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SYMPLECTIC BALL PACKINGS IN HIGHER DIMENSIONS
a complete proof
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1. THE ANSWER
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Fix an integer n >= 3 (the "complex dimension"; the real dimension
is 2n). Fix a number R > 0 (the target capacity) and numbers
R_1, ..., R_k > 0 (the capacities of the balls we want to pack).
THEOREM (Siegel-Yao, Conjecture A, now proved).
The k closed balls of capacities R_1,...,R_k can be placed
disjointly inside the open ball of capacity R by one
symplectic map -- each ball allowed a small buffer of room --
if and only if BOTH of these hold:
R_1^n + R_2^n + ... + R_k^n < R^n (volume)
R_i + R_j < R for every pair i != j (two-ball)
There are no other conditions, for any n >= 3, any k >= 1,
and any positive capacities.
The two inequalities are obstructions, so they are necessary
(Section 3). The whole rest of the file proves they are also
sufficient (Sections 4 through 12). The strict "<" reflects a
convention: the closed balls must land inside the OPEN interior
of the target, so no volume may touch the boundary. Each embedding
is defined on a small neighborhood of its closed source ball;
"buffer of room" above refers to exactly this.
Real dimension four (n = 2) is genuinely different and is not
covered; Section 13 explains why. Complex dimension one (n = 1)
is just ordinary area and is boring.
2. THE BASIC OBJECTS
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2.1 Complex coordinates.
A point of C^n is z = (z_1,...,z_n) with z_j = x_j + i y_j.
As a real vector space C^n is R^{2n}. Write |z_j|^2 = x_j^2+y_j^2.
2.2 The standard ball of capacity R.
B^{2n}(R) = { z in C^n : pi (|z_1|^2+...+|z_n|^2) <= R }.
This is a round Euclidean ball of radius sqrt(R/pi). Its
volume is R^n / n!. We measure balls by capacity R, never by
radius, because symplectic geometry sees area, and the
"symplectic area" of a complex line through the origin is
exactly R.
2.3 The symplectic form.
omega_0 = dx_1 ^ dy_1 + dx_2 ^ dy_2 + ... + dx_n ^ dy_n.
Each term dx_j ^ dy_j is the signed-area form of the j-th
complex coordinate plane. So omega_0 records signed area on
each coordinate plane separately and adds them up.
2.4 Symplectic embedding.
A smooth one-to-one map f from an open set U into an open set
U', with f^* omega_0 = omega_0 (the pulled-back area form
equals the original). This is much stricter than preserving
the single top-degree volume form omega_0^n / n!, because it
pins every 2-plane area, not only the total 2n-volume. The
whole subject is the gap between "symplectic" and "volume".
2.5 Moment and angle coordinates (used constantly).
For z_j = r_j e^{2 pi i theta_j}, set
p_j = pi |z_j|^2 = pi r_j^2, theta_j = (arg z_j)/(2 pi).
Then omega_0 = sum_j d p_j ^ d theta_j. We call p_j the
moments. They satisfy p_j >= 0, and the ball B^{2n}(R) is
exactly { sum_j p_j <= R }. The phases theta_j live on a
circle (angle mod 1). Every computation below is easy in
(p, theta) coordinates and smooth across the axes z_j = 0.
2.6 A "downward" polytope and its toric domain.
Fix m >= 1. Let
Delta = { p in R^m : p_j >= 0, b_v . p <= c_v, v=1..N }
with every b_v >= 0 (nonnegative coefficients) and c_v > 0.
Because coefficients are nonnegative, lowering any single p_j
keeps a point inside Delta; we call this "downward". The
faces b_v . p = c_v are the OUTER faces; the coordinate faces
p_j = 0 are NOT outer faces. The toric domain is
V = { z in C^m : p(z) in Delta },
and (p,theta) coordinates turn V into Delta x (circle)^m with
the standard form sum_j d p_j ^ d theta_j.
2.7 Capacity bookkeeping.
Symplectic forms are additive over disjoint pieces, so
"volume of B^{2n}(R)" is R^n/n! and the volume of the shell
{ a < sum p_j < 1 } is (1 - a^n)/n!. Dilating a source and
the target by the same positive factor multiplies every
capacity by that factor and cancels inside the pullback; hence
throughout the sufficiency proof we may and do set R = 1.
3. WHY THE TWO INEQUALITIES ARE NECESSARY
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We prove that if a packing exists, both inequalities must hold.
This direction is short.
3.1 Volume.
A symplectic map preserves omega_0, hence preserves the
volume form omega_0^n / n!, hence preserves volume. The
closed images of the k balls are disjoint and sit inside the
open target, whose volume is R^n / n!. The total volume of
the k balls is (R_1^n+...+R_k^n)/n!. Therefore
R_1^n + ... + R_k^n < R^n.
(Strict, because the images are compact inside the open ball,
so a little of the target's volume is never used.)
3.2 The two-ball obstruction (Gromov 1985).
This is the surprising inequality R_i + R_j < R. It comes
from pseudo-holomorphic ("J-holomorphic") curves, a genuine
theorem of symplectic topology. We state exactly the input
we use; this is the one deep tool imported into the proof.
GROMOV'S TWO-BALL RIGIDITY. Fix n >= 1. Suppose the
disjoint union of two closed balls B^{2n}(a) and B^{2n}(b)
embeds symplectically into the interior of B^{2n}(R). Then
a + b <= R, and if the sources are closed and the
target open, a + b < R.
WHY THIS IS TRUE (idea only; the real proof needs the theory
of J-holomorphic spheres, which is beyond this file).
The ball B^{2n}(R) carries a compatible almost-complex
structure J (a rule for multiplying tangent vectors by i).
With J one can draw "J-holomorphic spheres": round 2-spheres
whose every tangent plane is J-complex. Gromov proved these
spheres are RIGID: their symplectic area is pinned by the
topology, and they cannot be pushed through obstacles. Now
take a J-holomorphic sphere through a point of the first
ball. Its area is at least the capacity a of that ball.
Since the two ball images are disjoint, this sphere cannot
enter the second ball. Tracking the largest such sphere, and
the smallest one the second ball forces, gives a + b <= R.
The point of the inequality is that it is NOT visible to
volume: two balls of capacities 0.6 and 0.6 have total
volume 2*0.6^n/n!, which for large n is far below 1/n!, yet
they cannot fit because 0.6+0.6 > 1.
We use Gromov's result as black box: it is standard, it is
quoted in the technical paper as [Gromov1985, Section 0.3.B]
in the capacity normalization of [SiegelYao2025, Thm 1.1],
and everything else in this file is proved from scratch.
3.3 All pairs.
Suppose we want to pack balls R_1,...,R_k. If the whole
packing fits, then for any two indices i, j the pair
(R_i, R_j) alone also fits (ignore the other balls; the
remaining images are still disjoint from these two). Apply
3.2 to that pair:
R_i + R_j < R for every i != j.
That gives all the pairwise inequalities at once.
Together, 3.1 and 3.3 prove the necessity half of the theorem:
volume and all pairs are forced. The hard direction -- that
these two conditions are also enough -- is the rest of the file.
5. HAMILTONIAN MAPS AND SORTING
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Part A changes fiber coordinates with a Hamiltonian map. Here
is exactly what that means, and the one-dimensional sorting
tool it relies on.
5.1 Hamiltonian vector fields and flows.
On C^m with form omega = sum_j d p_j ^ d theta_j, every
smooth real function H (the "Hamiltonian") defines a vector
field X_H by
iota_{X_H} omega = -d H.
(iota means "plug X_H into the first slot".) Its flow, run
for time t, is the map that slides every point along X_H.
The fundamental fact, which we use repeatedly:
the time-t flow of ANY H is a symplectic map.
Proof (Moser's computation, one line): along the flow,
d/dt (flow_t^* omega) = flow_t^* (d iota_{X_H} omega) =
flow_t^* (-d d H) = 0. So the pulled-back form never changes
from omega. A flow of a time-dependent family H_t is
likewise symplectic.
5.2 Toric Hamiltonians only rotate the phases.
If H = H(p) depends on the moments alone ("toric"), then
working in (p,theta) coordinates one checks
X_H = sum_j (dH/dp_j) d/dtheta_j .
So along the flow every moment p_j is CONSTANT, and only the
angles move: theta_j -> theta_j + t (dH/dp_j). A toric
Hamiltonian therefore stays inside each fiber
{p = const} (each "torus") and just turns it. This is why
a Hamiltonian can "re-sort" the inside of each fiber without
changing which points exist or where the fiber's boundary is.
The functions dH/dp_j may jump at corners of Delta; the flow
is still well defined because we always work on a slightly
smaller polytope with room to spare.
5.3 The increasing rearrangement on an interval.
Let f : [0,1] -> R be continuous and convex. For h in
[0,1] define
g(h) = min over 0 <= a <= 1-h of max{ f(a), f(a+h) }.
Read it as: among all windows [a,a+h] of width h, take the
one whose larger endpoint value is smallest; that smallest
"worst value in a window of width h" is g(h). This is the
INCREASING REARRANGEMENT of f: it re-sorts the heights of f
into nondecreasing order while keeping the same spread of
values. For a general convex F of several variables we
apply this in one coordinate at a time with the others
frozen (Section 6.2).
LEMMA 5.4 (properties of g).
For continuous convex f : [0,1] -> R, the function g is
(i) continuous, with g(0)=min f and g(1)=max f;
(ii) convex;
(iii) nondecreasing;
(iv) equal-integral: int_0^1 g = int_0^1 f;
(v) order/translation preserving: f <= h pointwise =>
g_f <= g_h; and g_{f+c} = g_f + c for constants c;
(vi) nonexpansive: |g_f - g_h| <= max|f - h| everywhere.
Proof.
(i) For fixed h, max{f(a),f(a+h)} is continuous in a on a
compact window, so the minimum g(h) is continuous in h
(the window shrinks continuously, and f is uniformly
continuous on [0,1]). At h=0 every window collapses to
a point, giving g(0)=min_a f(a). At h=1 the only window
is [0,1], giving g(1)=max{f(0),f(1)}. For a convex f
the maximum on [0,1] is at an endpoint (convexity), so
g(1)=max f. Done.
(ii) Take any two widths h0,h1 and a number 0<=t<=1. Let
[a0,a0+h0] and [a1,a1+h1] be the minimizing windows
giving g(h0) and g(h1). Their convex combination
[ (1-t)a0+t a1 , (1-t)(a0+h0)+t(a1+h1) ]
is a window of width (1-t)h0 + t h1. Its two endpoints
are convex combinations of the endpoints of the two
windows. By convexity of f,
f(A) <= (1-t) f(a0) + t f(a1),
f(B) <= (1-t) f(a0+h0) + t f(a1+h1) .
Taking the larger of the two endpoints and using
max{(1-t)x0+t x1, (1-t)y0+t y1}
<= (1-t)max{x0,y0} + t max{x1,y1},
we get
g((1-t)h0 + t h1) <= (1-t) g(h0) + t g(h1),
because g of the combined width is the minimum over
windows, hence at most the value on this particular one.
That is convexity.
(iii) A convex f on [0,1] is "cup shaped": the set
{f <= c} is a single closed interval (possibly empty).
An interval-window of width h catches a value <= c for
the most central possible window, and a wider window
only makes the smallest achievable "worst value" larger
or equal. Equivalently, g(h) is exactly "the value c
such that the down-set {f<=c} has length h" and that
value rises with h. So g is nondecreasing.
(iv) On [0,1], sort the values of f. For each level c, the
length of {f < c} equals the length of {g < c} (that is
what re-sorting preserves); integrating these lengths
over c (the "layer cake" identity
int f = int_0^infty length{f>c} dc) gives equal
integrals.
(v) f <= h makes every window's worst value no larger, so
the minima satisfy g_f <= g_h. Adding a constant c to
f adds c to both endpoint values, hence to their max,
hence to the min: g_{f+c} = g_f + c.
(vi) From (v): f - delta <= h <= f + delta with
delta = max|f-h| gives
g_f - delta <= g_h <= g_f + delta. QED.
6. PART A: THE HAMILTONIAN COMPARISON LEMMA
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This is the heart of the proof. It takes a statement about an
AVERAGE over a polytope and upgrades it to a statement that holds
at EVERY point, by sliding fibers with a Hamiltonian map.
SETTING. Let m >= 1 and let
Delta = { p in R^m : p_j >= 0, b_v . p <= c_v, v=1..N }
be a compact downward polytope (Section 2.6), full dimensional,
with all b_v >= 0 and c_v > 0. Let V = {z : p(z) in Delta} be
its toric domain with form omega_V = sum d p_j ^ d theta_j.
Functions on Delta we read as toric functions on V.
THE LEMMA 6.1 (Hamiltonian comparison).
Let P : Delta -> R be continuous and concave. Suppose
(H1) the average of P over Delta is positive:
(1/area) int_Delta P dp > 0; and
(H2) for some 0 < tau < 1,
P > 0 on Delta minus (tau Delta),
where tau Delta is Delta scaled toward the origin
by tau (its outer faces move to b_v . p = tau c_v).
Then there exist a smooth toric function K on Delta and a
Hamiltonian diffeomorphism q of V such that
(K - P)( q u ) < K(u) for every u in V. (6.2)
Moreover q comes from a Hamiltonian isotopy that preserves V
and, near each outer face b_v . p = c_v, is generated by
Hamiltonians whose Poisson bracket with b_v . p vanishes
uniformly in time.
Meaning of (6.2). "K-P composed with q is strictly below K"
says: after the fiber-slide q, the leftover function K-P is below
K with a uniform margin. That margin is exactly what Part B
spends to build disjoint layers. The outer-face condition says
the slide never pushes points across the slanted walls of Delta,
so the model keeps its shape.
6.2 Rearrangement in one coordinate.
For a continuous convex F on Delta and a coordinate j, let
R_j F be the result of applying Lemma 5.4 to the slice of F
at fixed other moments y=(p_i, i != j). The slice is the
segment 0 <= p_j <= L(y), where
L(y) = min over v with b_{v,j} > 0 of
( c_v - sum_{i!=j} b_{v,i} y_i ) / b_{v,j} .
(This is the furthest p_j can go before hitting an outer wall;
it exists because Delta is bounded.) By Lemma 5.4, each R_j
sends continuous convex functions to continuous convex
functions, preserves integrals, order, translation, is
nonexpansive, and makes the output nondecreasing in p_j while
keeping any monotonicity in the other coordinates. Applying
R_1,R_2,...,R_m in turn gives R = R_m o ... o R_1, whose
output is nondecreasing in EVERY coordinate.
6.3 The contraction fixed point.
For 0 < lambda < 1 consider the map on the (complete, uniform
norm) space of continuous convex functions on Delta:
F |-> R( lambda F - P ).
Since R is nonexpansive with constant 1 and lambda<1, this map
is a contraction with constant lambda, so it has a unique fixed
point K_lambda:
K_lambda = R( lambda K_lambda - P ). (6.4)
Integrals are preserved by R, and the integral of P is fixed,
so taking integrals in (6.4) gives
int K_lambda = lambda int K_lambda - int P,
i.e. int K_lambda = - (int P)/(1 - lambda). (6.5)
Since int P > 0 by (H1), the average of K_lambda tends to
minus infinity as lambda -> 1. Also K_lambda is coordinatewise
nondecreasing, so its minimum over Delta is K_lambda(0).
6.4 A uniform gap appears.
Pick a small number d with
0 < d < min{ mean(P), min over Delta minus tau Delta of P },
which is possible by (H1) and (H2).
CLAIM: for lambda sufficiently close to 1,
M_lambda := max_Delta K_lambda < - d/(1 - lambda). (6.6)
Proof of the claim. Suppose instead M_lambda >= -d/(1-lambda).
Two cases cover Delta.
OUTER BAND (p in Delta minus tau Delta). There P >= d, and
K_lambda <= M_lambda, so
lambda K_lambda - P <= lambda M_lambda - P
<= (M_lambda + d) - d
= M_lambda ,
where we used lambda M_lambda <= M_lambda + d, an immediate
rearrangement of M_lambda >= -d/(1-lambda). The inequality is
strict because P > d there.
INNER CORE (p = tau v in tau Delta, v in Delta). Convexity of
K_lambda gives
K_lambda(tau v) <= (1-tau) K_lambda(0) + tau M_lambda ,
and K_lambda(0) <= average(K_lambda) = -mean(P)/(1-lambda), so
K_lambda(p) <= tau M_lambda - (1-tau) mean(P)/(1-lambda).
Multiply by lambda, subtract P(p) >= P_min = min_Delta P:
lambda K_lambda(p) - P(p)
<= tau M_lambda - lambda(1-tau)mean(P)/(1-lambda) - P_min.
Subtract M_lambda and use -M_lambda <= d/(1-lambda) (again
from the supposition):
(tau-1) M_lambda - lambda(1-tau)mean(P)/(1-lambda) - P_min
<= (1-tau) [ d - lambda mean(P) ] / (1-lambda) - P_min .
Since d < mean(P) and lambda -> 1, the bracket d - lambda mean(P)
is eventually negative, say <= -c0 < 0, so the last line tends
to -infinity; hence for lambda close enough to 1,
lambda K_lambda - P < M_lambda
on the core as well. Both bands give max( lambda K_lambda - P )
< M_lambda. But K_lambda = R(lambda K_lambda - P) and
max(RF) <= max F (R never raises the maximum, by the window-min
formula), so max( lambda K_lambda - P ) >= max K_lambda =
M_lambda, a contradiction. Hence (6.6) holds.
Fix such a lambda. Multiply (6.6) by (1-lambda): we have
(1-lambda) K_lambda <= -d pointwise on Delta. (6.7)
Apply R to both sides of the identity (K_lambda-P) =
lambda K_lambda - P + (1-lambda) K_lambda:
R(K_lambda - P) <= R(lambda K_lambda - P) - d
= K_lambda - d, (6.8)
using translation invariance (the (1-lambda)K_lambda <= -d
part shifts the argument down by at least d) and (6.4).
So the abstract re-sorting R already produces a strict gap d.
6.5 Realizing R with actual Hamiltonian maps.
R is abstract. We realize it by genuine Hamiltonian maps,
one coordinate at a time, with arbitrarily small error.
This is the content of:
LEMMA 6.9 (lifted rearrangement). For every continuous
convex F on Delta, every coordinate j, and every eps>0,
there is a Hamiltonian diffeomorphism q_j of V with
F( q_j u ) <= (R_j F)(u) + eps for all u in V,
generated by an isotopy that preserves V and, near each
outer face, has Hamiltonians Poisson-commuting with
b_v . p (uniformly in time).
We prove Lemma 6.9 in Section 6.6 below; for now take it as a
black box with error eps.
Choose q_j for j=1,...,m with errors eps_1,...,eps_m whose
total is < d/2. Let q = q_1 o q_2 o ... o q_m (applied in
this order). Chaining the inequalities,
(K_lambda - P)(q u) <= R(K_lambda - P)(u) + sum eps_j
< K_lambda(u) - d + d/2
< K_lambda(u).
Finally approximate K_lambda uniformly on Delta by a smooth
function K, with error < d/4 (polynomial approximation works).
The last comparison changes by < d/4 + d/4, still leaving
(K - P)(q u) < K(u).
The isotopy preserves V and all outer-face moments on small
collars by construction (each q_j does, and we take common
smaller collars for the finitely many j). This proves
Lemma 6.1.
6.6 Proving the lifted rearrangement (Lemma 6.9).
The whole difficulty is one-dimensional, on a disk. Write
D(1) = { z in C : pi|z|^2 < 1 }.
STEP 1 (planar realization). LEMMA 6.10. Let f_y:[0,1]->R
be a jointly continuous family of convex functions (y in a
compact parameter set), with rearrangements g_y. For every
eps>0 there is a smooth family of compactly supported
Hamiltonian isotopies of D(1), with time-one maps psi_y, such
that
f_y( pi |psi_y(z)|^2 ) <= g_y( pi |z|^2 ) + eps.
Idea. On D(1) the quantity pi|z|^2 is the area of the
concentric disk through z; "sorting f" means moving points so
that larger f sits at larger radius. Concretely:
- choose a fine grid 0<h_1<...<h_l<1 and, for each h_i, a
nested sublevel interval I_i(y) = {v : f_y(v) < g_y(h_i)+
eps'}, each of length > h_i (that is what g_y means);
- inside I_i(y) place a smooth disk of area h_i, nested in i,
chosen smoothly in y (a partition of unity preserves the
strict nesting margins);
- by the "nested-disk" lemma below, a Hamiltonian isotopy of
D(1) carries the standard concentric disks of areas h_i
onto these chosen disks;
- then any point of area v lands below the next grid value,
giving f_y <= g_y + const*eps'.
The NESTED-DISK LEMMA (standard; see the technical paper,
Lemma cmp:planar-disks). Given smooth families of strictly
nested closed disks E_1(y)<<...<<E_l(y)<<D(A) of areas
a_1<...<a_l, there is a smooth family of compactly supported
Hamiltonian isotopies of D(A) carrying each centered disk of
area a_i onto E_i(y). The proof: first build ordinary
diffeomorphisms carrying the disks (shrink, move, expand -
always possible in 2D), then correct them to be area
preserving by Moser's trick using a primitive form whose
restriction to each boundary circle is exact; the correction
leaves the disks fixed, and on a disk a compactly supported
area-preserving isotopy is Hamiltonian. This is the one
place we lean on a named standard construction; everything
else is short.
STEP 2 (lift to the fiber). In the j-th factor of C^m, a
slice has length L(y). Choose a smooth l(y) <= L(y), very
close to L(y) and >= eta/2 on the region L(y)>=eta, via a
smooth "soft minimum" of the affine functions defining L,
shifted down a little. Apply Lemma 6.10 to
f_y(v) = F(y, l(y) v) and lift the generating Hamiltonian
B_t(y,.) on D(1) to the whole fiber by
H(y, z_j) = l(y) * B_t( y, z_j / sqrt(l(y)) ).
Because H does not involve the angles of the OTHER
coordinates, those moments are unchanged; only the j-th angle
moves. On slices of length <= l(y) the inequality
F(q u) <= R_j F + eps holds by Lemma 6.10; on the outer strip
l(y) < p_j <= L(y) we can take the identity map (there R_j F
already dominates F to within the modulus of continuity of F,
by the window formula), and the identity is smooth. Choose
eta, then the scale error l vs L, then the planar error eps,
in that order, to make the total < eps. The result is a
smoothed Q, and a Hamiltonian flow tangent to V.
STEP 3 (outer faces). If b_{v,j}=0 the j-th motion does not
change b_v . p at all (it commutes). If b_{v,j}>0, a point on
the face b_v.p=c_v has p_j = L(y) its maximum, while the
construction keeps p_j <= (1-delta) l(y) with l(y)>=eta/2, so
b_v.p stays strictly below c_v near the face; the motion is
supported away from the face and preserves it as a set. This
gives the uniform outer-face control promised in Lemma 6.1.
QED (Lemma 6.1 and Lemma 6.9).
WHAT WE HAVE. A single integral condition (mean positive) plus
a positivity-away-from-center condition (H2) yields a strict
pointwise gap (6.2), delivered by an honest Hamiltonian map that
respects the slanted walls of the polytope. Part B converts that
gap into disjoint layers.
7. PART B: PACKING OVER A SURFACE WITH A HANDLE
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Part A gave a strict pointwise gap in the fiber. Part B spends
that gap: an essential loop ("handle") in a base surface lets a
closed one-form with period 1 cut the model into disjoint slabs,
one ball per slab.
7.1 The model.
Let C^o be a closed connected oriented surface of genus >= 1
with at least one point removed, and let
Sigma_1 << Sigma_2 << ...
be an exhaustion by compact bordered subsurfaces. Suppose
that on a neighborhood of each Sigma_l x V (V the toric
domain of Section 2.6) there is a two-form
Omega_l = omega_V + d Gamma_l,
where Gamma_l is a toric horizontal one-form: it kills every
vector tangent to the fiber V, and its coefficients depend
smoothly on the base point and on p. Write d_C for the base
differential (p held fixed). Suppose
d_C Gamma_l = kappa_l(p) is a POSITIVE area form on
Sigma_l for every p, and
A_l(p) := int over Sigma_l of kappa_l(p)
converges to H(p) UNIFORMLY in p, H continuous > 0.
(The models for different l need not fit together.) The
identity that makes "positive curvature" mean "symplectic" is
(omega_V + d Gamma)^{m+1}
= (m+1) omega_V^m ^ d_C Gamma, (7.2)
for any toric horizontal Gamma on a surface; it holds because
the only other possible top-degree term would need two moment
one-forms squeezed into one slot. So d_C Gamma > 0 makes
Omega_l symplectic. (This also extends smoothly across the
fiber axes p_j = 0.)
7.3 THE SURFACE PACKING LEMMA 7.4.
Let r_1,...,r_k > 0, with each full simplex
{ p >= 0 : sum_j p_j <= r_i } lying strictly inside Delta and
disjoint from every outer face. Put
C_r(p) = sum_i ( r_i - sum_j p_j )_+ ,
P(p) = H(p) - C_r(p).
Suppose (i) P is concave; (ii) the average of P over Delta is
positive; (iii) P > 0 on Delta minus tau Delta for some
0 < tau < 1. Then for any r_i' < r_i there is some l for
which Int(Sigma_l x V) contains a disjoint packing of the
closed balls B^{2m+2}(r_i'), each embedding defined on a
neighborhood of its ball.
7.5 Step 1: pin the gap on a finite surface.
Lemma 6.1 gives a toric K and a Hamiltonian q with
(K-P)(q u) < K(u). Writing u = q^{-1} v,
K(q^{-1}v) - K(v) + H(p(v)) - C_r(p(v)) > 0
for every v in V. The minimum of the left side over the
compact V is some g > 0 (the uniform gap). Choose
0 < eps < (1/8) min( g, min over Delta of H ).
Take l so large that sup over p of |A_l(p) - H(p)| < eps
(uniform convergence) and Sigma_l contains a torus with a
disk removed (genus >= 1 makes a handle). From now on H is
the limiting area; the finite A_l only approximates it.
7.6 Step 2: move the base area onto a handle.
Inside the handle choose nested annuli A << A+ << Int(Sigma),
with positive coordinates (s,t) on A+, t in R/Z (t goes
around), and a transverse annulus B. There is a smooth
closed 1-form beta supported in B with
beta = f(t) dt on A+, f >= 0, int_0^1 f(t) dt = 1.
So beta has period exactly 1 around the essential loop: this
"handle form" is what will measure out the layers.
Choose a smooth nondecreasing S(s), 0 near the bottom of A
and 1 near the top, with S' supported inside A. Then
chi_A = S'(s) ds ^ beta
is a nonnegative 2-form, supported in A, of integral 1.
Next choose a positive area form kappa_sm(p) that agrees
with kappa on a thin boundary collar, is independent of p on
A+, and has tiny total area:
sup over p of int over Sigma of kappa_sm(p) < eps.
(Fix any reference area lambda, write kappa = a(y,p) lambda
with a bounded, blend to a small constant away from A+.)
Define
H_*(p) = int over Sigma of ( kappa(p) - kappa_sm(p) ),
kappa_1(p) = kappa_sm(p) + H_*(p) chi_A .
Then H_* > 0, |H_* - H| < 2 eps, kappa_1 > 0, and
kappa_1 - kappa has total integral zero and compact interior
support.
RELATIVE AREA PRIMITIVE LEMMA 7.5. Let eta_p be a smooth
family of 2-forms on a bordered surface, supported in a fixed
interior compact set, with int_Sigma eta_p = 0 for all p.
Then there is a smooth family of 1-forms alpha_p, supported
in a fixed interior set, with d_C alpha_p = eta_p.
Proof sketch. In one coordinate chart with eta = h dx ^ dy
and int h = 0, the explicit
alpha = A dy - rho(x) B(y) dx,
A(x,y)= int_{-inf}^x ( h(v,y) - rho(v) h_0(y) ) dv,
h_0(y)= int h(x,y) dx, B(y)= int_{-inf}^y h_0(v) dv,
gives d alpha = eta with compact support (rho a fixed bump of
integral 1). Patch finitely many charts with a partition of
unity and move the "common bump" part across overlaps; the
zero total integral kills the leftover common coefficient.
This is a fixed linear operator, so it is smooth in p.
Apply Lemma 7.5 to eta = kappa_1 - kappa, getting a toric
horizontal alpha with d_C alpha = kappa_1 - kappa. Set
Gamma_1 = Gamma_0 + alpha with Gamma_0 = Gamma_l, then
normalize on the annulus A. Choose a base one-form gamma on
A+ with d gamma = e lambda (e the constant value of kappa_sm
on A+); a primitive exists on a bipunctured annulus. The
form delta(p) = Gamma_1 - gamma - S(s) H_*(p) beta is
base-closed on A+; subtract its period c(p) times beta and an
exact correction d_C(zeta_A F), and one obtains Gamma_1
globally horizontal, toric, equal to Gamma_0 near the base
boundary, with
d_C Gamma_1 = kappa_1, Gamma_1 = gamma + S(s) H_*(p) beta on A.
Because (7.2) is linear in d_C Gamma and d_C Gamma_1 > 0, a
symplectic interpolation from Gamma_0 to Gamma_1 is immediate.
7.7 Step 3: build the layer path (the exact deformation).
Choose smooth positive toric h_i on Delta with
h_i(p) > ( r_i - sum_j p_j )_+ and 0 < sum_i h_i - C_r < eps
(use a smoothed clamp (x + sqrt(x^2+eta_i^2))/2 to round the
corner of (.)_+ upward). Put
J(v) = K(v) - H_*(p(v)),
R_*(v) = K(q^{-1}v) - J(v) - sum_i h_i(p(v)).
Then R_* >= g - 3 eps > 0: on each fiber there is a positive
"residue" left over after the k bumps h_i are accounted for.
Let q_s := q_{S(s)} be the comparison isotopy, the identity
near the bottom of A and q near the top. Define a suspension
map D(s,u)=(s,q_s u). For a Hamiltonian q_s generated by Q_s
the standard suspension identity is
D^* omega_V = omega_V + d( (Q_s o q_s) ds ). (7.6)
(Differentiate in s; both slots agree.) Now form, on A x V,
Omega_lambda = d gamma + D^*( omega_V + d(G_lambda beta) )
gluing to omega_V + d Gamma_1 outside A. By (7.2) and (7.6),
Omega_lambda is symplectic as soon as d_s G_lambda >= 0; at
the ends G_lambda matches Gamma_1, so it extends globally.
Refine to put the bumps in STRICTLY ORDERED, disjoint
s-intervals I_1,...,I_k and a residue interval I_*: choose
nondecreasing step functions rho_i, rho_* and set
G_f(s,v) = J(v) + sum_i rho_i(s) h_i(p(v)) + rho_*(s) R_*(v).
Its s-derivative is >= 0; its bottom value is J and its top
value is (H_*+J) o q^{-1} = K o q^{-1}. On the i-th interval
it is a toric expression
G_f = a_i(p(v)) + rho_i(s) h_i(p(v)), a_i = J + sum_{j<i} h_j .
This is a genuine exact path of symplectic forms from the
model Omega_0 to a final form Omega_f. (This suspension "lift"
is the same device Schlenk uses in symplectic folding.)
7.8 Step 4: undo the deformation with Moser.
Every variation primitive above is horizontal and is
supported away from the base boundary. Near each outer face
the comparison isotopy Poisson-commutes with the defining
moment mu_v = b_v . p (that was Lemma 6.1's boundary clause),
so all the functions used are invariant under the face's
Hamiltonian vector field. Thus
MOSER-WITH-OUTER-FACES LEMMA 7.7. Suppose Omega_t is a
smooth family of symplectic forms on a neighborhood of
Sigma x V with d_t Omega_t = d alpha_t, alpha_t horizontal
and vanishing in a fixed collar of d Sigma x V, and near
each outer face iota_{X_{mu_v}} Omega_t = -d mu_v. Then the
Moser isotopy preserves Sigma x V and its interior, is the
identity near the base boundary, and is defined on a
neighborhood of the compact region.
Proof. Solve iota_{Y_t} Omega_t = -alpha_t. Horizontality
gives alpha_t(X_{mu_v}) = 0, so d mu_v(Y_t)=0 near the face;
trajectories cannot cross any outer level. The Moser
computation d/dt(psi_t^* Omega_t) = psi_t^*( d alpha_t +
d iota_{Y_t} Omega_t ) = 0 finishes it.
Applying 7.7 yields a diffeomorphism psi, defined on a
neighborhood, with
psi^* Omega_f = Omega_0,
preserving Sigma x V and its interior. We will undo it with
psi^{-1} at the end.
7.9 Step 5: one closed ball per layer.
Work in (s,t,v) coordinates on the i-th interval. Where
f > 0 the coordinate T with dT = beta runs over (0,1). Write
d gamma = ell(s,T) ds ^ dT, L(s,T) = int_{s_i^-}^s ell d sigma,
with s_i^- the bottom of I_i; then ell>0 and dL ^ dT = d gamma.
Set
x = L + G_f - a_i = L + rho_i(s) h_i(p(v)).
At fixed (T,v) this is strictly increasing in s (ell>0), so it
is a coordinate. In (x,T,v) the form becomes
omega_V(v) + dx ^ dT + d a_i ^ dT .
Set v' = phi_{a_i}^T v, the time-T flow of the toric
Hamiltonian a_i (its angles move, its moments are preserved).
This turns the last display into the clean product
omega_V(v') + dx ^ dT .
The available region in the i-th layer is therefore
0 < T < 1, 0 < x < h_i(p(v')).
Now fit the ball B^{2m+2}(r_i') into the product. The planar
fact we need (a corollary of the nested-disk lemma 6.10): for
any target area A and any delta > 0, a neighborhood of the
closed planar disk of area A embeds symplectically into the
(x,T)-plane so that a point of radial area w lands at
0 < T < 1, 0 < x < w + delta. (7.8)
Choose 0 < delta < r_i - r_i' and use (7.8) for the LAST
complex coordinate of B^{2m+2}(r_i'), keeping the first m
coordinates as v'. A point of the closed ball then has
w + sum_j p_j(v') <= r_i' (it is in the ball), so
x < w + delta < r_i - sum_j p_j(v') < h_i(p(v')) (two
uses above). Hence the whole closed ball lands compactly
inside the available region, away from the outer faces, with
uniform positive slack; so the embedding extends to a tubular
neighborhood of the ball. Different layers use disjoint base
intervals I_i, hence disjoint images. Undoing psi (Step 4)
returns the packing to the original model Omega_0. This
proves Lemma 7.4.
WHAT WE HAVE. From "P has positive mean and is positive away
from center" plus a handle, we get an actual disjoint packing of
the requested balls in the abstract surface-times-fiber model,
with honest symplectic embeddings and neighborhoods. Part C
builds such models inside the real ball and carries the packing
home.
8. PART C: KAEHLER MODELS AND SYMPLECTIC TRANSFER
----------------------------------------------------------------
Part C is the bridge. It builds, inside projective space, the
"surface times toric fiber" models Part B needs, and it carries a
packing done in such a model back into the real ball. We present
five tools; Sections 9-11 assemble them.
We fix one convention. Write PP^N for complex projective space
and O(k) for its line bundles; a projective LINE measured by
O(1) has area 1. "U" always means U = c_1(O(1)) of a projective
bundle's tautological line, the "vertical size". All our models
live on complex projective varieties and carry Kähler forms whose
cohomology class is a positive rational combination of these
classes.
8.1 Tool 1: Moser's lemma, relative version.
The classical Moser trick: if a smooth path of Kähler forms
omega_t = omega_0 + d alpha_t stays Kähler, solve
iota_{X_t} omega_t = -alpha_t and integrate X_t to an isotopy
psi_t with psi_t^* omega_t = omega_0. So two Kähler forms in
the same cohomology class are isotopic.
We need to keep some complex submanifolds fixed as SETS.
RELATIVE KAEHLER MOSER LEMMA 8.1. Let X be compact
complex, omega_0, omega_1 cohomologous Kähler, and let S_1,...,
S_e be pairwise disjoint closed complex submanifolds. There
is an isotopy psi_t from the identity with
psi_1^* omega_1 = omega_0 and psi_t(S_j) = S_j .
If a compact Lie group acts holomorphically and the forms and
all S_j are invariant, the isotopy can be made equivariant,
and then preserves every connected piece of the fixed locus.
Proof. omega_t = (1-t)omega_0 + t omega_1 is Kähler for all t.
Pick alpha with d alpha = omega_1 - omega_0. Because S_j is
complex, T X restricted to S_j splits as T S_j plus a normal
bundle N_{j,t} = (T S_j)^{omega_t} (the omega_t-orthogonal
complement). Prescribe the first derivative of a corrector
f_t along S_j by
f_t = 0 and df_t = 0 on T S_j, df_t = -alpha on N_{j,t}.
This is a well-defined smooth family of covectors (the split
is a direct sum); realize it as a function in a fixed tube
around S_j and multiply by a cutoff equal to 1 near S_j.
Add up over j (the tubes are disjoint, compactness of the
parameter interval fixes the tubes). Set alpha_t = alpha+df_t.
Then iota_{X_t} omega_t = -alpha_t is tangent to each S_j,
because the right side kills N_{j,t}. Its flow therefore
preserves each S_j, and the standard computation
d/dt(psi_t^* omega_t) = psi_t^*( dd alpha_t + d iota_{X_t} omega_t )
= 0 gives psi_1^* omega_1 = omega_0. The equivariant claim:
average alpha over G; the normal splits are invariant; average
f_t; the field becomes invariant, hence equivariant, and its
flow fixes the fixed-point locus. Done.
8.2 Tool 2: transport inside a projective degeneration.
A flat family f : X -> B over a smooth curve B (with a marked
point 0) has fibers that are projective varieties, all but
finitely many equal to a fixed one, and near 0 possibly a
singular or reducible central fiber X_0 whose components we
understand.
PROJECTIVE TRANSPORT LEMMA 8.2. Let L be a relatively
ample rational polarization. Let Y be a smooth projective
component of X_0. After shrinking B around 0:
(1) sections on Y extend, so a relative Kähler form can be
chosen whose restriction to Y is the Fubini-Study form
of Y in L; if a torus acts, the basis can be chosen by
weights so the restriction is torus-invariant.
(2) any compact K in the smooth ("submersion") locus of f
transports, by a genuine symplectic map, into a nearby
smooth fiber X_t, avoiding any closed subset K avoids.
(3) if a compact group acts and the total form is invariant,
this transport is equivariant; for a Hamiltonian circle
action it preserves the moment map up to an additive
constant on each connected transport domain (zero if
one point's moment matches).
Proof idea. Relative Serre vanishing R^1 f_*( I_Y L^N ) = 0
for large divisible N means every section of L^N on Y lifts to
a section over a neighborhood of 0. Lifting a basis of Y and
adding sections that vanish on Y gives a relative projective
embedding; its pullback Fubini-Study form (divided by N) is the
desired relative Kähler form Omega, equal to the Y-form on Y.
To move points: at each point of the submersion locus, lift a
base tangent vector to be Omega-orthogonal to the fiber. That
lift exists uniquely (the fiber restriction is symplectic).
Flow along it a short base path from 0 to t. The contraction
of the horizontal lift with Omega is zero on fiber vectors, so
the fiber forms are preserved: the flow is the symplectic
transport. Compact K flows for a uniform time. Invariance of
Omega makes the horizontal lift equivariant, hence the flow;
the moment difference mu_t circ Phi_t - mu_0 is d-closed, hence
constant on a connected domain.
8.3 Tool 3: the normal-cone transfer.
This is the central geometric move: "zoom into a submanifold
Z, read off the projective normal bundle, do the packing there,
zoom back".
NORMAL-CONE TRANSFER PROPOSITION 8.3. Let T be smooth
projective, Z in T a smooth closed submanifold of positive
codimension (possibly disconnected), L ample rational. Set
T' = Bl_Z T, Y = PP_Z( 1 + N_{Z/T} ),
D_inf = PP_Z( N_{Z/T} ) (the "infinity section" of Y).
Here N_{Z/T} is the normal bundle of Z in T; "1 + " adds a
trivial line. Suppose for some rational d > 0 the two classes
pi^* L - d E_Z on T' (E_Z = exceptional divisor),
L|_Z + d U on Y
are ample. Let F in T be a finite union of disjoint smooth
complex submanifolds (a set to dodge), with a suitable
disjointness of its preimage. Let omega_T be any Kähler form
in class L and omega_Y any Kähler form in class L|_Z + dU
that is invariant under rotating the normal line.
Then every compact subset of
Y minus ( D_inf union pi^{-1}(F cap Z) )
has a NEIGHBORHOOD admitting a symplectic embedding into
( T minus F , omega_T )
carrying the given form omega_Y. In particular a compact
packing done in Y transfers, with neighborhoods, into T.
Proof. Form the deformation to the normal cone,
X = Bl_{Z x {0}} ( T x A^1 ) -> A^1 .
Its fiber away from 0 is T. Its central fiber (fiber at 0)
is the reduced union of T' and Y, glued along D_inf. (This
is classical intersection theory: in local coordinates the
blow-up of the ideal (x_1,...,x_r,t) has a t-chart where
x_i = t v_i and Y is the exceptional at t=0, and x_1-chart
giving T'; one checks smoothness and flatness chart by chart.)
Put the rational polarization L - d[Y] on all of X. On the
two components it restricts to the two ample classes in the
statement. Ampleness on both components gives ampleness on
the (reduced) central fiber, because the natural map from the
disjoint union of components to the central fiber is finite
and surjective; ampleness is then open in t, so the
polarization is ample on X over a neighborhood of 0.
The nonzero fibers are T with class L, as desired.
Choose (Lemma 8.2) a relative Kähler form whose Y-restriction
is invariant; adjust omega_Y to it by the relative Moser
Lemma 8.1 (which preserves D_inf and the vertical forbidden
pieces). Now a compact K in Y off D_inf and off pi^{-1}(F)
sits in the submersion locus and its image in X is disjoint
from the inverse image of F; Lemma 8.2 transports a
neighborhood into a nearby T minus F with a Kähler form in L.
Finally Lemma 8.1 on T (preserving F) matches that form to
omega_T. All maps are defined on neighborhoods; shrink at
each of the finitely many compositions. Done.
INTUITION. A projective T with a submanifold Z has a "zoom
limit" where it looks like the projective bundle Y over Z. If Y is
a nicer place to pack (a surface times toric fiber, thanks to Part
B), we pack there, then run the zoom backwards into T. The
forbidden set F (often a hyperplane, or curves) is what survives
as "excluded" after the zoom.
The remaining two tools (8.4, 8.5) give the explicit forms and
identify the shell; they occupy the next sections of Part C.
8.4 Tool 4: the explicit normal-bundle form.
This is where the abstract "surface times toric fiber" model
becomes concrete numbers.
BUNDLE-FORM LEMMA 8.4. Let A be an ample line bundle on a
smooth projective Z with a Hermitian metric whose Chern form
is sigma > 0 (a positive 2-form). Suppose the normal bundle is
N = A^b (+) A^b (+) ... (+) A^b (l summands),
b a positive integer. Let d > 0 with b d < 1. Then the
class sigma + d U on Y = PP_Z(1 + N) has an invariant
Kähler representative of the affine shape
omega_Y = sigma + d( sum_{j=1}^l p_j alpha_j ),
p_j >= 0, sum_j p_j < d, (8.5)
where alpha_j is the period-1 angular connection of the j-th
normal line (curvature -b sigma) and p_j are the standard
fiber moments. The term p_j alpha_j is smooth across the
axis z_j = 0. If Z is a CURVE and Sigma in Z is a compact
bordered subsurface, the truncation of this model to any
compact moment polytope strictly inside sum p_j < d admits
a toric horizontal primitive (as in Part B), with base
curvature
kappa(p) = ( 1 - b sum_j p_j ) sigma . (8.6)
For bordered exhaustions of C minus finitely many points the
integrated base areas converge uniformly (on that polytope) to
H(p) = (deg A) ( 1 - b sum_j p_j ) . (8.7)
Proof. In the affine chart N in PP(1+N), with normal
coordinates w_1,...,w_l and their Hermitian norms ||.||, set
omega_Y = sigma + dd^c[ d log( 1 + sum_j ||w_j||^2 ) ].
(dd^c = i d d-bar up to a constant; it is a real (1,1) form.)
The bracket is d times the local weight of the dual
tautological metric, so its curvature represents dU globally.
The normal-circle moments come out
p_j = d ||w_j||^2 / ( 1 + sum_l ||w_l||^2 ),
and using radii sqrt(p_j/pi) identifies each affine fiber with
the open ball sum_j p_j < d. Differentiating the potential
and writing the metric connection gives (8.5); in a unitary
frame alpha_j = d theta_j + A_j and p_j d theta_j is the
standard smooth angular part, so all is smooth at the axes.
To see positivity, at any base point pick a holomorphic frame
whose metric has zero first derivative; the form then splits
into a positive VERTICAL Fubini-Study piece plus the
HORIZONTAL piece (1 - b sum_j p_j) sigma. This still holds
on the projective infinity chart, where sum_j p_j = d, and
there the horizontal factor is at least 1 - b d > 0. So
omega_Y is Kähler. For rational d a divisible multiple is the
curvature of a positive Hermitian line bundle, hence ample by
Kodaira's theorem.
For Z a curve: every complex line bundle on a bordered
surface is smoothly trivial (the obstruction lives in
H^2(Sigma) = 0), so pick unitary frames on a slightly larger
bordered neighborhood, giving alpha_j = d theta_j + A_j, and a
base primitive gamma of sigma. Then
omega_Y = sum_j d p_j ^ d theta_j + d Gamma_0,
Gamma_0 = gamma + sum_j p_j A_j,
which is toric horizontal with base differential (8.6).
Integrating over Sigma, and using int_Sigma sigma -> int_C
sigma = deg A (uniform because the coefficient is continuous
and bounded on the fixed compact polytope), gives (8.7).
Done.
REMARK. Formula (8.7) is EXACTLY the "limiting area H(p)"
demanded by the surface packing Lemma 7.4: H is affine in
p = sum p_j, positive as long as sum p_j < 1/b, and its
"slope" -b controls how fast the base area shrinks as moments
grow. The negative caps (r_i - sum p_j)_+ are then subtracted
from this H to form P. This is the whole reason the proof
works: the normal-bundle model IS a surface-times-toric-fiber
model with an explicitly known H.
8.5 Tool 5: the target ball and its exterior shell.
Two identities turn projective data back into honest balls in
C^n.
8.5.1 The unit ball is PP^n minus a hyperplane.
In affine coordinates w in C^n the Fubini-Study moments and
phases are
p_j = |w_j|^2 / (1 + sum_l |w_l|^2), theta_j = arg(w_j)/2pi,
the form is sum_j d p_j ^ d theta_j, and the map
w |-> w / sqrt( pi(1 + sum_l |w_l|^2) ) (8.8)
is a symplectomorphism from PP^n minus the hyperplane
{z_0 = 0} onto the open unit ball { z : pi sum_j |z_j|^2 < 1 }.
(Check the form identity where all w_j are nonzero, then
extend smoothly across the axes.)
8.5.2 The shell is a blow-up minus two divisors.
SHELL LEMMA 8.5. Let n >= 2, o a point of PP^n, H_inf a
hyperplane not through o. Let T = Bl_o PP^n, E the
exceptional divisor, H = pi^* c_1(O_{PP^n}(1)). For 0 < a < 1
there is a Kähler form omega_a in class H - aE such that
( T minus (E union H_inf) , omega_a )
is symplectomorphic to
( { z in C^n : a < pi sum_j |z_j|^2 < 1 } , omega_0 ),
and the same holds for EVERY Kähler form in that class.
Also, for rational A > B > 0, the class AH - BE is ample.
Proof. Use the symplectic cut (Lerman). Put homogeneous
coordinates o = [1:0:...:0], H_inf = {z_0 = 0}. On
PP^n x C let the circle act by
([z_0:z'], v) |-> ([z_0 : e^{2 pi i t} z'], e^{-2 pi i t} v),
with Hamiltonian, for unit Fubini-Study on PP^n and the
standard form on C,
F = mu - pi|v|^2, mu = |z'|^2/(|z_0|^2 + |z'|^2).
The level F = a (with 0<a<1) is compact and regular and the
circle acts freely on it (if v != 0 it is free in v; if v = 0
then mu = a in (0,1) where the projective scalar action is
free). So the symplectic reduction is a smooth compact Kähler
manifold: the "cut" of PP^n at level a. Its complex model is
exactly T: the holomorphic map ([z_0:z'],v) -> ([z_0:vz'],[z'])
has values in the graph resolution of projection from o (which
is Bl_o PP^n) and its fibers are the free complex scalar
orbits; each such orbit meets the level circle in exactly one
point, so the quotient is the blow-up. On the open
mu > a part, choose v = sqrt((mu-a)/pi) > 0; the extra
complex-line form vanishes on that real slice, so the
Fubini-Study form is unchanged and (by 8.5.1) the retained open
region is precisely the shell of the statement. Reduction of
the level mu = a (v = 0) gives E = PP^{n-1} of line area a,
matching E; and the proper transform of H_inf is retained with
unit line area. These two restrictions fix the class H - aE.
Any other Kähler form in H - aE is related to omega_a by the
relative Moser Lemma 8.1, which preserves the disjoint
divisors E, H_inf and hence their complement.
Ampleness of AH - BE: the graph model is a closed subvariety
of PP^n x PP^{n-1}; the two hyperplane classes pull back to
H and to H - E (projection from o is by hyperplanes through
o). Restricting a positive rational product polarization is
ample, and
AH - BE = (A-B) H + B (H - E)
exhibits AH - BE as a sum of a positive multiple of H (ample)
and a positive multiple of H - E (ample). Done.
WHAT PART C SUPPLIES. Concrete models: blow up a point, look at
the projective normal bundle over a hyperplane or over a curve,
and Lemma 8.4 gives the model an explicit H(p) = (deg A)(1 - b
sum p_j); Lemma 8.3 moves a packing from that model into PP^n, and
Lemma 8.5 plus 8.5.1 identify the result as the open unit ball, or
as the exterior shell of a central ball. Sections 9-11 do this,
case by case, and check the surface-packing hypotheses (concavity,
positive mean, positivity away from center) in each case.
9. CASE 1: EVERY CAPACITY BELOW 1/2
----------------------------------------------------------------
Here all requested capacities are < 1/2 and their n-th powers sum
to < 1. We pack them into the unit ball using a curve that is a
complete intersection of quadrics, whose normal-bundle model
captures essentially all the volume.
9.1 Two integral formulas we use.
For p = (p_1,...,p_m) >= 0 put t = p_1+...+p_m.
(a) int_{sum p <= h} f(t) dp = (1/(m-1)!) int_0^h f(t) t^{m-1} dt,
because {sum p <= h} has (m-1)-dimensional cross-sections
of volume (h-t)^{m-1}/(m-1)!; differentiating in h gives
the stated formula.
(b) int_{R^m_{>=0}} (r - t)_+ dp = r^{m+1}/(m+1)!,
by taking f(t) = r - t on [0,r] in (a) and using
int_0^r (r-t)t^{m-1} dt = r^{m+1}/(m(m+1)) / ((m-1)!).
These are the "layer cake" of the unit simplex; the second one
is exactly the volume of a ball of capacity r, which is why
caps and balls match.
9.2 The quadrics complete intersection.
LEMMA 9.3 (curves cut by quadrics). Let N >= 3 and let
F in PP^N be a hyperplane. There is a smooth connected curve
C, the complete intersection of N-1 quadrics, meeting F
transversally. With A = O(1)|_C,
deg A = 2^{N-1}, N_{C/PP^N} = (A^2)^{oplus (N-1)},
g(C) = 1 + (N-3) 2^{N-2}. (9.4)
Moreover, with H the hyperplane pullback and E_C the
exceptional divisor of Bl_C PP^N, the class H - d E_C is ample
for every rational 0 < d < 1/2, and the class A + d U is ample
on PP_C( 1 + (A^2)^{oplus(N-1)} ) for the same d.
Proof. Bertini's theorem, applied to the (basepoint-free on
each smooth intermediate step, plus on F) quadratic Veronese
system, gives N-1 quadrics whose common intersection is smooth
and meets F transversally. The Koszul resolution of O_C has
terms in negative twists O(-2j), 1 <= j <= N-1; the vanishings
H^i(PP^N, O(t)) = 0 (0<i<N) and the absence of sections of
negative twists give H^0(O_C) = C, so C is connected. The
intersection class is (2H)^{N-1}, so deg A = 2^{N-1}. The
defining equations form a regular sequence, so the conormal
bundle is (A^{-2})^{oplus(N-1)}; dualizing gives the normal
bundle. Adjunction K_C = K_{PP^N}|_C (x) det N_{C/PP^N}
= A^{N-3}, giving 2g(C)-2 = (N-3)2^{N-1}, i.e. (9.4).
Ample class H - dE_C: the quadric equations generate the ideal
I_C(2), so the Rees algebra (+)_j I_C^j(2j) is a quotient of
the polynomial algebra; its relative Proj is Bl_C PP^N, and
that quotient map is a closed embedding
Bl_C PP^N -> PP^N x PP^{N-2}.
The two hyperplane classes pull back to H and to 2H - E_C. For
0<d<1/2,
H - d E_C = (1 - 2d) H + d (2H - E_C)
is the restriction of a positive rational product polarization,
hence ample (a projective embedding after clearing denominators).
The class A + dU: Lemma 8.4 (with b = 2, l = N-1) gives it an
explicit Kähler representative whose horizontal coefficient is
1 - 2 sum_j p_j >= 1 - 2d > 0 everywhere, including infinity;
a rational class with a Kähler representative is ample by
Kodaira. Done.
9.3 The limiting base area.
Now specialize N = n, m = n-1, and apply Lemma 8.4 to the
normal bundle (A^2)^{oplus m}: the exponent is b = 2 and
deg A = 2^{n-1} = 2^m. Formula (8.7) gives the limiting base
area of the normal-bundle model Y = PP_C(1 + (A^2)^{oplus m}):
H_0(p) = 2^m ( 1 - 2 sum_{j=1}^m p_j ). (9.5)
Its truncation to Delta = { p >= 0 : sum_j p_j <= h }, for
0 < h < 1/2, integrates to
J_m(h) = int_{sum p <= h} 2^m (1 - 2t) dp
= (2^m h^m / m!) ( 1 - 2mh/(m+1) ), (9.6)
by 9.1(a). As h -> 1/2, J_m(h) -> (2^m (1/2)^m/m!)(1 -
2m(1/2)/(m+1)) = (1/m!)(1/(m+1)) = 1/(m+1)! = 1/n!, which is
EXACTLY the volume of the unit ball B^{2n}(1). So as h
approaches 1/2 the model captures all of the target's volume.
9.4 Fixing the parameters.
Let rho_1,...,rho_k be the requested capacities, all < 1/2,
with sum rho_i^n < 1. Increase them slightly to RATIONAL
r_i > rho_i with r_* := max r_i < 1/2 and sum r_i^n < 1 (open
conditions, so rationals are dense enough). By 9.6 and the
strict volume inequality choose RATIONAL parameters
r_* < h < d < 1/2, with J_m(h) > (1/n!) sum_i r_i^n . (9.7)
(First push h close to 1/2 to make J_m(h) exceed the sum, then
insert d strictly between h and 1/2.)
9.5 The surface-packing check.
Define, on Delta = {sum_j p_j <= h},
P(p) = H_0(p) - sum_i ( r_i - sum_j p_j )_+ .
We verify Lemma 7.4's three hypotheses.
CONCAVE: H_0 is affine in t = sum p_j; each cap (r_i - t)_+
is concave (a minimum of two affine functions); a sum of
concave functions is concave. Also H_0 > 0 on Delta because
t <= h < 1/2 implies 1 - 2t > 0.
MEAN POSITIVE: by 9.1(b) and 9.7,
int_Delta P dp = J_m(h) - sum_i r_i^{n}/n! > 0.
POSITIVE AWAY FROM CENTER: choose tau with r_* / h < tau < 1.
If p is in Delta but NOT in tau Delta, then t = sum p_j > tau h
> r_*, so every cap (r_i - t)_+ vanishes (since r_i <= r_*
< t); hence P = H_0 >= 2^m (1 - 2h) > 0 there. So P > 0 on
Delta minus tau Delta, with a uniform lower bound.
The auxiliary simplices {sum p_j <= r_i} lie strictly inside
Delta (r_i < h) and away from the outer face t = h. All
hypotheses of Lemma 7.4 hold.
9.6 Producing and transporting the packing.
Take any r_i' with rho_i < r_i' < r_i. Lemma 7.4 packs
disjoint closed balls B^{2m+2}(r_i') = B^{2n}(r_i') into
Int(Sigma_l x V) for the normal-bundle model, with neighborhood
embeddings. (This model sits inside Y = PP_C(1+(A^2)^{oplus
m}); it is truncated at t <= h < d, so it lies in the affine
part, away from the infinity section D_inf and away from the
fibers over the finitely many points C cap H_inf.)
Apply the normal-cone transfer Proposition 8.3 to the pair
(T, Z) = (PP^n, C), forbidden set F = H_inf. Its hypotheses:
the two classes H - d E_C and A + dU are ample by Lemma 9.3
since d < 1/2. So the compact packing transfers, with
neighborhood embeddings and using the prescribed Fubini-Study
form, into
PP^n minus H_inf .
(Avoiding D_inf and the vertical fibers over C cap H_inf is
exactly what keeps the image inside PP^n minus H_inf and off
its boundary.) Finally, by 8.5.1 the map (8.8) identifies
PP^n minus H_inf symplectically with the open unit ball. Since
rho_i < r_i' < r_i < h < 1/2 at every step there is slack, so
each closed ball lands with a compact neighborhood inside the
target. This packs the rho_i (equivalently the requested
R_i) into Int B^{2n}(1) and proves Case 1.
REMARK ON "most of the volume". The model at level h<1/2 captures
J_m(h) of the 1/n! total, and J_m(h) -> 1/n! as h -> 1/2. That is
the precise sense in which a thin normal neighborhood of a
degree-2^{n-1} curve can hold almost the entire volume of
projective space. The reason: the curve's normal bundle is
O(2)^{oplus(n-1)}, so its fiber polytope {sum p_j < d} of size
d up to nearly 1/2 fills almost the whole moment simplex.
10. CASE 2: ONE LARGE CAPACITY, n >= 4
----------------------------------------------------------------
Now exactly one capacity, call it A, is >= 1/2 (uniqueness is
Section 4.2). The remaining capacities are the rho_i, and by
hypothesis
A^n + sum_i rho_i^n < 1, A + rho_i < 1 for all i.
The idea: put a slightly larger ball of capacity a > A at the
center, and pack the rho_i into the shell { a < pi sum |z_j|^2 < 1}
of the central ball. This is a two-stage construction: first a
HYPERPLANE inside T = Bl_o PP^n, then a QUADRICS CURVE inside that
hyperplane.
10.1 Choosing the parameters.
The inequalities are strict and finite in number, so we can
perturb up to RATIONAL numbers
A < a < 1, 1/2 < a, rho_i < r_i < s := 1 - a,
sum_i r_i^n < 1 - a^n. (10.1)
Note 0 < s < 1/2. Let q = n - 2 (so q >= 2 since n >= 4) and
r_* = max_i r_i. Later we will have parameters
0 < r_* < h_1 < d_1 < s, 0 < r_* < h_2 < d_2 < 1/2. (10.2)
10.2 The shell model for the central ball.
By the shell lemma 8.5, the complement of (E union H_inf) in
T = Bl_o PP^n, with a Kähler form omega_a in the class
H - aE, is symplectomorphic to the shell {a < pi sum|z_j|^2<1}.
So if we pack the small balls into a compact piece of this
shell we can add the central ball of capacity A < a. We now
build a surface-times-toric-fiber model of that shell, inside T,
carrying class H - aE.
10.3 Stage 1: a hyperplane inside T.
E and H_inf are two disjoint smooth divisors in T (E is the
blow-up divisor, H_inf the proper transform of a hyperplane
through nothing special). Pick a hyperplane D0 in PP^n that
does not pass through o and differs from H_inf; its strict
transform D = D0 in T is isomorphic to PP^{n-1}, lies in the
class [D] = H (it is disjoint from E), and meets H_inf in a
hyperplane F of D. The normal line of D in T is O_D(1), and
L|_D = O_D(1).
Deform to the normal cone of D with a rational parameter d_1.
Component classes are
(i) on the blow-up Bl_D T: (1 - d_1) H - a E ,
(ii) on Y_1 = PP_D(1 + O_D(1)): O_D(1) + d_1 U_1 .
Class (i) is (1 - d_1) H - a E, and on the blow-up of PP^n at
one point x H - y E is ample for x > y > 0 (Lemma 8.5); here
x = 1 - d_1 > a = y because d_1 < s = 1 - a. So (i) is ample.
Class (ii): use Lemma 8.4 with A = O_D(1) (b = 1, l = 1,
deg A = 1); with d_1 < s < 1 it is Kähler of the affine shape
omega_D + d(u alpha_1), dd alpha_1 = -omega_D, 0 <= u < d_1,
(omega_D = unit Fubini-Study on D) with base area
H_0^(1)(u) = 1 - u . (10.3)
The forbidden preimage in Y_1 lies over F (= D cap H_inf);
the preimage of E is empty.
10.4 Stage 2: a quadrics curve inside D = PP^{n-1}.
Apply the quadrics Lemma 9.3 in D with N = n-1 (needs
n-1 >= 3, i.e. n >= 4) and forbidden hyperplane F. Get a
smooth curve C, the complete intersection of n-2 quadrics,
meeting F transversally, with
deg C = 2^{n-2} = 2^q, N_{C/D} = (O_C(2))^{oplus q},
g(C) = 1 + (n-4) 2^{n-3} >= 1 .
Deform to the normal cone of C in D with parameter d_2. By
Lemma 9.3 both component classes are ample when d_2 < 1/2.
The normal-bundle component Y_2 = PP_C(1 + (O_C(2))^{oplus q})
has, by Lemma 8.4 with sigma = a positive curvature form of
O_C(1) of total integral 2^q (b = 2), the affine model
omega_X = sigma + d( sum_{j=2}^{n-1} v_j alpha_j ),
dd alpha_j = -2 sigma, sum_{j=2}^{n-1} v_j < d_2, (10.4)
with base area
H_0^(2)(v) = 2^q ( 1 - 2 sum_{j=2}^{n-1} v_j ). (10.5)
10.5 Assembling the two stages.
The whole point: the stage-2 model lives over the curve C,
which sits inside D, which sits inside the first normal line in
T. Explicitly, exhaust C minus F by bordered surfaces Sigma;
for a large one, the truncation sum v_j <= h_2 (< d_2) is a
compact region in Y_2 away from the infinity section and away
from the finitely many forbidden fibers over C cap F. The
normal-cone transfer Proposition 8.3, applied to (D, C) with
forbidden set F, embeds a neighborhood X of that region
symplectically into (D minus F, omega_D). Call it phi : X -> D.
(All line bundles on bordered curve pieces are smoothly framed,
so no obstruction; phi need only be symplectic.)
Now pull back the first normal line of (10.3) along phi. The
disk bundle X over Sigma has H^2(X;Z) = H^2(Sigma;Z) = 0, so the
pulled-back normal line and the trivial line with the curve's
own O_C(1)-connection alpha_{1,C} are equivalent; the
difference of their connections is a closed one-form xi on X.
Writing eta = i^*xi on the curve (i the zero section) and
contracting the disk fibers gives xi = pi^*eta + df for a
function f; a gauge change removes df. Hence the pulled-back
first connection reads exactly
alpha_{1,C} - sum_{j=2}^{n-1} v_j alpha_j + eta ,
with eta generally NOT exact (its periods are kept). These are
smooth unitary operations, so they fix the first moment u.
The lifted (10.3) therefore becomes, on X,
sigma + d( u(alpha_{1,C}+eta) + (1-u) sum_{j=2}^{n-1} v_j alpha_j ). (10.6)
10.6 The moment change and the combined model.
Use disk coordinates z_1,...,z_{n-1} on the fibers of X with
u = pi|z_1|^2, v_j = pi|z_j|^2. The smooth change of variables
zeta_1 = z_1, zeta_j = sqrt(1 - u) z_j (2 <= j <= n-1) (10.7)
is a diffeomorphism on u < 1 with a smooth inverse there,
including all coordinate axes. Its moments are
p_1 = u, p_j = (1 - u) v_j (2 <= j <= n-1),
with the same angles. Put w = p_2 + ... + p_{n-1}. The
truncations u <= h_1 and sum_{j>=2} v_j <= h_2 become
Delta = Delta(h_1,h_2) = { p >= 0 : p_1 <= h_1, w + h_2 p_1 <= h_2 }. (10.8)
In the new variables the horizontal coefficient of (10.6) is
1 - u - 2w = 1 - p_1 - 2w ,
and using w + h_2 p_1 <= h_2 (so w <= h_2(1-p_1)) together with
p_1 <= h_1 < 1, it stays positive on Delta:
1 - p_1 - 2w >= (1 - p_1)(1 - 2 h_2) >= (1 - h_1)(1 - 2 h_2) > 0. (10.9)
So the combined model is a genuine surface (Sigma) times toric
fiber (over Delta), and its limiting base area is
H_0(p) = 2^q ( 1 - p_1 - 2w ). (10.10)
10.7 Enough volume.
Integrate H_0 over Delta. For fixed p_1 = u, the q = n-2
remaining moments range over the simplex {w >= 0 : w <=
h_2(1-u)}; using 9.1(a) and the substitution s = (1-u)sigma,
int_{w <= h_2(1-u)} 2^q (1 - u - 2w) dw
= B_q(h_2) (1 - u)^{q+1},
B_q(h_2) = 2^q ( h_2^q/q! - 2q h_2^{q+1}/(q+1)! )
-> 1/(q+1)! as h_2 -> 1/2.
Then integrating p_1 from 0 to h_1,
int_Delta H_0 dp
= B_q(h_2) * (1 - (1 - h_1)^n)/n
-> (1 - a^n)/n! as h_1 -> s = 1-a, h_2 -> 1/2. (10.11)
And (1 - a^n)/n! is EXACTLY the volume of the shell
{ a < pi sum |z_j|^2 < 1 } (the target minus the central ball
of capacity a). So with h_1 close to s and h_2 close to 1/2 the
model captures all the shell volume.
Fix these parameters rational, by (10.1), so close to the limits
that BOTH
r_* < h_1 < d_1 < s, r_* < h_2 < d_2 < 1/2, and
int_Delta H_0 dp > (1/n!) sum_i r_i^n (10.12)
hold simultaneously (the last is an open condition near the
limits; d_1,d_2 then fit between the h's and their limits).
10.8 The surface-packing check.
Define on Delta
P(p) = H_0(p) - sum_i ( r_i - p_1 - w )_+ , w = sum_{j>=2} p_j.
CONCAVE: H_0 is affine in (p_1, w); each cap (r_i - p_1 - w)_+
is concave. So P is concave.
SIMPLICES INSIDE: if p_1 + w <= r_i then p_1 <= r_i < h_1 and
w + h_2 p_1 <= (p_1 + w) <= r_i < h_2, so each auxiliary simplex
{ p_1 + w <= r_i } lies strictly inside Delta and away from both
outer faces u = h_1 and w + h_2 u = h_2.
MEAN POSITIVE: by 9.1(b) and (10.12),
int_Delta P dp = int_Delta H_0 dp - sum_i r_i^{n}/n! > 0
(each cap integrates to a full ball of capacity r_i).
AWAY FROM CENTER: choose tau with max{ r_*/h_1, r_*/h_2 } <
tau < 1. If p is not in tau Delta, either p_1 > tau h_1 > r_*
or w + h_2 p_1 > tau h_2 > r_* > p_1 + w; in either case every
cap vanishes and P = H_0 >= 2^q (1 - h_1)(1 - 2 h_2) > 0 (no
worse than (10.9)). So P > 0 on Delta minus tau Delta with a
uniform bound. All hypotheses of Lemma 7.4 hold.
10.9 Producing and transporting the packing.
Pick r_i' with rho_i < r_i' < r_i. Lemma 7.4 packs the disjoint
closed balls B^{2m+2}(r_i') (here 2m+2 = 2(n-1)+2 = 2n) into one
large model of the form (10.6), with neighborhood embeddings,
with all moments below p_1 <= h_1 < d_1 and total sum below
h_2 < d_2. The chain of transfers, each with neighborhood:
(a) The model lies in the stage-2 part, over (D minus F), by
the lifting 10.5. The first moment u = p_1 < h_1 < d_1
keeps it inside the affine part of Y_1 (below the first
normal infinity) over D minus F.
(b) Normal-cone transfer Proposition 8.3 with (D, C), forbidden
F, moves it into (D minus F) inside D = PP^{n-1}, i.e.
puts the packing in the first-stage fiber over D minus F.
(This is the phi of 10.5 read backwards.)
(c) Now the packing sits in Y_1 = PP_D(1+O_D(1)), with
u < h_1 < d_1 < s, off infinity and off the preimage of F.
Apply the first normal-cone transfer Proposition 8.3 with
(T, D), forbidden F union (points over D cap H_inf). Its
component classes are ample: (i) (1-d_1)H - aE (shown
ample above) and (ii) O_D(1)+d_1 U_1. So the packing moves
into T minus (the forbidden set), inside T minus (E union
H_inf).
(d) By the shell lemma 8.5, T minus (E union H_inf) with the
Kähler form in class H - aE is symplectomorphic to
{ a < pi sum |z_j|^2 < 1 }.
So the rho-balls land in the shell with compact closure. Add
the central closed ball of capacity A < a in the middle of the
target ball. It is disjoint from the shell packing. Every
capacity had slack A < a and rho_i < r_i' < r_i < s, and every
transfer was defined on a neighborhood of a compact set, so the
closed balls land inside the open unit ball with neighborhood
embeddings. This proves Case 2.
WHY TWO STAGES. The hyperplane step converts the outer shell
{ a < sum < 1 } into "one normal line over D". Since D is just
PP^{n-1}, and n-1 >= 3, we can re-run the Case-1 quadrics
construction INSIDE D to get a base curve of positive genus
g = 1+(n-4)2^{n-3}, exactly as before. When n = 3 there is no
"n-1 >= 3" to spend: the plane inside PP^3 is PP^2, and a conic
has genus 0. That is the sole reason Case 3 (Section 11) needs a
different base curve.
11. CASE 3: ONE LARGE CAPACITY, n = 3
----------------------------------------------------------------
This is the one six-real-dimensional case that Case 2 cannot
reach. By Section 4.2 exactly one capacity A is >= 1/2; the rest
are the rho_i, with
A^3 + sum_i rho_i^3 < 1, A + rho_i < 1.
We again reserve a central ball of capacity a > A and pack the
rho_i into the shell { a < pi sum_{j=1}^3 |z_j|^2 < 1 }. The
difficulty (Section 10's "WHY TWO STAGES"): the hyperplane inside
PP^3 is PP^2, and a conic in PP^2 has genus 0, no handle. We
replace the conic by a plane CUBIC through two marked points,
which has genus 1, and we splice the pieces with a KAHLER CUT.
11.1 Parameters.
Choose rational
1/2 < a < 1, s := 1 - a (so 0 < s < 1/2),
b := (2 - a)/3 = (1 + s)/3 . (11.2)
Then 0 < s < b < a < 1 (check b < a and s < b are both easy
from s < 1/2). Choose the rho_i's padded up to rationals
A < a, rho_i < r_i < s, sum_i r_i^3 < 1 - a^3 . (11.3)
The target is the shell of Lemma 8.5:
{ z in C^3 : a < pi sum_{j=1}^3 |z_j|^2 < 1 }.
11.2 A quadric normal model of the shell.
Let T = Bl_o PP^3, o not in H_inf, and L = H - aE. By Lemma
8.5, T minus (E union H_inf) in class H - aE is the shell.
Choose a smooth quadric surface through o, general with respect
to H_inf, and let D be its strict transform in T.
LEMMA 11.4 (the quadric model). D is isomorphic to the
plane blown up at two points q_1, q_2:
D = Bl_{q_1,q_2} PP^2.
Let M be the pullback of the plane hyperplane class and e_i the
exceptional classes over q_i. Then
H|_D = 2M - e_1 - e_2, E|_D = M - e_1 - e_2, (11.5)
hence (the normal class of D in T is [D]|_D = (2H - E)|_D, and)
N_{D/T} = O_D( 3M - e_1 - e_2 ), L|_D = 3b M - s(e_1+e_2). (11.6)
For every rational 0 < c < s the two normal-cone component
classes
(1 - 2c) H - (a - c) E on the blow-up Bl_D T, (11.7)
3b M - s(e_1 + e_2) + c U on Y = PP_D(1 + O_D(3M - e_1 - e_2)), (11.8)
are ample. Also D cap E and D cap H_inf are two disjoint smooth
plane curves (a LINE and a smooth CONIC), both through q_1, q_2.
Proof. Projection from o restricted to the quadric becomes a
morphism after blowing up o; it contracts the strict transforms
of the two ruling lines through o (each has self-intersection
-1). In coordinates the quadric is x_0 x_1 = x_2 x_3 with
o = [1:0:0:0]; projection to [x_1:x_2:x_3] has the quadratic
inverse [u:v:w] -> [vw:u^2:uv:uw], whose resolution is the
two-point blow-up at [0:1:0] and [0:0:1]. This gives (11.5),
and [D] = 2H - E gives (11.6). Class (11.7) is ample because
a - c > 0 and (1-2c)-(a-c) = s - c > 0 (Lemma 8.5 criterion).
Class (11.8) is ample by the scalar-invariant-cycle argument:
subtracting a small ample class leaves a class that is
nonnegative on every invariant curve (the two fixed sections
and the fibers, all with positive sizes), hence nef, hence
ample after adding the small class back. The curves D cap E
and D cap H_inf are the plane line and plane conic through the
two marks; for a general quadric the conic is smooth and the
line and conic meet only at q_1, q_2 with distinct tangents.
Done.
11.3 The lower Kaehler cut.
There is an auxiliary threefold whose moment sublevel IS the
Y of (11.8). Put
V_0 = PP_{PP^2}( 1 + O_{PP^2}(3) ), (a PP^2-bundle of
projective LINES), and
V_hat = Bl_{(q_1,0),(q_2,0)} V_0, (blow up the two points
(q_i, the zero in the fiber)).
Write U_1 for the pullback of the tautological class of V_0 and
Ehat_i for the point-exceptional divisors. A circle turns the
O(3) summand; it lifts to V_hat.
LEMMA 11.9 (the cut). Fix s < d_1 < b and suppose V_hat
carries a circle-invariant Kähler form in the class
A_class = 3b M + d_1 U_1 - s(Ehat_1 + Ehat_2). (11.10)
Normalize its first moment p_1 to be 0 on the strict bottom
section D = Bl_{q_1,q_2} PP^2. For every rational 0 < c < s,
the symplectic (Kähler) cut retaining p_1 <= c is the complex
projective bundle Y of (11.8), with an invariant Kähler form in
the class 3bM - s(e_1+e_2) + c U. The retained open region
{ p_1 < c } sits in V_hat and is symplectomorphic to the affine
part of Y, in a way that preserves the projection to D on the
attracting basin of the bottom section (so it preserves
avoidance of the preimages of the plane line-and-conic).
Proof. The circle is Hamiltonian (H^1(V_hat;R) = 0: the
PP^2-bundle has vanishing H^1 and point blow-ups preserve it).
At either blown-up point the tangent weights are 0,0,1 (two
plane directions, one vertical), so the fixed loci on the
exceptional PP^2 are the projectivized base directions (a line
in the bottom section) and the pure vertical point; a line
joining them has area s and weight 1, giving moment gap s. An
ordinary projective fiber has area d_1 and weight one at the
bottom. So the fixed levels are exactly
0, s, d_1 . (11.11)
The attracting basin of the bottom section under complex scalar
contraction is the total space of the normal line
N' = O_D(3M - e_1 - e_2)
(checked in local charts at a marked point: y_2 = y_1 v,
z = y_1 w, contraction w -> lambda w; the other chart w' = w/v
is a fiber coordinate of pi^*O(3)(x)O_D(-e_1-e_2)). It is an
attracting BASIN, not a moment sublevel, but it contains
{p_1 < s}: contraction strictly lowers the moment on every
nonconstant orbit, and a point of moment < s must contract to
the bottom (levels 11.11).
Now form the Kähler reduction of V_hat x C by the diagonal
circle at level
p_1 + pi|v|^2 = c . (11.12)
Every point of the level lies over the basin (its first moment
is at most c < s). On any nonzero pair (w,v) in N' + 1 the
complex scalar orbit meets (11.12) exactly once (total moment
rises from 0 at contraction to beyond c at expansion), and the
circle acts freely there (fiber weights one). The holomorphic
quotient of the nonzero pairs is
PP_D( N' + 1 ) = Y ,
which is exactly (11.8); its Kähler structure is the quotient
of the invariant form. The reduction class restricts to
3bM - s(e_1+e_2) on the bottom and has projective fiber area c,
determining 3bM - s(e_1+e_2) + cU. A point of {p_1 < c} maps
to [x, sqrt((c - p_1(x))/pi)] symplectically (the added real
coordinate contributes no form), preserving the base point in
D. Done.
11.4 The cubic degeneration, and where the cut's Kähler form
comes from.
Lemma 11.9 assumed V_hat has a circle-invariant Kähler form in
the class (11.10). That form is produced by deforming V_hat
itself along a family whose special fibre contains a model over
a plane CUBIC. Choose a smooth plane cubic C through q_1, q_2,
general enough that: it meets each of the line l and conic Q in
finitely many points, and it is smooth at q_1, q_2. Such
cubics exist (the cubics through q_1, q_2 form a linear system,
and avoiding the finitely many bad conditions is open and
nonempty). With M = O_{PP^2}(1)|_C, this C is an elliptic curve
of degree deg_C M = 3.
Build the family
Z = Bl_{C x {0}} ( PP^2 x A^1 ) -> A^1 .
Its fibre away from t=0 is PP^2; its central fibre is the
reduced union of the strict plane P and
Z_0 = PP_C( 1 + O_C(3) ) .
Write U_2 for the dual tautological class on Z_0 and put
N = O_Z( 3M - [Z] ).
Since [Z] restricts to -U_2 on Z_0 and to 3M on P,
N|_{Z_0} = O_Z(3M + U_2), N|_P = trivial . (11.14)
Now take the projective bundle PP_Z(1 + N) and blow it up
along the two parameter-sections arising from q_1, q_2 in the
zero section; call the result f : X -> A^1. These two sections
are disjoint, meet the central fibre in its smooth locus, and
every NONZERO fibre of f is exactly V_hat. The central fibre
has two smooth components
W = Bl_{(q_1,0,0),(q_2,0,0)} PP_{Z_0}( 1 + O_Z(3M+U_2) ),
W' = P x PP^1 ,
glued along the first projective bundle over the infinity
section of Z_0. The family is flat (smooth total space with
local rings torsion-free over the parameter's local ring).
Choose rational parameters s < d_1 < d_2 < b and the relative
rational class
A_class = 3b M - d_2[Z] + d_1 U_1 - s(E_1 + E_2), (11.15)
where U_1 is the dual tautological class of the first bundle and
E_1,E_2 are the family blow-up divisors. Its restrictions are
A_class|_W = 3b M + d_2 U_2 + d_1 U_1 - s(E_1+E_2),
A_class|_{W'} = 3(b - d_2) M + d_1 U_1 . (11.16)
CLAIM 11.17 (both component classes are ample).
W': the class 3(b-d_2)M + d_1 U_1 is a positive product class
(b-d_2 > 0 and d_1 > 0), so ample. W: use the two scalar
factors, the second lifted to the first bundle by its
linearization on U_2. An ordinary surface fibre over C is
PP_{PP^1}(1 + O_{PP^1}(1)); its toric diagram is
Q = { p_1, p_2 >= 0 : p_1 <= d_1, p_1 + p_2 <= d_2 },
with vertices (0,0), (d_1,0), (d_1, d_2-d_1), (0,d_2) and
boundary curve areas d_1, d_2-d_1, d_1, d_2, all positive.
Over either marked point the fibre adds the exceptional plane
of size s; the strict surface fibre is then the corner (0,0)
cut by p_1 + p_2 >= s, with five positive edge lengths
s, d_1-s, d_2-d_1, d_1, d_2-s (all positive by 11.15).
A horizontal invariant irreducible curve must be a strict
vertex section (intersecting a generic fibre over C in a
torus-fixed finite set); the four vertex sections have classes
9b - 2s, 9(b-d_1), 9(b-d_2), 9(b-d_2) against A_class|_W, all
positive (the first is 3+s because 9b-2s = 9(1+s)/3-2s =
3+3s-2s = 3+s). Only the bottom vertex section passes through
the two blow-up centres, losing 2s. Now fix any ample class B
on W; for small epsilon > 0 every listed curve has
nonnegative degree against A_class|_W - epsilon B. Every curve
on W is numerically equivalent to a torus-invariant effective
cycle (the classical invariant-cycle theorem; the torus is
connected and solvable, finitely many orbits not required),
so A_class|_W - epsilon B is nef; adding the small ample class B
gives ample. Ampleness on the reduced central fibre is
ampleness on each component, and relative ampleness is open, so
A_class is relatively ample near 0. Done.
Finally, the first circle acts on the WHOLE family X and
preserves the blow-up centres. Averaging any relative Kähler
form over this circle keeps it closed, Kähler on the fibres, and
invariance-preserving. The nonzero fibres V_hat therefore
acquire a circle-invariant Kähler form in the class (11.10), as
Lemma 11.9 required. This closes the loop: the cut of 11.3 is
fed by the cubic degeneration of 11.4.
11.5 The toric model over the cubic, and the base area.
We must compute, for the cubic model, the limiting horizontal
area H_0 as a function of the two fiber moments. The idea is
that the two marked points q_1, q_2 force certain coefficient
sections to vanish, and that vanishing is what shrinks the base
area.
The weight decomposition.
On the affine chart of Z_0 = PP_C(1 + O_C(3)) the two affine
line coordinates w_1, w_2 both have transition bundle O_C(3).
WEIGHTS LEMMA 11.18. For the full projective system (of a
large divisible power N times A_class) the weight (k_1,k_2)
runs over 0 <= k_1 <= N d_1, 0 <= k_2 <= N d_2 - k_1, and
its coefficient space on C is
H^0( C, O_C((3bN - 3k_1 - 3k_2) M)(-m_k q_1 - m_k q_2) ),
m_k = ( sN - k_1 - k_2 )_+ . (11.18)
These systems are basepoint-free for large divisible N.
Proof. Expand first in the U_1-coordinate: a term of degree k_1
leaves coefficient class (3bN-3k_1)M + (Nd_2-k_1)U_2. Expand
in the U_2-coordinate: this gives the stated range and the
coefficient line. Sections on the blow-up are exactly sections
before the blow-up vanishing to total order >= sN at each center;
in local coordinates (y, w_1, w_2) the coefficient of
w_1^{k_1} w_2^{k_2} must vanish in y to order >= m_k. Writing
t = (k_1+k_2)/N, the degree of that line, divided by N, is
9(b - t) - 2(s - t)_+ , (11.19)
which is decreasing in t on 0 <= t <= d_2 and at least
9(b - d_2) > 0; the degrees therefore grow uniformly in the
weights as N grows. On a genus-one curve every line of degree
>= 2 is basepoint-free (evaluate at a point; H^1 vanishes after
subtracting it). Done.
The reduced potential and the base form.
TORIC-MODEL LEMMA 11.20. Let Delta be a compact downward
polytope strictly below the two upper boundaries of Q (the
diagram (11.17)). Over every compact bordered subsurface
Sigma of C^o = C minus {q_1, q_2}, the affine moment region
has smooth product disk coordinates (including the lower axes)
in which the form is
Omega = omega_V + d Gamma, d_C Gamma = kappa(p) > 0,
with Gamma horizontal and toric, and for a bordered exhaustion
of C^o the integrated areas converge UNIFORMLY on Delta to
H_0(p) = 9(b - p_1 - p_2) - 2( s - p_1 - p_2 )_+ . (11.21)
The same limit holds after deleting finitely many further
points.
Proof (the parts that matter).
In a local frame of M|_C the projective potential is
B(y, rho) = (1/N) log( sum_k g_k(y) e^{k . rho} ),
rho_j = log|w_j|^2 ,
where g_k is the sum of squared lengths of a basis of the
coefficient space (11.18). The moments are p = d_rho B; their
gradient is a diffeomorphism onto the interior of Q and is
smooth at every lower face (the coefficient system contains
weights with k_j = 1, so the Jacobian is block-triangular with
invertible diagonal blocks there). Taking the partial Legendre
transform G(y,p) = inf_rho(B - p.rho) gives, in moment and
phase coordinates, the form sum d p_j ^ d theta_j +
d(d_y^c G), whose base part kappa(p) = d_y d_y^c G is positive
(a Schur-complement of a positive Hermitian Hessian). If the
frame changes by e' = h e with lambda = log|h|^2, the potential
transforms as G' = G - 3(b - p_1 - p_2) lambda; since
d^c lambda = d(arg h /2pi), the primitive changes by an exact
term. Hence the local base forms have the transition law of
the real class
3(b - p_1 - p_2) c_1(M|_C) . (11.22)
On a bordered Sigma (where H^2 = 0) we can pick a global frame,
and a radial-homotopy argument produces Gamma with
d Gamma = Omega - omega_V, proving the smooth model.
The base AREA is the spatial integral of kappa. Near a marked
point y=0, sN - k_1 - k_2 > 0 forces the coefficient system to
vanish to order m_k, so g_k(y) = |y|^{2m_k} a_k(y) with a_k > 0
smooth, and the potential differs by a bounded amount from
max over weights of (k/N . rho + lambda . (s - t)_+), lambda =
log|y|^2. Taking the Legendre transform gives
G(y,p) = ( s - p_1 - p_2 )_+ log|y|^2 + O(1) . (11.23)
So G has, at each marked point, an ATOM of the current d d^c G
of mass (s - p_1 - p_2)_+ (subtracting the log term leaves a
bounded subharmonic function, which has no atom). By (11.22)
the TOTAL mass of the class is int_C 3(b-t) c_1(M|_C) =
3(b-t) deg_C M = 9(b-t). Removing the two atoms (one per
marked point) leaves the smooth base area
H_0 = 9(b - t) - 2(s - t)_+ , t = p_1 + p_2 ,
which is (11.21). Uniform convergence of the bordered
integrals to H_0 is Dini's theorem (continuous positive forms
increasing to a continuous limit on a compact polytope). Done.
11.6 The volume of the model equals the shell volume.
The limiting moment domain is
Delta_inf = { p_1, p_2 >= 0 : p_1 <= s, p_1 + p_2 <= b }.
Because b > s, direct integration of (11.21), with u = p_1,
gives
int_{Delta_inf} H_0 dp
= int_0^s ( (9/2)(b-u)^2 - (s-u)^2 ) du
= (3/2)( b^3 - (b-s)^3 ) - s^3/3
= s/2 - s^2/2 + s^3/6 .
Substituting s = 1 - a, a one-line check gives
s/2 - s^2/2 + s^3/6 = (1 - a^3)/6 . (11.24)
And (1 - a^3)/6 is EXACTLY the volume of the shell
{ a < pi sum_{j=1}^3 |z_j|^2 < 1 } = B^{6}(1) minus B^6(a).
Meanwhile the full cap of size r_i (< s) integrates to a ball:
int_{p_1,p_2>=0} (r_i - p_1 - p_2)_+ dp = r_i^3/6 ,
which is the volume of B^6(r_i). So at the limiting domain the
model carries exactly the shell volume, and each requested ball
is exactly a cap of the H_0 profile.
CONCAVITY of H_0. As a function of t = p_1 + p_2, H_0 is
piecewise affine with slopes -7 (for t < s: derivative of
9(b-t) - 2(s-t) is -9 + 2 = -7) and -9 (for t > s). The slope
drops at t = s, so H_0 is CONCAVE. This is the crucial sign:
the proof's engine (Part A) needs P concave, and the base area
of a cubic with two marks is concave precisely because the two
atoms have positive mass.
11.7 Packing and the transfer chain for Case 3.
Fix rational parameters
max_i r_i < h_1 < c < s < d_1 < d_2 < b, s < h_2 < d_2, (11.25)
so close to h_1 = s and h_2 = b that on
Delta = { p_1, p_2 >= 0 : p_1 <= h_1, p_1 + p_2 <= h_2 }
the function
P(p) = H_0(p) - sum_i ( r_i - p_1 - p_2 )_+
has positive integral (hence positive mean). This is possible
because of (11.24), the convergence of Delta to Delta_inf, and
the fact that the caps already fit inside every sufficiently
close truncation.
Surface-packing check (Lemma 7.4).
H_0 > 0 on Delta (it is decreasing in t and t <= h_2 < b). H_0
is concave (slopes -7 then -9); each negative cap is concave, so
P is concave. The auxiliary simplices {p_1 + p_2 <= r_i} lie
strictly inside Delta and away from both outer faces. Choose
tau < 1 with tau h_1 > max_i r_i and tau h_2 > max_i r_i. If
p is not in tau Delta, either p_1 > tau h_1 or p_1 + p_2 >
tau h_2, so all caps vanish and P = H_0 has a uniform positive
lower bound. So all three hypotheses of Lemma 7.4 hold.
Produce the packing.
Construct the cubic family of 11.4 with the chosen d_1, d_2.
Remove from its base curve C the two marked points AND all
intersection points with l union Q (finitely many). The
remaining curve still has genus one and admits connected
bordered exhaustions of genus one. Lemma 11.20 supplies the
toric models there, with uniform limiting area H_0. Lemma 7.4
then packs the closed balls of capacities r_i' (with rho_i <
r_i' < r_i) disjointly in one such model, with neighborhood
embeddings. The whole chosen model has p_1 <= h_1 < c, avoids
the double locus (p_1+p_2 = d_2) and the two blow-ups, and
avoids the inverse images of l union Q under the map to the
plane.
MOMENT-PRESERVING TRANSPORT LEMMA 11.26. Let Sigma be a
compact bordered surface in the unmarked base, K the compact
moment region over it for a polytope strictly below the two
upper faces of Q. For sufficiently small nonzero parameter,
symplectic parallel transport from the central component W to a
nearby fibre V_hat is defined on a neighborhood of K and
preserves p_1 EXACTLY, when both moments are normalized to
vanish on their bottom sections. If K avoids the inverse image
of a closed set, its image does too.
Proof. K avoids the central double locus and the two blow-ups,
so it lies in the submersion locus. Parallel transport for the
averaged total form exists for short time near this compact set
(Lemma 8.2). The connection is invariant under the first
circle, so the transport Phi is symplectic and equivariant; hence
d(p_1^{V_hat} circ Phi) = d p_1^{W}. The difference is thus
constant on the connected region. The region includes points
with first affine coordinate zero, which are fixed on the first
bottom section; a short equivariant transport keeps them there,
so both normalized moments are zero and the constant is zero.
The avoidance is preserved because the excluded set is closed
and transport is short. Done.
Finish the chain.
(a) Transport the model from W into a nearby V_hat via Lemma
11.26: p_1 <= h_1 < c is kept, avoidance of l union Q is
kept.
(b) Apply the lower Kähler cut of Lemma 11.9 (which requires
c < s and the invariant form of 11.4): the balls move into
the projective bundle Y of (11.8), off the infinity section
and off the two forbidden vertical submanifolds over D cap E
and D cap H_inf. The cut form is invariant and is in the
class of Lemma 11.4.
(c) Apply the normal-cone transfer Proposition 8.3 with
(T, D) = (Bl_o PP^3, quadric D), forbidden set F = E union
H_inf (whose D-traces are the line l and conic Q handled in
step (a)). Component classes (11.7) and (11.8) are ample
for 0 < c < s by Lemma 11.4. So the packing transfers into
T minus (E union H_inf), with a Kähler form in class H - aE,
with neighborhood embeddings.
(d) By the shell lemma 8.5, T minus (E union H_inf) with that
form is symplectomorphic to { a < pi sum_{j=1}^3 |z_j|^2 <
1 }.
Add the central closed ball of capacity A < a. It is disjoint
from the shell packing. Every capacity had strict slack
(A < a, rho_i < r_i' < r_i < s) and every transfer was defined on
a neighborhood of a compact set, so the closed balls land inside
the open unit ball with neighborhood embeddings. This proves
Case 3.
12. ASSEMBLY: THE THEOREM
----------------------------------------------------------------
We now combine everything.
CLAIM 12.1 (sufficiency). Let n >= 3, and let R_1,...,R_k > 0
with sum_i R_i^n < 1 and R_i + R_j < 1 for all i < j. Then the
closed balls of capacities R_i pack into Int B^{2n}(1) with
neighborhood embeddings.
Proof. By 8.5.1 normalize the target to capacity 1. If k = 1,
then R_1 < 1 and inclusion packs it (Section 4.1). Assume
k >= 2. By 4.2 at most one capacity is >= 1/2.
- If every capacity is < 1/2, Section 9 (Case 1) applies: all
its hypotheses (rho_i < 1/2, sum rho_i^n < 1) are exactly
the assumed ones.
- Otherwise exactly one capacity, A, is >= 1/2.
If n >= 4, Section 10 (Case 2) applies: its hypotheses
A^n + sum rho_i^n < 1 and A + rho_i < 1 (all i) are
exactly the assumed ones.
If n = 3, Section 11 (Case 3) applies with the same
hypotheses.
In each case the preceding section produces the packing with
neighborhood embeddings. So sufficiency holds.
THEOREM 12.2. Theorem 1 holds: for every n >= 3, k >= 1, and
positive R, R_1,...,R_k, the packing exists iff both the volume
inequality and all pairwise inequalities hold.
Proof. Necessity is Section 3 (volume plus Gromov's two-ball
rigidity, then all pairs). Sufficiency is Claim 12.1. Together
they prove the "if and only if" in every n >= 3. QED.
13. DIMENSION FOUR, AND DIMENSION ONE
----------------------------------------------------------------
The theorem is stated for n >= 3 (real dimension 6 and up). The
two excluded cases are genuinely different, not laziness.
n = 1 (real dimension 2).
Here omega_0 = dx ^ dy is just the ordinary area form, and
"symplectic" = "area preserving". Packing is pure area:
the k disks fit iff R_1 + ... + R_k < R. No pairwise condition
is needed (2-dim packing of round disks is flexible). The
second inequality is vacuous. Nothing deep happens.
n = 2 (real dimension 4).
This is the rich case. Gromov (1985) proved the two-ball
obstruction R_1 + R_2 < R here too. But McDuff and Polterovich
(1994) showed volume + pairwise is NOT enough: there are
infinitely many further obstructions, coming from exceptional
classes of blow-ups of CP^2 and the J-holomorphic curves that
represent them. These act as extra rigid rulers that forbid
configurations the two-ball rule would allow. Partial
understanding: Biran (1997, 1999) proved packing STABILITY
(asymptotically, on a closed rational 4-manifold, many equal
balls fill almost the whole volume); Buse-Hind (2011, 2013)
extended equal-ball stability to higher dimensions and to all
closed rational symplectic manifolds; Buse-Hind-Opshtein (2016)
proved strong stability for unequal balls, provided every
capacity is sufficiently small. But the EXACT criterion (which
configurations fit) is still open in dimension four.
Our proof uses, in Cases 2 and 3, a base surface of positive
genus as the engine (the "handle"). In dimension 4 the natural
analogue has no room to build such a handle with the required
volume, and the extra J-holomorphic obstructions are real. So
dimension 4 needs different ideas.
The moral: once there are at least 3 complex dimensions, the extra
"floppiness" makes every obstruction disappear except volume and
the two-ball rule.