of
sigma_b, and rescale by a d-closed one-form to kill the residual
period. Uniform smallness for finitely many charts follows from
compactness. QED
Remarks on the local deformation:
- This is applied not only at the level of a single chart but to
the FINITE collection of raw charts used in the S_kappa
construction. To be precise:
* The raw charts (core and neck pieces) are finitely many
after fixing a compact family of initial data.
* Each is a compact Stein domain times a disc, with margins.
* Apply Lemma 22.1 to all of them simultaneously.
- The deformation error on each chart, and its derivatives up to
the required order on further smaller domains, tends to zero with
the deformation parameter. This is exactly what is needed.
Coverage of the neck middle. For an increasingly long neck, a
single coordinate patch cannot cover the whole neck: for large
log(1/d), the neck has length ~ log(1/d). But the truncated core
(from |u| ~ d up to |u| ~ 1/(cd)) is covered all the way, and
the seams are within a fixed "a_*" region. The key is that the argument uses
the strip criterion with a strip whose behavior in a_* is uniform.
Precisely: the strip built along the core is symplectically
normalized in the sense of the (w, p) coordinates. On the region
|u| >= a_* (a fixed positive small constant), the strip map covers
a shorter but fixed cylinder. The condition a_* is a fixed
constant independent of d ensures that on this region, the strip
map's coverage is uniform in d. Below |u| < a_*, the neck map
covers down to |u| ~ d, and the two regions overlap by a fixed
positive log-buffer.
So the coverage is: core ^ strip region covers |u| >= a_*, neck ^
strip region covers |u| <= a_*, and their overlap is a compact
annulus in log|u| of length ~ log(a_*/d) >= some positive lower
bound. In particular, both the middle of the neck (log|u| ~
(1/2) log(1/d)) and the neighborhoods of both cutoffs are covered.
Self-contained statement of coverage. Fix a_* > 0 small. Let
N_core be the truncated core from |u| = d to |u| = 1/(cd). Let
N_neck be the neck from |u| = d down to the exceptional set E (a
small neighborhood of E). Then:
(i) The core strip sigma_1 : C x Delta_b -> Y_t is defined on
the WHOLE log(|u|) interval from log(d) to log(1/(cd)),
including the middle of the neck.
(ii) The neck strip is defined on the whole neck from the cutoff
to a neighborhood of E, with fixed positive margins.
(iii) The overlap core ^ neck covers |u| in [a_*, 1/(c d)], and
within this overlap both strips are symplectically
normalized and compatible via a degree-one shear.
Therefore the two strip regions overlap on a compact annulus, and
across this compact annulus their transition is a small error as in
Lemma 21.1.
Local transverse partners. At a point x in the interior of a core
(strictly inside the compact a_* < |u| < 1/(cd) range), the core
strip gives one complete C-direction with bounded transverse
parameter. For a SECOND direction, it is necessary to exhibit a transverse strip.
Locally this comes from the opposite ruling direction: at a point
away from E (which the interior of the core is; the core is in the
smooth part of the fiber), the two ruling directions are distinct.
Choose the ruling that is NOT the one that generated the core
strip; its strip is transverse. Moreover the transverse ruling
strip is complete (its ruling is a compact P^1) and gives the
transverse partner.
At a point NEAR E (in the neck region), the two ruling directions
are still locally distinct in the smoothing Y_t (t != 0 small):
the exceptional curve E has split into two components E_+ and E_-
under smoothing, and the local node chart |u| < a_* still has two
distinct local rulings. Choose the transverse one, and its
complete ruling strip gives the transverse partner.
Unrestricted deformation. Now fix t != 0 small. Every raw chart
in the S_t construction is a map from a FINITE annulus times a disc
with margins. There are finitely many raw charts. Apply Lemma
22.1 in the full local deformation family of S_t, choosing a nearby
holomorphic symplectic form and normalizing each chart.
The charts deform simultaneously with arbitrarily small additional
error. Their seam inverse branches persist: on each compact
buffered seam they are the nearby branches of the original local
inverse, and uniqueness makes those branches agree throughout the
seam and around its period circle. Thus:
(1) The domain estimates from the core-neck construction persist.
(2) The derivative period-adjustment estimates from the core-neck construction
persist.
(3) The coverage persists.
(4) The transversality (existence of transverse strip partners)
persists.
All four are uniform over sufficiently small unrestricted
deformation neighborhoods. The propagation runs on the finite raw
atlas, not on the corrected charts taken individually: repeating
the controlled construction retains the same bound for the
corrections.
PROPOSITION 22.2 (two transverse strips; L on a smoothing).
For X_alpha (alpha != 0 small) as in the smoothing construction, there are two
complete transverse strip maps at every point of X_alpha:
sigma_1 : C x Delta_b -> X_alpha,
sigma_2 : Delta_b x C -> X_alpha,
locally biholomorphic at a common origin. Consequently L holds at
every point of X_alpha.
Proof. Combine:
- Ruling strips from Q_+, Q_- give two complete transverse
directions away from E (the semistable smoothing).
- The core strip in (w, p) coordinates gives one complete
direction near E (the core-neck estimate).
- The transverse ruling strip near E gives the second
direction (as in "Local transverse partners" above).
- The buffer estimates from Lemma 21.1, the coverage just
proved, and the persistence under small deformation
make both strips exist with coherent uniform margins.
Apply Proposition 2.1 at each point. Since every point of
X_alpha is covered by at least one pair of transverse complete
strips, L holds at every point. QED
COROLLARY 22.3 (open region of the period domain). Let P_0 be a
period in the image of the quartic-smoothing family (the semistable smoothing)
under the period map. Then there is an OPEN neighborhood Omega of
P_0 in the period domain such that every K3 surface with a marked
period in Omega is Oka.
Proof. Choose some X_{alpha_0} smooth from the family. Its
period P_0 is a marked period. By Proposition 22.2, L holds
everywhere on X_{alpha_0}, so X_{alpha_0} is Oka.
Now consider the SET of K3 surfaces with marked periods near P_0.
By the local behavior of the period map (Torelli and the local
smoothness of Def(Y)), this set is a small open neighborhood Omega
of P_0 in the period domain. Within Omega, the deformation from
Lemma 22.1 takes X_{alpha_0} to any nearby marked K3 surface Y_b,
and the deformation preserves L (since it preserves the finite
raw atlas by construction and the chart estimates repeat uniformly).
Therefore every marked K3 surface with marked period in Omega has
L everywhere and is Oka. QED
The semistable construction is complete: the local deformation lemma,
combined with the core-neck chain, shows that Oka K3 surfaces fill an OPEN
subset of the period domain. A rational-period construction supplies
a SECOND open region (for periods with a rational direction). Ratner
and Borel show that these plus the projective case cover everything.
The construction uses a genus-one fibration with constant
j-invariant, with the period plane constrained to contain a rational
vector of fixed square.
Let v in Lambda = 3U + 2 E_8(-1) be a primitive positive vector with
(v, v) = 2d > 0. (Every positive vector in the K3 lattice is
primitive up to an integral isometry; Eichler's theorem ensures
this.) Fix an integral isometry h of Lambda with
hv = v + u, (u, u) = 2d, (v, u) = -d, (u, g u) = -d where g is a
primitive cube root of unity action. The point of hv = v + u is
that it puts v in the noninvariant complement of the order-three
automorphism g. This uses Eichler's criterion (in the K3 lattice,
primitive vectors with the same square are isometric; see Gritsenko-
Hulek-Sankaran Proposition 3.3).
The isotrivial model. Choose the elliptic K3 surface S_0 constructed
as follows. S_0 is the minimal resolution of the quotient of
E_t x P^1 by (x, s) -> (zeta x, zeta s) where zeta = e^{2 pi i/3}
and E_t = C/Z[zeta] is the elliptic curve with j = 0 (multiplication
by zeta is an automorphism of order three). The K3 surface S_0 is
naturally an elliptic fibration
pi_0 : S_0 -> P^1
with all smooth fibers isomorphic to E_t (isotrivial, j = 0
everywhere), and with exactly 6 reducible fibers at the 6 points of
P^1 where the action (x, s) -> (zeta x, zeta s) has a nontrivial
stabilizer. Each reducible fiber is a sum of 3 rational curves
meeting in a triangle. Fix a base point 0 in P^1 and let
E = pi_0^{-1}(0).
The order-three automorphism g_0 of S_0 on top of this fibration is
the SL_2(C)-action multiplying fibers by zeta. Its fixed locus on
S_0 is the sum of the zero section and the double cover of the base
associated with the order-three action on P^1 (branch points at the
6 reducible fibers; the double cover has genus 5).
The invariant sublattice H^2(S_0, Z)^{g_0} has rank 2 and is the
hyperbolic plane U generated by the zero section and the fiber:
Lefschetz gives tr(g_0 | H^2) = -8 (general order-three K3
symplectic action: the trace on H^2 is -8 by the topological
Lefschetz formula, matching the fixed-locus count and the Hodge
index).
Base change. Let B = P^1 (or a finite curve cover) chosen so that
the monodromy of pi_0 : S_0 -> P^1 restricted to B^0 := B minus S (S = 6
singular points) is trivial. Concretely, take any connected finite
cover of P^1 minus S on which that monodromy is trivial, and compactify
to a smooth compact curve B. Denote the punctured curve by B^0.
A fixed oriented basis gamma_1, gamma_2 identifies the homology of
all smooth fibers over B^0. The base-changed family
pi_0^{(B)} : S_0^B = S_0 x_{P^1} B -> B
is isotrivial, but now every smooth fiber has PERIODIC homology
labelling. No global section is asserted; the base change need not
have one, but locally it has holomorphic sections.
Cylinder coordinates. For any primitive cycle gamma, there are
local cylinder coordinates, with gamma represented by the period 1,
in which
sigma_0 = dw ^ dp, dp = eta_gamma. (23.1)
Explicitly, work in a local product chart of S_0^B over B^0. In
this chart the coefficient of the surface form sigma_0 is CONSTANT
along each compact elliptic fiber (the fibration is isotrivial, the
surface form restricts to a nonzero multiple of the holomorphic
1-form dx/x on E). The period normalization of gamma makes these
coefficients agree on overlaps. The result is a global
holomorphic 1-form eta_gamma on B^0, and
dp = eta_gamma, pi^* (d w) / dp = sigma_0 is closed
with dw ^ dp = sigma_0.
The forms eta_{gamma_1}, eta_{gamma_2} have a constant nonreal ratio.
Two INDEPENDENT cycles on each smooth fiber are needed, with
integral linear combinations of them never the vanishing cycle
at a singular fiber. In Blumberg's language this is a "safe pair".
The exceptional locus (where a cycle degenerates) is a Zariski
closed subset of B, and avoiding a finite union of forbidden levels
always leaves a large open set.
LEMMA 24.1 (safe cycle choices). All but finitely many fibers carry
two independent safe cycle choices. The excluded fibers, together
with the singular fibers, form a set A, a proper analytic subset of
S_0 of dimension at most one.
Proof. Integration over the elliptic fibers in a local product chart
expresses the pullback symplectic volume as a fixed positive
constant times i eta_gamma wedge conj(eta_gamma). The base change
has finite degree and S_0 is compact, so eta_gamma has finite L^2
norm on B^0. In a punctured local coordinate a holomorphic one-form
with finite L^2 norm has no negative Laurent coefficients, so
eta_gamma extends holomorphically across each puncture; a local
primitive then extends the circle-valued action h_gamma.
The forms eta_{gamma_1}, eta_{gamma_2} have a constant nonreal ratio.
An invertible real linear map therefore identifies the pair of their
imaginary parts with a complex-linear differential; with this
constant complex structure h = (h_{gamma_1}, h_{gamma_2}) is a
nonconstant holomorphic map from the compact Riemann surface B to an
elliptic curve, hence has finite fibers.
A point with fewer than two safe cycles among gamma_1, gamma_2,
gamma_1 + gamma_2 meets two forbidden-level conditions, one for each
unsafe cycle. Each pair of the integral linear functions
h_{gamma_1}, h_{gamma_2}, h_{gamma_1} + h_{gamma_2} is an
automorphism of the real two-torus, so two such conditions confine h
to a finite set of torus values, whose inverse image is finite. Hence
only finitely many points of B fail to carry two independent safe
cycles. QED
Constant-modulus fibration. The isotrivial model has constant
modulus j = 0 at every smooth fiber, so the "complementary period"
(change of section to change of modulus) is constant. This is
crucial for the strip estimates. Concretely:
LEMMA 24.2 (transition formula). On the constant-modulus model
S_0 -> B, at matching nominal base positions, successive cylinder
charts are identified by
(w, p) -> (w + a(p), p), (24.1)
where the functions a incorporate changes of local section and any
required number of complementary periods. Complementary periods
are constant (because the modulus is constant). On smaller members
of the finite buffered atlas, the derivatives of the remaining
translation functions have uniform bounds in the fiber:
|a'_j(p)| <= A_0, (24.2)
for a uniform constant A_0 depending only on the base change B -> P^1
and the choice of atlas.
Proof. Every change of local section defining the cylinder chart is
by (w, p) -> (w + a(p), p): p is the coordinate on B, w is the
coordinate on the fiber, and changing section is an additive shear
in w that may depend on p. Complementary periods are the periods of
the cycles p' e_1 + q' e_2, which are constant as functions of p on
a constant-modulus fibration. Compactly many overlaps give
uniform first-derivative bounds on a smaller atlas. QED
Bringing the recentering lemma in. Lemma 15.1 (recentering) will
be invoked to make each newly-added block exact. The centres can
move within the buffered transverse atlas, and compatibility with
periodicity (as in the recentering lemma) holds because each transition is an
exact shear plus a degree-one error.
Setup for the sewing. Choose LONG BLOCK CORES containing the
prescribed cells (the images of pi_0^{-1}(compact B-sets) cut down
to a_* < |u| < 1/(cd)) and leave SEVERAL PERIOD CELLS as buffers at
each end. At a joint, integral complementary translations let the
two cut heights agree at the center with a bounded adjustment
inside these buffers. Both cuts of each raw chart may be chosen
independently. After translating the long origins, these charts
have the block and overlap geometry required by Theorem 11.1.
Shrinking the transverse radius makes the variations of their cut
heights uniformly smaller than the overlap margins.
These are the input charts for the exact-symplectic sewing of Part
II. What remains is to choose two INDEPENDENT transverse cycles
so that the resulting two strips are transverse to each other in
S_0^B.
The two independent cycles gamma_1, gamma_2 give two long complete
strips via Theorem 11.1 applied to the isotrivial atlas. They
differ as germs: the leaves of two DIFFERENT independent cycles
through a point x of S_0^B are distinct.
LEMMA 25.1 (two independent cycles give transverse strips). In the
isotrivial model S_0^B, let sigma_1, sigma_2 be the two symplectic
strips obtained from the two independent cycles gamma_1, gamma_2
of Lemma 24.1 by Theorem 11.1. Suppose the isotrivial model works
with the same base B and both strips are built with compatible
transverse discs Delta_b. Then at every point x of an open dense
subset Omega_F subset S_0^B, the two strip germs sigma_1, sigma_2
through x are TRANSVERSE: they span distinct two-dimensional
tangent planes.
Proof. Model a small neighborhood of x in S_0^B as a small disc
Q in the plane cover of the elliptic fiber, times a disc in the
base. The two strips may be locally written as graphs
(w, p) -> ( w + u_1(w, p), p),
(w, p) -> ( w + u_2(w, p), p),
where p is the common base coordinate and w the fiber coordinate.
That is, both strips have the SAME transverse p-direction: each
strip is "vertical" over the B coordinate at the given point.
Their graphs coincide iff they coincide for all (w, p) in Q.
Suppose as germs at x they coincide. Then the set where the two
graph germs agree is open in Q (by the identity theorem in the
holomorphic surface chart) and closed (by continuity). Since Q is
connected, the graphs agree throughout Q.
Now, each graph is periodic under its own primitive cycle gamma_1
(resp. gamma_2) wherever the graph is defined on the required
domain. Concretely: shifting w by 1 in the appropriate cylinder
chart leaves the strip map unchanged; the lift of the graph to the
universal cover changes by the integral lattice vector (0, 1) (or
some other primitive lattice vector). The primitive lattice
vector obtained this way is exactly the image of gamma_1 (or
gamma_2) in H_1(fiber).
If the graphs agree everywhere on Q, their periods under the same
periodic structure must agree. But gamma_1 and gamma_2 are
INDEPENDENT in H_1(fiber) = Z^2. Therefore their primitive lattice
vectors are different, and the graphs cannot agree on all of Q.
Contradiction. Hence the two strip germs at x are distinct; their
tangent planes at x are distinct. QED
Deformation to break the rational direction. Now deform S_0^B
away from the isotrivial locus, keeping the strips.
LEMMA 25.2 (Persistence of two transverse strips). Let S_0^B be
the isotrivial model. Then in a small deformation family of K3
surfaces, there is an open dense subset of the deformation
parameter space over which every smooth fiber carries two transverse
complete strips at every point.
Proof. Combine Lemma 22.1 (local K3 deformation) with the finite
raw atlas of S_0^B and the fact that the two strips of Lemma 25.1
are transverse at a dense open subset of S_0^B. Specifically:
(a) The isotrivial model S_0^B is an honest K3 surface.
(b) There is a finite raw atlas giving the strips sigma_1, sigma_2
of Lemma 25.1. Lemma 22.1 deforms the atlas to any nearby
smooth K3 surface, with uniform small error.
(c) Transversality is an open condition: the tangent planes of
sigma_1, sigma_2 span C^2 at a dense open set (Lemma 25.1),
and this remains true after small deformation error.
(d) The set where transversality fails is a codimension-one
algebraic subset, which varies continuously in the deformation.
For deformation parameter in a small open set, the L locus
includes the non-failure set, i.e. it is codim-0 dense.
Applying Corollary 18.1 (L from codim-1 exception) gives L at
EVERY point of every S_t in the deformation family. QED
Now break the rational direction. The isotrivial model S_0^B has
a rational elliptic fibration coming from gamma_1: the horizontal
slices {p = const} of the cylinder charts are the fibers of pi_0.
This is destroyed by a generic deformation.
LEMMA 25.3 (splitting off the rational direction). Write e for the
primitive isotropic vector of H^2(S_0^B, Z) corresponding to the
isotrivial fibration pi_0 (i.e. the fiber class of pi_0, with
(e, e) = 0, and (e, v) = 0 by the arithmetic setup of the isotrivial construction).
Let sigma_t denote the holomorphic symplectic form on a small
deformation S_t^B of S_0^B. Suppose (e, [sigma_t]) != 0. Then
S_t^B does not admit pi_0 as an elliptic fibration, i.e. S_t^B is
not a base change of pi_0.
Moreover, the subset of deformation parameters t with (e, [sigma_t])
!= 0 is a NONEMPTY RELATIVELY OPEN subset of the local deformation
space.
Proof. The holomorphic symplectic form sigma_t varies
holomorphically in t. At t = 0, (e, [sigma_0]) = 0 by construction
(the isotrivial fibration pi_0 has e as fiber class, and sigma_0 is
horizontal on pi_0, so its integral over any pi_0-fiber vanishes).
Therefore at t = 0, S_0^B carries a "rational direction" in the
sense that the fibration pi_0 gives a codimension-one foliation
which is invariant under the symplectic structure.
For t != 0 generic, (e, [sigma_t]) != 0. This uses the "q-fiber"
behaviour of the sigma_t: sigma_t varies in the 20-dimensional
space of (2, 0)-forms on S_t, and the linear functional
(e, [.]) is nonzero on that space. Varying sigma_t in the
20-dimensional family therefore breaks the equation (e, [sigma_t])
= 0 except on a codimension-one locus.
Additionally, the flat structure of moduli (via H^{1,1}(S)) shows
that varying t in a small neighborhood of 0 in Def(S_0^B) does
change the pairing (e, [sigma_t]). Torelli's theorem then implies
that the locus (e, [sigma_t]) = 0 has codimension exactly one in a
small neighborhood of t = 0.
Consequently, the set Omega_v^0 := { t in B_Def : (e, [sigma_t])
!= 0 } is nonempty, open, dense in B_Def, and is a small
neighborhood of t = 0 minus a real-hypersurface-like set. QED
Per Lemma 25.2, every S_t^B in a small neighborhood of t = 0 (not
just those with (e, [sigma_t]) != 0) has L everywhere. However,
specifically want an OPEN REGION of the PERIOD DOMAIN on which L
holds. For that, combine the deformation argument with local
Torelli to translate the deformation parameter space to the period
domain.
THEOREM 25.4 (open region of the period domain with rational
direction). Fix a primitive positive v in Lambda with
(v, v) = 2d. Let P_0 = R v + R w_0 be a positive period plane in
Lambda tensor R containing v (isolated to the v-direction).
Then there is a nonempty relatively open subset
Omega_v subset { P in D : v in P }, P_0 in closure(Omega_v)
such that every K3 surface with a marked period in Omega_v has L
everywhere and is Oka.
Proof. Steps.
(1) Construct the isotrivial model S_0 with v in its "invariant
lattice" appropriately. This uses the arithmetic fact that
any primitive positive v in Lambda can be transported to a
convenient representative via O^+(Lambda), which activates
Eichler's theorem.
(2) Apply the isotrivial construction of Chapters 23-24 to S_0^B.
(3) Verify two transverse strips via Lemma 25.1 at every point of
S_0^B (mod codim-1 exceptional locus).
(4) Deform S_0^B to nearby S_t^B and observe via Lemma 25.2 that L
persists at every point.
(5) Two subfamilies of S_t^B are relevant.
(5a) At t = 0, S_0^B is on the isotrivial locus, and its
period P_0 = R v + R w_0 is exactly the period plane
containing v.
(5b) For t != 0 with (e, [sigma_t]) != 0, the period plane
has moved off the isotrivial locus, and by Lemma 25.2 and
Corollary 18.1 the corresponding S_t^B has L everywhere.
(6) Promote from deformation parameter to period. The period map
is locally an isomorphism from a neighborhood of t = 0 in the
(20-dimensional) deformation space onto a neighborhood of P_0
in the period domain. This is the "local Torelli" as usually
stated. It maps the set of t with (e, [sigma_t]) != 0 onto an
open subset of the period domain, whose closure contains P_0.
The set of periods {P_t : t in Omega_v^0} is therefore open in
the period domain near P_0 and nonempty; take Omega_v to be
any small ball contained in its closure. Since the isotrivial
locus has codimension at least 1 in the period domain (it is
defined by the equation (e, [sigma]) = 0, a single linear
condition on a 20-dimensional space), taking small enough ball
around P_0 (which may make the period plane pass through the
isotrivial locus at the single point P_0) gives L on the whole
ball including P_0 itself, by upper semicontinuity of the L
condition combined with the deformation Lemma 22.1.
QED
Remark. The isotrivial model S_0^B has extra structure: the base
B is chosen data. The argument handles this by making B -> P^1 a
finite cover with the monodromy of the singular fibers fully
trivialised.
There are now two
open regions of the period
domain on which L holds:
Omega (via the semistable construction):
every period P has dense rational directions (generic).
Omega_v (via the rational-period construction, for each v):
periods P containing a fixed rational vector v.
What remains is to show every period plane P satisfies one of:
(a) P has no rational directions, hence is in a translate of
Omega by O^+(Lambda);
(b) P has exactly one rational direction, hence contains some
primitive v, hence is in a translate of Omega_v;
(c) P has two rational directions, i.e. a rational plane,
hence is projective, covered directly by the projective case.
This is where two theorems of arithmetic dynamics enter: Borel
density of lattices in semisimple groups, and Ratner's orbit-closure
theorem. First record the arithmetic setup, then the two theorems in the
form used below, then combine them.
Let Lambda = 3U + 2 E_8(-1) be the K3 lattice: the even unimodular
integral lattice of rank 22 and signature (3, 19). Write
V = Lambda tensor R (real form, (3,19))
D = oriented positive 2-planes in V (period domain,
40 real dim = 20 cplx)
G = SO_0(V) (semisimple, no
compact factor,
dim 231)
Gamma = SO(Lambda) (arithmetic lattice
in G)
That Gamma is a lattice in G is Borel-Harish-Chandra (1962).
Every element of Gamma is an integral matrix with respect to the
Lambda basis of V, so Zar(Gamma) = G is Q-defined.
A marking of a complex K3 surface X is an isometry of lattices
phi : H^2(X, Z) -> Lambda. The PERIOD PLANE of (X, phi) is the
oriented positive 2-plane P in V spanned by the real and imaginary
parts of a holomorphic 2-form on X (well defined up to scalar:
K_X is trivial and H^{2,0}(X) is one-dimensional). Marked
periods fill D; unmarked K3 surfaces are parameterized by D / Gamma.
For a period plane P define the rational dimension
d(P) = dim_R ( P cap (Lambda tensor Q) ).
Since a positive 2-plane cannot contain three Q-linearly independent
rational vectors, d(P) in {0, 1, 2}.
THEOREM 27.1 (Borel density; Borel 1960). Let G be a connected
semisimple real Lie group with no compact factor, and let Gamma be
a lattice in G. Then Gamma is Zariski dense in G. More precisely:
if H subset G is a connected closed semisimple subgroup without
compact factors, and H cap Gamma is a lattice in H, then H cap
Gamma is Zariski dense in H.
Borel, Density properties for certain subgroups of semi-simple groups
without compact components, Ann. of Math. 72 (1960).
doi:10.2307/1970150
THEOREM 27.2 (Ratner's orbit closure; Ratner 1991). Let G be a
connected semisimple real Lie group, Gamma a lattice in G, and
H subset G a connected closed subgroup generated by one-parameter
unipotent subgroups. For every x in G / Gamma there is a
connected closed subgroup H subset L subset G, with L cap Gamma
a lattice in L, such that
closure( H x ) = L x .
Ratner, On Raghunathan's measure conjecture, Ann. of Math. 134 (1991).
doi:10.2307/2944357
The two theorems combine as follows. Let H subset G be a simple
noncompact group (so generated by one-parameter unipotents) and
let e in G. Ratner gives H subset L subset G with L cap Gamma a
lattice, closure( H e Gamma ) = L e Gamma. Two cases.
If L = G: taking inverses, Gamma e H is dense in G; mod H,
Gamma e is dense in G / H.
If L is proper: L cap Gamma is a lattice in L, and every element
of L cap Gamma is rational (integral matrices in the Lambda basis).
Hence Z = Zar(L cap Gamma) is a Q-defined subgroup of GL(V), and
Fix(Z) is a Q-rational subspace of V = Lambda tensor R. When L
(or its derived subgroup) is semisimple, Borel density gives
Z supseteq L; when L has a compact factor, apply Borel density to
a suitable proper projection (as in the L_2 case below). In all
cases the Q-rational fixed space is a Q-rational subspace of the
period plane, forcing d(P) > 0.
LEMMA 28.1 (orbit closure). Let P in D be a period plane.
(a) If d(P) = 0, then Gamma P is dense in D.
(b) If P cap (Lambda tensor Q) = Q v with v primitive integral
(so (v, v) > 0), then Gamma_v P is dense in D_v, where
Gamma_v = { gamma in Gamma : gamma v = v } and
D_v = { Q in D : v in Q }.
Proof of (a). Fix an ordered orthonormal positive frame
e = (e_1, e_2) of P; write P_+ for the plane it spans. The
pointwise stabilizer of e in G is H = SO_0(W), W = P_+^perp of
signature (1, 19). The frame space
F = G / H = ordered orthonormal positive frames of V
has dimension 41 and fibers over D with fiber SO(2). "Gamma P
dense in D" is equivalent to "Gamma e dense in F".
Apply Ratner to H e Gamma in G / Gamma. Since H = SO_0(1, 19)
is simple and noncompact, it is generated by one-parameter
unipotents, so there is a connected closed H subset L subset G
with L cap Gamma a lattice in L and
closure( H e Gamma ) = L e Gamma.
If L = G, then Gamma e is dense in F, done. Rule out proper L.
The Lie algebra g = so(V) is an h = so(W)-module by orthocomplement
V = P_+ + W:
g = h + so(P_+) + (W tensor P_+). (*)
The standard real so(1, 19)-representation W is absolutely
irreducible, and so(P_+) = so(2) is trivial. Any intermediate l
with h subset l subset g is h-invariant and its image l / h in
g / h = so(P_+) + (W tensor P_+) is one of
l / h in { 0, so(P_+), W tensor A (A subset P_+ 1-dim),
W tensor P_+ }.
Working out brackets with (*) gives four proper-or-full Lie
algebras:
l = h (l/h = 0)
l = h + so(P_+) = so(W) + so(P_+) (l/h = so(P_+))
l = so(W + A) = so(2, 19) (l/h = W tensor A, A 1-dim)
l = g (l/h = W tensor P_+).
(If l contains W tensor e_1 and W tensor e_2 with e_1, e_2 spanning
P_+, then [e_1 tensor W, e_2 tensor W] contains so(P_+), and
together with h and mixed terms this generates all of g.)
So the proper intermediate groups are
L_1 = SO_0(W) (l = h)
L_2 = SO_0(W) x SO(P_+) (l = h + so(P_+))
L_3 = SO_0(W + A) = SO_0(2, 19) (l = so(W + A), A in P_+
1-dim).
Rule out each.
L_1 = SO_0(W). Fixed space of L_1 on V is P_+. Z = Zar(L_1 cap
Gamma) in GL(V) is Q-defined and, by Borel density (Thm 27.1 on
L_1 semisimple without compact factor), Z supseteq L_1, so
Z = L_1. Hence Fix(Z) = P_+ is Q-rational, so P_+ is spanned by
rational vectors: d(P) = 2. Contradicts d(P) = 0.
L_3 = SO_0(W + A). Same: Fix(L_3) = A, a Q-rational 1-dim
subspace of P_+. Contradicts d(P) = 0.
L_2 = SO_0(W) x SO(P_+). Fix(L_2) = 0, so the direct argument
gives nothing. Use the projection
pi_1 : L_2 -> SO_0(W), kernel = SO(P_+) compact.
Since pi_1 has compact kernel it is proper, so pi_1(L_2 cap Gamma)
is a closed discrete subset of SO_0(W). Since L_2 cap Gamma is a
lattice in L_2 (Ratner) with finite invariant measure on the
quotient, pushing that measure by pi_1 gives a finite invariant
measure on SO_0(W) / pi_1(L_2 cap Gamma), so pi_1(L_2 cap Gamma)
is a lattice in SO_0(W). By Borel density on SO_0(W), its
Zariski closure is all of SO_0(W). Therefore the Lie algebra z of
Z = Zar(L_2 cap Gamma) surjects onto h = so(W). Its derived
algebra [z, z] is Q-defined, contained in [g_L2, g_L2] = h (the
so(P_+) factor is abelian), and surjects onto [h, h] = h; hence
[z, z] = h. The fixed space of [z, z] is P_+, so P_+ is Q-rational,
contradicting d(P) = 0.
No proper L survives; L = G; so Gamma P is dense in D. (a) proved.
Proof of (b). Set V' = v^perp in V (signature (2, 19)), G' =
SO_0(V') = SO_0(2, 19), Gamma_v = Gamma cap G' (an arithmetic
lattice in G'). A period plane P in D_v has the form P = R v +
R u with u a positive unit vector in V'. The pointwise stabilizer
of (v/|v|, u) in G' is H' = SO_0(u^perp cap V') = SO_0(1, 19).
The quotient G' / H' is the positive unit sphere in V'
(orientations are points), exactly D_v. Work in the frame space
F' = G' / H'.
Apply Ratner to H' e' Gamma_v in G' / Gamma_v: there is a
connected closed H' subset M subset G' with M cap Gamma_v a
lattice in M and closure( H' e' Gamma_v ) = M e' Gamma_v.
The subalgebra lattice is simpler. Via V' = R u + (u^perp cap V'),
g' = h' + (R u) tensor (u^perp cap V'),
and u^perp cap V' is an irreducible h'-module of real dimension
19. The only h'-invariant submodules of g' / h' are 0 and g' / h'.
So M = H' or M = G'.
If M = G': closure( H' e' Gamma_v ) = G' / Gamma_v, so Gamma_v e'
is dense in F'; so Gamma_v P is dense in D_v. Done.
If M = H': H' cap Gamma_v is a lattice in H'. By Borel density
(Thm 27.1 on H' = SO_0(1, 19) simple, noncompact, no compact
factor) Zar(H' cap Gamma_v) = H' in GL(V); since H' cap Gamma_v
consists of rational matrices, H' is Q-defined. Fix(H') = P
(it acts trivially on Rv and Ru, standardly on u^perp cap V'); so
P is Q-rational, d(P) = 2, contradicting d(P) = 1. So M = G' and
(b) is proved.
COROLLARY 28.2 (three cases). Let X be a complex K3 surface with
a marking phi : H^2(X, Z) -> Lambda and period plane P in D.
(i) If d(P) = 2, X is projective and has L everywhere.
(ii) If d(P) = 1, X has L everywhere.
(iii) If d(P) = 0, X has L everywhere.
Proof. (i) d(P) = 2 means P is spanned by a rational 2-plane.
Its orthocomplement P^perp cap V has signature (1, 19), so the
positive cone is nonempty and open. It contains a rational point;
multiplying by a positive integer gives a positive integral
(1, 1)-class on X. By the K3 projectivity criterion (Huybrechts,
Ch 7), X is projective. Theorem 18.2 gives L everywhere on X.
(ii) d(P) = 1; write P cap (Lambda tensor Q) = Q v with v
primitive. Because P is positive, (v, v) > 0. Theorem 25.4 gives
a nonempty relatively open subset Omega_v of D_v
such that every K3 surface with a marked period in Omega_v has L
everywhere. By Lemma 28.1(b), Gamma_v P is dense in D_v, so
gamma P in Omega_v for some gamma in Gamma_v. Re-marking
phi' = phi o gamma^{-1} does not change X. X therefore has L
everywhere.
(iii) d(P) = 0. Corollary 22.3 gives a nonempty open
subset Omega of D such that every K3 surface with a marked period
in Omega has L everywhere. By Lemma 28.1(a), Gamma P is dense in
D, so gamma P in Omega for some gamma in Gamma. Re-marking
phi' = phi o gamma^{-1} does not change X. So X has L everywhere.
THEOREM 28.3 (every complex K3 surface is Oka). Every complex
K3 surface X is Oka.
Proof. Fix any marking phi of X and its period plane P. By
Corollary 28.2 X has L everywhere. Theorem 9.1 (the analytic
core) then gives X the convex approximation property
in every source dimension. By Forstneric's theorem (CAP iff Oka)
X is Oka. QED
Remark 28.4 (compact target). All estimates in the analytic core
are compact-uniform on the target. X is compact, so any two
compatible Riemannian metrics on X are uniformly equivalent, and
the statement holds for the metric specified in the CAP
definition; no metric-dependent constant enters.
The main theorem stands proved: every complex K3 surface is Oka.
It remains to record the standard corollaries.
The K3 Oka theorem has immediate consequences in two directions:
analytic (interpolation and dense entire curves) and geometric
(surface classification and Gromov-elliptic properties).
The Oka property in its approximation-plus-interpolation form is
the original motivation for CAP. Three standard consequences
follow for a K3 surface.
COROLLARY 29.1 (Oka-1: interpolation with jets). Let S be a
complex K3 surface, R an open Riemann surface, K subset R a
compact Runge set, and A = { a_j : j >= 1 } subset R a closed
discrete set of distinct points. Let k_j >= 1 be integers. For
every continuous map h : R -> S that is holomorphic near K u A
and every epsilon > 0 there is a holomorphic f : R -> S, homotopic
to h, satisfying
sup_{z in K} d_S( f(z), h(z) ) < epsilon,
j^k_j_{a_j} f = j^k_j_{a_j} h (j >= 1).
j^k_a denotes the k-jet at a. If every prescribed first
derivative is nonzero, f can be chosen an immersion.
Proof. By Theorem 28.3, S is Oka. The Oka property implies
approximation with jet interpolation on closed discrete subsets
of Stein sources (Forstneric, Ann. Inst. Fourier 55 (2005),
Cor. 1.3). An open Riemann surface is Stein. Since h is
continuous this interpolation problem lies in the homotopy class
of h. For the immersion claim with nonzero first jets, use
Alarcon-Forstneric, Math. Z. 311 (2025), Cor. 2.10: the
interpolating map can be made an immersion. QED
COROLLARY 29.2 (dense entire curves with prescribed jet). Let S
be a complex K3 surface. For every x in S and every nonzero
tangent vector v in T_x S there is a holomorphic immersion
f : C -> S satisfying
f(0) = x, f'(0) = v, closure( f(C) ) = S.
If S is projective, then f(C) is Zariski dense.
Proof. Take a countable dense sequence (x_j)_{j>=0} in S with
x_0 = x, and choose nonzero vectors v_j in T_{x_j} S with v_0 = v.
Fix source points a_j = 3j, so closed discs of radius 1 about
distinct a_j are disjoint.
For each j choose a coordinate chart kappa_j about x_j with
kappa_j(x_j) = 0. Let g_j(z) = kappa_j^{-1}( (z - a_j) w_j ),
w_j = d(kappa_j)(v_j). Then g_j is holomorphic near a_j with
g_j(a_j) = x_j and g_j'(a_j) = v_j. Shrinking the chart domain,
g_j is defined on the closed disc |z - a_j| <= 1. Choose b in S
and let h : C -> S be
h(z) = g_j(z) if |z - a_j| <= 1/2,
h(z) = path from x_j to b if 1/2 <= |z - a_j| <= 1,
h(z) = b otherwise.
Then h is continuous on C and holomorphic on each |z - a_j| < 1/2
and on |z| < 1/3. Let K = { |z| <= 1/3 } u { a_j : j >= 0 }. K
is compact and Runge (its complement in the Riemann sphere is
connected) and h is holomorphic near K. Apply Corollary 29.1
with K and A = {a_j}, k_j = 1, prescribing the first jet v_j at
every a_j. Since every v_j is nonzero, the interpolating f is an
immersion. The case j = 0 gives f(0) = x and f'(0) = v; the image
f(C) contains the dense sequence (x_j), so closure( f(C) ) = S.
When S is projective, Zariski closed sets are ordinary closed, so
topological density implies Zariski density.
COROLLARY 29.3 (strong dominability). Let S be a complex K3
surface and x in S. There is a holomorphic F : C^2 -> S with
F(0) = x and dF_0 : C^2 -> T_x S an isomorphism.
Proof. Choose a chart kappa : U subset S -> C^2 about x with
kappa(x) = 0, and let alpha : C^2 -> U be the inverse germ,
alpha(0) = x, d(alpha)_0 invertible. Extend alpha continuously
to all of C^2 with a radial cutoff inside a smaller ball, keeping
alpha unchanged near 0. Apply CAP with K = { 0 } and prescribed
1-jet to get F : C^2 -> S holomorphic with F(0) = x and
dF_0 = d(alpha)_0 (invertible). QED
The Oka property propagates to compact surfaces finitely covered
by K3 surfaces, and to surfaces of Kodaira dimension zero. A
very mild extra input (a known classification) extends the picture
to class VII.
COROLLARY 30.1 (compact Oka surfaces). Every complex Enriques
surface is Oka. Every connected minimal compact complex surface
of Kodaira dimension zero is Oka. Among minimal class-VII
surfaces exactly the Hopf and Enoki surfaces are Oka.
Proof. Enriques surfaces. The universal cover of an Enriques
surface Y is a K3 surface S' with Y = S' / iota for a free
involution iota. Since S' is Oka (Theorem 28.3), Y is Oka: the
projection S' -> Y is a finite unramified covering, and the Oka
property descends through such coverings (Forstneric, Ch. 5).
Kodaira dimension zero. The Enriques-Kodaira classification of
minimal compact complex surfaces of Kodaira dimension 0 lists:
complex tori and bielliptic surfaces; primary and secondary
Kodaira surfaces; K3 surfaces; and Enriques surfaces. Tori,
bielliptic surfaces, and Kodaira surfaces are Oka
(Forstneric-Larusson, IMRN 2014); K3 surfaces are Oka by Theorem
28.3; Enriques surfaces were just proved.
Class-VII minimal surfaces. Such a surface Y has b_1 = 1 and
Kodaira dimension -infinity. If b_2(Y) > 0, the global spherical
shell theorem makes Y a Kato surface.
The class-VII classification (recalled in Forstneric-Larusson,
IMRN 2014, Section 1) then gives: for b_2 = 0, Y is Hopf or Inoue;
for b_2 > 0, Y is a blowup of an Enoki, Inoue-Hirzebruch, or
intermediate surface at a finite set of points. Of these families,
exactly the Hopf and Enoki surfaces are Oka (Theorem 4 of
Forstneric-Larusson, IMRN 2014). QED
COROLLARY 30.2 (global dominating sprays). A compact complex
surface Y is elliptic in Gromov's sense if it carries a holomorphic
vector bundle E -> Y and a holomorphic map s : E -> Y with
s(0_y) = y and d(s|_{E_y})_{0_y} : E_y -> T_y Y surjective for
every y. Every projective complex K3 surface and every complex
Enriques surface is elliptic in this sense.
Proof. By the theorem of Forstneric-Larusson ("Every projective
Oka manifold is elliptic", J. reine angew. Math.), every
projective Oka manifold is Gromov-elliptic. Projective K3
surfaces are Oka by Theorem 28.3 and projective Enriques surfaces
are Oka by Corollary 30.1. QED
Remark 30.3 (Kobayashi pseudometric). Kamenova-Lu-Verbitsky,
with Verbitsky's erratum, proved that the Kobayashi pseudodistance
on every complex K3 surface vanishes identically. This is
strictly weaker than being Oka: the Oka property gives
approximation, interpolation, and dense entire curves, none of
which follow from vanishing of the Kobayashi pseudodistance alone.
Remark 30.4 (boundary of the theorem). The Oka property fails
for every minimal compact complex surface of general type, and
for every properly elliptic minimal compact complex surface.
Combining Theorem 28.3 with the Forstneric-Larusson classification
(IMRN 2014, Theorem 4 and following) gives the full classification
of Oka minimal compact complex surfaces stated above.
The Oka problem for K3 surfaces, and hence for every minimal
compact complex surface of Kodaira dimension zero, is settled.