INTEGRABILITY OF SPLIT TANGENT BUNDLES ON RATIONALLY CONNECTED MANIFOLDS
A self-contained proof, written to be read.
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ABSTRACT
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If the tangent bundle of a complex manifold splits as a direct sum of two
subbundles, it is natural to ask when the manifold is a product and, in
particular, when each summand is "consistent with itself" (integrable).
We prove, for a smooth projective rationally connected manifold, that the
answer is forced: for every specified splitting into two positive-rank
subbundles, BOTH summands are integrable. This settles a conjecture of
Horing. A classical product theorem of Horing then upgrades integrability to
an actual product decomposition respecting the given splitting.
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HOW TO READ THIS PAPER
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There is one idea, used four times, and the rest is bookkeeping.
The idea: a split tangent bundle lets us try to "fill in" a rational curve
into a whole surface (a "tangent box"), one coordinate direction running
inside E_1 and the other inside E_2. On that surface each summand behaves
like a bundle pulled back from a single P^1 factor. Testing the failure of
integrability (the "bracket") then becomes a statement about a section of a
TRIVIAL bundle on P^1, and trivial bundles are rigid: a constant section
that vanishes once vanishes always. Rational connectedness is exactly what
supplies the vanishing our test needs.
The steps, in order:
Section 3 Build the tangent box F: P^1 x P^1 -> X with prescribed
tangencies. Three substeps: a formal solution along the diagonal,
turning that formal solution into a genuine rational map, and
removing the remaining indeterminacy.
Section 4 Show that on the box each summand E_i pulls back from its own
P^1 factor (this is where "one direction runs inside E_i" pays off).
Section 5 Combine the box with rational curves to force the bracket of E_i
to vanish, first on an open set, then everywhere.
Section 6 Apply the theorem to general fibers of the rational quotient.
Two appendices supply the only two external facts we use: resolution of a
rational map from a surface by point blowups (Appendix A) and a precise form
of the "two-point" property of rational connectedness (Appendix B).
Nothing here uses positivity, Fano, or rank hypotheses. The only input about X
is rational connectedness, and it enters exactly once, in Section 5.
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1. INTRODUCTION AND MAIN THEOREM
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Let X be a smooth connected projective complex manifold of dimension n.
Write T_X for its tangent bundle.
DEFINITION 1.1 (Splitting). A holomorphic splitting is a decomposition
T_X = E_1 (+) E_2 into two holomorphic subbundles whose fibers are in direct
sum at every point. Equivalently it is a pair of holomorphic bundle maps
P_i : T_X -> E_i with P_i P_i = P_i, P_i P_j = 0 for i != j, and
P_1 + P_2 = id. The P_i are the projectors belonging to the splitting; the
embeddings E_i -> T_X are part of the data. We always assume rk E_1 >= 1 and
rk E_2 >= 1.
DEFINITION 1.2 (Integrability). A subbundle E of T_X is integrable if the Lie
bracket of any two local holomorphic sections of E is again a section of E.
Equivalently, in the language of the next section, the bracket tensor of E is
zero.
DEFINITION 1.3 (Rational connectedness). X is rationally connected if two
general points lie on a rational curve, i.e. there is a nonempty Zariski open
subset of X x X such that every pair in it is the image of some pair of points
under a single morphism g: P^1 -> X. In symbols, g(a) = p and g(b) = q for
fixed distinct a, b in P^1.
Our paper proves:
MAIN THEOREM 5.2. Let X be a smooth connected projective complex manifold
of dimension n >= 2, with a specified splitting T_X = E_1 (+) E_2 into
positive-rank subbundles. If X is rationally connected, then both E_1 and
E_2 are integrable.
The theorem says the local condition (closedness under bracket) is automatic
once the manifold is rational enough. The global consequence, due to Horing,
is the following.
COROLLARY 1.4 (Compatible product). Under the hypotheses of the Main
Theorem there exist smooth projective X_1, X_2 and an isomorphism
X = X_1 x X_2 under which E_i = pr_i^* T_{X_i} for i = 1, 2.
We prove the Main Theorem in Sections 3-5. The Corollary is deduced from it
immediately below using Horing's product theorem. Section 6 records the
application to the rational quotient.
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Proof of Corollary 1.4 from the Main Theorem.
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Horing's product theorem (Theorem 1.4 of his 2007 paper on uniruled varieties
with split tangent bundle) says: if X is a smooth connected projective
rationally connected manifold with a specified splitting T_X = E_1 (+) E_2
and at least one summand is integrable, then X is a product and the summands
are the factor tangent bundles. The Main Theorem supplies both summands, so
the two hypotheses hold. His theorem yields the isomorphism and the
identification E_i = pr_i^* T_{X_i}. Each factor X_i is smooth; it is
projective because it sits as a slice of the product X, a closed complex
submanifold of a projective manifold, and a projective embedding restricts to
one.
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2. NOTATION AND THE BRACKET TENSOR
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Throughout, for i in {1,2} we let j denote the other index. Fix the splitting
with projectors P_i as in Definition 1.1.
For vector fields (or local sections) xi in E_j and e in E_i, define the
projected bracket
(2.1) nabla_i(xi, e) = P_i [xi, e].
This measures exactly how the bracket of an E_j field with an E_i field fails
to stay inside the common subbundle.
LEMMA 2.2. For holomorphic functions f, the projected bracket satisfies
nabla_i(f xi, e) = f nabla_i(xi, e), (tensorial in xi)
nabla_i(xi, f e) = f nabla_i(xi, e) + xi(f) e. (Leibniz in e)
Proof. Expand [f xi, e] = f[xi, e] - e(f) xi. Applying P_i,
P_i (f[xi,e]) = f P_i[xi,e], and P_i(e(f) xi) = e(f) P_i xi = 0
since xi lies in E_j and P_i vanishes there. This is the first identity.
Expand [xi, f e] = f[xi,e] + xi(f) e and apply P_i, noting P_i e = e.
The second identity follows.
Lemma 2.2 says nabla_i is a "partial connection": it is a connection along
directions in E_j, acting on E_i. It uses no bracket of two E_j fields and
therefore requires neither summand to be integrable.
Now the obstruction to integrability of E_i itself. Define the bracket tensor
(2.3) B_i : Lambda^2 E_i -> E_j, B_i(e, e') = P_j [e, e'],
where e, e' are local sections of E_i. That this is tensorial (a bundle map,
independent of the chosen sections) is the content of Lemma 2.2 applied with
the roles swapped: the scalar terms in a bracket of f e with e' land inside
E_i and are killed by P_j.
LEMMA 2.4 (Bracket criterion). E_i is integrable if and only if B_i = 0.
Proof. If B_i = 0 then P_j[e,e'] = 0 for all sections of E_i, i.e. [e,e']
lies in E_i, which is integrability. Conversely if E_i is integrable then
[e,e'] lies in E_i, so P_j[e,e'] = 0.
So proving the Main Theorem is precisely proving that both tensors B_1 and
B_2 vanish. By Lemma 2.4 we may work with B_i pointwise along carefully chosen
curves; see Section 5.
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Notation for the projective line and the square.
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Write C = P^1 and S = C x C. Let pr_1, pr_2 : S -> C be the projections, with
coordinates s on the first factor and t on the second. The diagonal is
Delta = { (s, s) : s in C } subset of S.
Write T^{(i)} = pr_i^* T_C subset of T_S for the copy of the tangent of C
running in the i-th factor direction. Let I subset of O_S be the ideal sheaf of
Delta; the diagonal has self-intersection 2 (it has bidegree (1,1)), so its
conormal bundle I/I^2 is O_C(-2). We will use the line bundles
O_S(m, m) = pr_1^* O_C(m) tensor pr_2^* O_C(m).
This is the whole notation. The next section constructs the tangent box.
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3. THE TANGENT BOX
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Everything rests on one construction.
PROPOSITION 3.1 (Tangent box). Let g: C -> X be a holomorphic map into a
complex manifold X with a holomorphic splitting T_X = E_1 (+) E_2. Then
there is a holomorphic map F: S = C x C -> X such that
F|_Delta = g, dF(T^{(1)}) subset F^*E_1, dF(T^{(2)}) subset F^*E_2.
Neither summand is assumed integrable.
We build F in three stages:
(I) A formal solution along the diagonal: a power series in the normal
direction whose constant term is g and which satisfies the tangency
conditions to infinite order. (Subsection 3.1.)
(II) A rational map R: S --> X that agrees with the formal solution near a
patch of the diagonal. (Subsection 3.2.)
(III) An extension of R across its base points to an honest holomorphic map
F on all of S. (Subsection 3.3.)
Here is why stage (I) is even possible. The two tangency conditions read, in
factor coordinates s on the first factor and t on the second,
F_s lies in F^*E_1 (equivalently P_2(F) F_s = 0),
F_t lies in F^*E_2 (equivalently P_1(F) F_t = 0).
The trick is to use diagonal/normal coordinates. Put
u = s, v = t - s.
Then d/dv = d/dt and d/du = d/ds + d/dt. The two conditions become a single
first-order equation in the "time" v:
(3.1) F_v = P_2(F) F_u, F(u, 0) = g(u).
Indeed F_t = F_v must lie in E_2, i.e. P_1(F) F_v = 0; and F_s = F_u - F_v must
lie in E_1, i.e. P_2(F)(F_u - F_v) = 0, i.e. P_2(F) F_u = P_2(F) F_v = F_v
(using F_v in E_2). Conversely, (3.1) recovers both. This is an evolution
equation: the v-derivative is prescribed in terms of the u-derivative and the
value of F, so the coefficients of the power series in v are determined one
order at a time.
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3.1 The formal Cauchy problem.
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We first produce F as a formal power series in v with holomorphic coefficients
in u.
Formal data. On a product affine chart with the coordinates above, a formal
solution along Delta is a series
F(u, v) = sum_{k>=0} f_k(u) v^k, f_0(u) = g(u),
where each f_k is holomorphic (we say: holomorphic coefficients, no convergence
in v is required). "Along Delta" means v = 0, where F restricts to g.
Recursion. The coefficients of the series of P_2(F) F_u can be computed order
by order: the coefficient of v^k involves only f_0, ..., f_k and their
u-derivatives, since P_2(F) is a convergent power series in F whose constant
term is P_2(g) and higher terms involve positive powers of the f_m (m >= 1).
Comparing with the coefficient of v^k in F_v = sum_{m>=1} m f_m v^{m-1}, the
term (k+1) f_{k+1} appears only on the left. Hence
(k+1) f_{k+1} = (coefficient of v^k in P_2(F) F_u),
which is determined by f_0, ..., f_k. Since k+1 is invertible over C, each f_k
is uniquely and algebraically determined in terms of the previous ones and
g. In particular the constant term gives
f_1 = P_2(g) g'. (first-order split of g')
Thus along the diagonal the E_2 derivative is P_2(g) g', and the E_1 derivative
is the complement g' - f_1 = P_1(g) g'. This is exactly what the two tangencies
prescribe to first order. Induction gives a unique formal solution satisfying
(3.1) to all orders.
Globalization. The recursion was written in one target coordinate chart and
one source chart. To define F globally along Delta we must check that the local
solutions agree on overlaps. Two kinds of change of chart occur.
Target coordinates. If z' = psi(z) is a change of target coordinates, Taylor
substitution sends F to F' = psi(F). The formal chain rule gives
D F' = d psi(F) D F for D = d/du or d/dv. The projector P_2(F) transforms by
conjugation under d psi, so equation (3.1) is carried to the same equation for
F'. Therefore target changes preserve the solution.
Source coordinates. If s' = h(s), t' = h(t) are simultaneous changes of the
source factors (so the diagonal is preserved), then
u' = h(u), v' = h(u+v) - h(u),
and v' = v times a unit, so the two formal completions are identified. The
derivatives transform as
d/dv' = b d/dv, d/du' = a d/du + (b - a) d/dv,
with a = 1/h'(u) and b = 1/h'(u+v). Since (3.1) puts F_v in the image of P_2(F),
these relations give
P_2(F) (d/du') F = b F_v = (d/dv') F.
So (3.1) is carried to the same equation in the new coordinates. The initial
condition transforms to the same diagonal map g.
Because the equation is uniquely solvable and invariant under both kinds of
change, the local series patch together into a single global formal solution
along Delta. This gives the first stage:
LEMMA 3.2 (Formal solution). There is a unique formal map F^ along Delta, with
F^|_Delta = g, satisfying (3.1), equivalently the two formal tangency
conditions of Proposition 3.1. Its coefficients are holomorphic and are
uniquely determined by g and the projectors.
"Formal" means we have a well-defined infinite Taylor object along Delta whose
pullback of any holomorphic function is a power series in v with holomorphic
coefficients. The next stage converts this combinatorial object into actual
geometry.
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3.2 From the formal solution to a rational map.
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We now turn the formal object of Lemma 3.2 into an actual rational map. The
engine is a counting argument. We bound the number of global formal sections of
a certain twisted bundle; we then exhibit more polynomial "test" sections than
the bound allows, so some nontrivial relation among them is forced. That
relation forces the projective coordinates of the formal map to be ratios of
polynomials, i.e. rational functions.
Embed X in a projective space, X subset Y := P^N. It is enough to produce a
rational map S --> P^N which lands in Y and agrees with the formal map; then it
is a rational map to X = Y.
LEMMA 3.3 (Rational reconstruction). Let G^ be a formal holomorphic map from
the formal neighborhood of Delta in S to a closed complex submanifold Y of
P^N. Then there is a rational map R: S --> P^N, holomorphic on a neighborhood
of some nonempty open patch of Delta, whose formal expansion there equals G^,
and whose values on that neighborhood lie in Y.
Proof. Write g_0 = G^|_Delta : Delta -> Y, so g_0 is identified with g via
Delta = C.
Step 1: the formal pullback of O(1). Cover g_0(Delta) by finitely many affine
projective charts U_a of P^N, with holomorphic trivializing frames e_a of
O_{P^N}(1) and transitions e_b = t_{ab} e_a. Restrict t_{ab} to Y and apply
the substitution G^#; this gives formal functions t^_ab on g_0^{-1}(U_a cap
U_b). Their constant terms are t_{ab} o g_0, which are nowhere zero, so each
t^_ab is a unit in the ring of formal series. The substitutions preserve the
cocycle identities, so the local free rank-one modules glue into a locally
free (rank one) formal sheaf L on the completion of S along Delta.
Step 2: global sections of L. If the homogeneous coordinate x_nu equals
c_{nu,a} e_a on U_a, substitute c_{nu,a}|_Y by G^# and multiply by e_a. The
local results agree on overlaps, giving N+1 global formal sections
S_0, ..., S_N of L. This is the formal replacement of "the coordinates of G^".
Step 3: identify the restriction of L to Delta. The quotient
L_0 = L / (I L) is the ordinary holomorphic line bundle g_0^*O(1) on
Delta = C, hence O_C(d) for some d >= 0. The reductions of S_0, ..., S_N have
no common zero, so after relabeling S_0|_Delta is not identically zero. Its
zero divisor identifies L_0 = O_C(d).
Step 4: bound the space of formal sections of a twist. For m >= 0 set
M_m = L tensor O_S(m, m) (restricted to the completion),
V = H^0(Delta, M_m), V_k = H^0(Delta, I^k M_m) for k >= 0.
Thus V = V_0, and V_k consists of global formal sections vanishing to normal
order >= k along Delta; passing to the class mod I^{k+1} M_m has kernel
V_{k+1}. This yields a leading-coefficient injection
(3.2) V_k / V_{k+1} -> H^0(C, O_C(d + 2m - 2k)).
Here is why the twist is d + 2m - 2k. The graded piece
I^k M_m / I^{k+1} M_m is L_0 tensor O_S(m,m)|_Delta tensor (I^k / I^{k+1}).
The conormal bundle I/I^2 is O_C(-2) (the diagonal has bidegree (1,1)), so
I^k/I^{k+1} = (I/I^2)^k = O_C(-2k); and O_S(m,m)|_Delta = O_C(2m). Multiplying
gives O_C(d + 2m - 2k), as claimed.
The filtration is separated: a formal section is a power series in the normal
variable v, so lying in every V_k forces every coefficient to vanish. For
k large the target of (3.2) is zero, so V_k = V_{k+1}; with separatedness,
V_K = 0 for large K. Summing the dimensions of the finitely many nonzero
graded pieces via (3.2),
dim V <= sum_{k>=0} max{ d + 2m - 2k + 1, 0 }.
For d = 2q the sum equals (m + q + 1)^2; for d = 2q + 1 it equals
(m + q + 1)(m + q + 2). Either way the leading term is m^2, so
dim V = m^2 + O(m).
Step 5: force a relation. Ordinary polynomial sections of O_S(m,m) restrict to
the completion. For each nu in {1, ..., N} consider the linear map
H^0(S, O_S(m,m))^{+(2)} -> V, (A, B) |-> A S_0 + B S_nu.
Its source has dimension 2(m+1)^2. For m large this exceeds the bound
dim V = m^2 + O(m), so there is a nonzero kernel pair (A_nu, B_nu), i.e.
(3.3) A_nu S_0 + B_nu S_nu = 0 as formal sections.
Step 6: read off rational coordinates. Fix a connected diagonal disk D on
which S_0|_Delta is nowhere zero. There S_0 is a unit formal frame, and
S_nu / S_0 equals the substituted affine coordinate x_nu / x_0. From (3.3),
q_nu := -A_nu / B_nu equals S_nu / S_0 as formal series on D.
First, B_nu is nonzero: if B_nu = 0 then A_nu S_0 = 0, so the ordinary section
A_nu vanishes to infinite order along D; its convergent Taylor series makes it
vanish on a neighborhood of D, and the identity principle on the connected S
makes A_nu = 0, contradicting (A_nu, B_nu) != 0. Since q_nu = -A_nu/B_nu is a
ratio of polynomial sections, the formula
R = [1 : q_1 : ... : q_N]
defines a rational map S --> P^N whose formal expansion on D is that of G^.
Step 7: R is holomorphic near a point of D. Use coordinates u = s, v = t - s
and an ordinary frame for O_S(m,m) on one product chart; write A_nu, B_nu as
holomorphic functions a_nu(u,v), b_nu(u,v), and expand in v along D with
coefficients a_{nu,k}(u), b_{nu,k}(u). For each nu the generic normal order
r_nu = min { k >= 0 : b_{nu,k} not identically zero on D }
is finite (else b_nu, then B_nu, would vanish identically). Each leading
coefficient b_{nu,r_nu} is a nonzero holomorphic function of u, with discrete
zeros; choose p in D avoiding all finitely many zero sets.
In the frame S_0 the relation reads a_nu + b_nu (S_nu/S_0) = 0 in O_D[[v]].
Because b_nu has normal order r_nu, the relation forces a_nu to have normal
order at least r_nu as well. Since a_nu and b_nu are ordinary holomorphic
functions whose convergent Taylor expansions start at order r_nu, they are
actually divisible by v^{r_nu}:
a_nu = v^{r_nu} A~_nu, b_nu = v^{r_nu} B~_nu, B~_nu(p) != 0.
Shrinking to a neighborhood of p where each B~_nu is a unit, cancel the common
v^{r_nu} (the ring of formal series is a domain) to get
q_nu = -A~_nu / B~_nu,
which is holomorphic near p. Hence R is holomorphic on this common
neighborhood of p.
Step 8: R lands in Y. Its affine coordinate functions match those of G^, so R
and G^ agree as formal maps. Let phi_alpha be holomorphic defining equations
of Y near g_0(p). The substitution phi_alpha o G^ is zero formally, so the
ordinary holomorphic function phi_alpha o R has zero Taylor series at p, hence
vanishes on a neighborhood of p. There R takes values in Y. This proves the
lemma.
REMARK 3.4. The whole argument is a dimension count: the positive normal bundle
of Delta (conormal O_C(-2)) makes formal sections grow only like m^2, while
the space of polynomial test sections grows like 2m^2. The gap produces the
relation. Readers who know the algebraic language will recognize the "G_3"
property of Hironaka-Matsumura; we do not need it, because the count works
directly with holomorphic formal series.
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3.3 Removing the indeterminacy.
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Lemma 3.3 gives R only away from a finite set B of base points in S. The two
tangencies, which held formally, now hold as honest holomorphic identities on
the dense open set where R is defined. We show this leaves no base point.
LEMMA 3.5 (Removal of indeterminacy). Let R: S --> X be a rational map which,
on its domain of definition, satisfies
dR(T^{(1)}) subset R^*E_1, dR(T^{(2)}) subset R^*E_2.
Then R extends to a holomorphic map S -> X.
Proof. Choose a projective embedding X subset P^N. By Lemma A.1 (Appendix A),
there is a finite set B subset S, a composite of point blowups mu: S' -> S
over B, and a holomorphic map F': S' -> P^N extending R o mu. Since X is
closed and F' maps a dense open set into X, the whole image of F' lies in X.
Claim: F' is constant on every exceptional curve. Let H be an irreducible
exceptional component, mapping to a point p in B, and let s, t be the factor
coordinates near p. Because the two tangency conditions are holomorphic
identities and mu is an isomorphism off the exceptional set, the two
bundle-valued two-forms
(3.4) (P_1 dF') wedge d(s o mu) = 0, (P_2 dF') wedge d(t o mu) = 0
vanish on a dense open set, hence everywhere by holomorphicity (both sides are
sections of the locally free sheaf Lambda^2 Omega^1_{S'} tensor F'^*T_X).
Let us decode them. A one-form P_1 dF' wedges to zero with d(s o mu) exactly
when it annihilates kernel(d(s o mu)) = T^{(2)}, i.e. when dF'(T^{(2)}) subset
F'^*E_2; similarly the second says dF'(T^{(1)}) subset F'^*E_1. So (3.4) is
the tangency, transported to S'.
Both functions s o mu - s(p) and t o mu - t(p) vanish along H (all of H maps
to p). Let their generic vanishing orders along H be alpha >= 1 and beta >= 1;
these exist because a coordinate function is not identically constant near H.
Work at a general point of H, away from other components and from the zeros of
the leading coefficients, and let r = 0 be a local equation of H. Then
s o mu - s(p) = r^alpha h, t o mu - t(p) = r^beta h~,
with h, h~ holomorphic units. The first identity of (3.4) becomes
r^{alpha-1} (P_1 dF') wedge (alpha h dr + r dh) = 0.
The local ring of the smooth surface is a domain, and the sheaf of
bundle-valued two-forms is locally free, so the factor r^{alpha-1} is a
nonzerodivisor and may be canceled. At r = 0, evaluate the remaining two-form
on a vector w tangent to H and a vector n with dr(n) = 1. The result is
alpha h (P_1 dF')(w) = 0, hence (P_1 dF')(w) = 0. Canceling r^{beta-1} in the
second identity likewise gives (P_2 dF')(w) = 0. Adding, dF'(w) = 0 for every
tangent vector w to H at a general point, hence at every point. Since H is
connected, F'|_H is constant.
Now the fibers of mu. Off B they are points. Over a point of B, a fiber is a
connected union of exceptional components (the first blowup makes an
exceptional line; each later blowup over the point glues in a new line meeting
the previous ones), see Lemma A.1. The constants taken by F' on intersecting
components agree, so F' is constant on each fiber.
Therefore F' factors through a map of sets F: S -> P^N. Since mu is a
continuous surjection from a compact space to a Hausdorff space (a quotient
map), F is continuous, and it is holomorphic off the finite set B. Near each
point of B, continuity places the image inside one target coordinate chart,
and the coordinate functions are continuous and holomorphic away from that
isolated point. To conclude holomorphy at the isolated point, use that a
function continuous on a neighborhood of the origin in C^2 and holomorphic
off the origin is holomorphic: restrict to lines through the origin and apply
the one-variable removable singularity theorem to get separate holomorphy,
then apply the iterated Cauchy integral formula. Hence F is holomorphic
everywhere. This proves the lemma.
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Proof of Proposition 3.1.
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Apply Lemma 3.2 to get the formal solution F^ along Delta, and Lemma 3.3 with
Y = X (a projective embedding of X) to get a rational map R: S --> X that is
holomorphic near a nonempty diagonal patch and agrees there with F^ formally.
We must transfer the formal tangencies to R. On the patch, differentiate R and
substitute: this identifies the pullback of P_1 R_t and P_2 R_s with the formal
expressions P_1(F^) (d/dt) F^ and P_2(F^) (d/ds) F^, which vanish formally by
Lemma 3.2. Being ordinary holomorphic bundle morphisms, they then vanish on the
whole patch.
Now spread these identities to all of the domain of R. Let B be the finite base
set of Lemma A.1. The complement S minus B is connected (perturb a path in the
connected real four-manifold S inside coordinate balls to dodge finitely many
points). The inverse image under R of the closed submanifold X is an analytic
subset of S minus B containing the nonempty open patch, hence all of S minus B.
The two vanishing identities P_1 R_t = 0 and P_2 R_s = 0 hold on the patch and
extend over the connected complement by the identity principle.
Now Lemma 3.5 produces a holomorphic extension F: S -> X. The tangencies extend
by holomorphicity, and F|_Delta = g, since R|_Delta = g on the patch and both
are holomorphic on the connected Delta = P^1. This is the tangent box.
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4. BUNDLES ON A TANGENT BOX
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Fix a tangent box F: S -> X of a curve g: C -> X, as in Proposition 3.1. The
tangencies say that moving in the first factor, F travels inside E_1, and moving
in the second factor it travels inside E_2. We now show that each summand of the
pullback F^*T_X, restricted along the box, comes from the matching factor of S.
The tool is the partial connection from Lemma 2.2. Recall nabla_i(xi, e) =
P_i[xi, e] for xi in E_j, e in E_i. Being tensorial in xi and Leibniz in e, it is
a connection in the E_j directions, acting on E_i, and it needs no integrability.
PROPOSITION 4.1 (Transport). Let F: S -> X be a tangent box of g: C -> X.
Parallel transport along the fibers of pr_i defines holomorphic bundle
isomorphisms
(4.1) Theta_i : pr_i^*(g^*E_i) -> F^*E_i, i = 1, 2,
whose restriction to the diagonal Delta is the identity (where the source is
identified with g^*E_i via pr_i|_Delta = id on C).
Proof. Fix i and let j be the other index. By the tangency, dF maps the vertical
bundle of pr_i, which is ker(d pr_i) = T^{(j)}, into F^*E_j. The partial
connection nabla_i can therefore be pulled back along F to a connection on
F^*E_i in the vertical (fiber) directions of pr_i, i.e. a relative connection
relative to pr_i : S -> C. We write it down because F may have critical points
and we must not divide by dF anywhere.
Take local frames e_lambda of E_i and xi_gamma of E_j, and write
nabla_i(xi_gamma, e_lambda) = sum_nu Gamma^nu_{gamma lambda} e_nu,
dF(V) = sum_gamma h_gamma F^*xi_gamma
for a local holomorphic vertical vector field V on S (vertical for pr_i). Define
(4.2) D_V( sum_lambda f_lambda F^*e_lambda )
= sum_lambda V(f_lambda) F^*e_lambda
+ sum_{gamma, lambda, nu} f_lambda h_gamma
(Gamma^nu_{gamma lambda} o F) F^*e_nu.
Because nabla_i is tensorial in xi and Leibniz in e (Lemma 2.2), and because of
the chain rule V(A o F) = dA(dF(V)), the expression (4.2) is independent of the
chosen frames: it transforms correctly under changes of the E_i frame, and the
tensoriality in xi handles changes of the E_j frame. The formulas therefore
agree on overlaps and define a holomorphic relative connection
D : O(F^*E_i) -> O( F^*E_i tensor Omega^1_{S/C} ).
Note that where dF(V) = 0 all h_gamma vanish and (4.2) still makes sense; no
inverse of dF is ever used.
Restrict D to a fiber pr_i^{-1}(c) = P^1. There D is a holomorphic connection
on the restriction of F^*E_i to that curve. Its curvature is a holomorphic
two-form with values in an endomorphism bundle on a curve; a curve has no
nonzero two-forms, so the curvature vanishes and D is flat. Equivalently,
horizontal sections obey an everywhere-nonsingular holomorphic linear ODE;
local existence and uniqueness hold, reverse transport inverts forward
transport, and since P^1 is simply connected the transport is path-independent.
Hence each fiber of pr_i determines a canonical trivialization of F^*E_i on it,
obtained by transporting from the diagonal point of that fiber. On the diagonal
fiber point the source pr_i^*(g^*E_i) and the target F^*E_i are both g^*E_i and
we normalize transport to be the identity there.
These identifications depend holomorphically on the base parameter. Fix a point
of S and a path in its fiber from the diagonal to that point. Cover the path by
finitely many coordinate disks with bundle frames, and shrink the base
neighborhood so the same chain works for all nearby fibers. In each disk the
coefficient matrix of the horizontal equation is holomorphic in the base and
fiber variables; Picard iteration converges normally on smaller disks (by the
exponential-series estimate) and produces a solution matrix holomorphic in the
base and the endpoint. Composing the local transports and frame changes gives
a holomorphic identification near the chosen point; path-independence makes the
local identifications agree globally. This is Theta_i, with the stated
normalization.
COROLLARY 4.2. On a fiber line L_i of pr_i (a copy of C in S), Theta_i
identifies F^*E_i with g^*E_i, while Theta_j identifies F^*E_j with the trivial
bundle on that line with fiber E_{j, g(b)} (at the point of the line where the
free coordinate equals b).
Proof. On any fiber of pr_i, pr_i^*(g^*E_i) is the pullback of g^*E_i along
pr_i|_fiber = identity, so Theta_i gives that identification. On a fiber line
L_i = { s_j = b } (in the notation of the next section), the map pr_j is
constant equal to b, so pr_j^*(g^*E_j) = O tensor E_{j,g(b)} is trivial.
==============================================================================
5. VANISHING OF THE BRACKET TENSORS
==============================================================================
We now combine the tangent box with rational curves to force B_i = 0.
Throughout, fix i and let j be the other index, and define once more
B_i : Lambda^2 E_i -> E_j, B_i(e, e') = P_j[e, e'].
By Lemma 2.4, E_i is integrable iff B_i = 0, so it suffices to prove B_i = 0.
Rational connectedness supplies curves with a strong two-point spanning
property. Its proof, which uses only polynomial maps from P^1 and the
constant-rank theorem, is deferred to Appendix B.
LEMMA 5.1 (Two-point variation). Fix distinct a, b in C = P^1. There is a
nonempty complex manifold T and a holomorphic map G: C x T -> X such that the
two-point evaluation
(ev_a, ev_b) : T -> X x X, tau |-> ( G(a, tau), G(b, tau) )
is a submersion. In particular, for the member g = G(., tau) and for either i,
the global sections of g^*E_i that vanish at a span E_i at g(b).
We use the following concrete consequence. Keep the notation of Corollary 4.2.
THEOREM 5.2 (Main Theorem). Let X be a smooth connected projective complex
manifold of dimension at least two, rationally connected, with a specified
splitting T_X = E_1 (+) E_2 into positive-rank subbundles. Then both E_1 and
E_2 are integrable.
Proof. Fix i and let j be the other index. It is enough, by Lemma 2.4, to show
B_i = 0.
Take the family G from Lemma 5.1 and any member g = G(., tau). Build a tangent
box F: S -> X from Proposition 3.1 (the construction may be made afresh for
each member; we do not need the boxes to vary holomorphically with tau).
Consider the fiber line
L_i = { (s_1, s_2) in S : s_j = b } = C,
the copy of C where the coordinate of the j-th factor is pinned at b; its free
coordinate is that of the i-th factor. Its point with free coordinate b is
(b, b) on the diagonal, and it carries the off-diagonal point (a, b) where the
free coordinate equals a.
Let u, w be any two global sections of g^*E_i that vanish at a in C. Pull them
back to the box along the identification of Corollary 4.2 and form the tensor
value
(5.1) sigma_{u,w} =
Theta_j^{-1} ( (F^*B_i)( Theta_i u wedge Theta_i w ) )
in H^0( C, O_C tensor E_{j, g(b)} ),
a holomorphic section of a trivial bundle on L_i = P^1. In (5.1) the outer
Theta_j^{-1} uses that, on L_i, F^*E_j is trivial with fiber E_{j,g(b)} (Cor.
4.2). A holomorphic section of a trivial bundle on P^1 is constant.
At the point of L_i with free coordinate a, the sections Theta_i u and
Theta_i w both vanish (because u, w vanish at a), so the wedge is zero there
and sigma_{u,w} = 0 at that point. Since sigma_{u,w} is constant, it is zero
everywhere on L_i, in particular at the point with free coordinate b, which is
(b, b) on the diagonal. At (b, b) we have F(b, b) = g(b) and, by Proposition
4.1, both Theta_i and Theta_j are the identity there. Unwinding (5.1),
(5.2) B_i(g(b)) ( u(b) wedge w(b) ) = 0 in Lambda^2 E_{i, g(b)} -> E_{j,g(b)}.
By Lemma 5.1 the values u(b) of sections of g^*E_i vanishing at a span the
fiber E_{i, g(b)}. Their pairwise wedges therefore span Lambda^2 E_{i,g(b)},
so (5.2) forces
(5.3) B_i(g(b)) = 0.
This holds for every member g of the family, hence for every point g(b) that
is the second marked point of some curve in the family. By Lemma 5.1 the map
ev_b: T -> X (the second component of the two-point evaluation, composed with
the projection to the second factor) is a submersion, so its image is a
nonempty open subset U of X. On U we have B_i = 0. A holomorphic bundle
morphism (B_i) that vanishes on a nonempty open subset of the connected manifold
X vanishes everywhere, by the identity principle. Hence B_i = 0.
Since i was arbitrary, both B_1 and B_2 vanish, so both summands are integrable.
This proves the Main Theorem.
==============================================================================
6. CONSEQUENCES FOR THE RATIONAL QUOTIENT
==============================================================================
The Main Theorem also applies to the general fiber of the rational quotient of a
uniruled compact Kahler manifold. That fiber is a smooth projective manifold even
when the ambient manifold is not projective, so the theorem makes its inherited
tangent splitting integrable, and this feeds Horing's structure theorem for
uniruled manifolds.
Recall the set-up of Horing's paper "The structure of uniruled manifolds with
split tangent bundle" (Osaka J. Math. 2008), whose numbering we follow. A
uniruled compact Kahler manifold has a rational quotient; a general fiber Z is a
smooth connected projective manifold (projectivity of a rationally connected
compact Kahler manifold is the theorem of Campana, recalled there).
COROLLARY 6.1. Let X be a connected uniruled compact Kahler manifold with a
holomorphic splitting T_X = V_1 (+) V_2. Let Z be a general fiber of its
rational quotient and put W_i = T_Z cap V_i|_Z. Then for each i there is an
almost-holomorphic fibration phi_i : X --> Y_i (allowing phi_i = id_X with
point fibers when W_i = 0) whose general fiber F_i is rationally connected and,
for compatible choices F_i subset Z, satisfies
T_{F_i} = W_i |_{F_i} subset V_i |_{F_i}.
If rk V_1 = 2, exactly one of the following holds:
(i) W_1 = V_1|_Z, and X has an analytic fiber bundle X -> Y with
rationally connected general fiber and T_{X/Y} = V_1;
(ii) W_1 is a line bundle, and there is an equidimensional holomorphic map
X -> Y whose general fiber F is a rational curve with T_F subset V_1|_F;
(iii) W_1 = 0, equivalently T_Z subset V_2|_Z.
Proof. Use the published version of Horing (2008), in which the definition of an
almost-holomorphic fibration includes the identity case. Choose Z over a regular
value in the fibration locus of the rational quotient (Horing, Definition 2.9 and
Theorem 2.10). Then Z is an actual smooth connected compact submanifold of X. It
inherits a Kahler form and is rationally connected; by Campana's theorem (recalled
after Horing's Definition 2.8) it is projective.
By Horing (Proposition 3.16), the ambient projectors preserve T_Z. Their
restrictions are complementary holomorphic idempotents on Z. An idempotent has
locally constant rank, so on the connected Z their images are holomorphic
subbundles W_i, and T_Z = W_1 (+) W_2.
If both ranks are positive then dim Z >= 2, and the Main Theorem makes both W_i
integrable. If one rank is zero, the pair is (0, T_Z), and both are integrable
trivially. In every case the general-fiber integrability hypothesis of Horing
(Theorem 1.3) holds, giving the fibrations phi_i and the tangent identities;
his proof includes the identity case when W_i = 0. When rk V_1 = 2, Horing
(Theorem 1.4) gives the three alternatives according as rk W_1 is 2, 1, or 0;
these ranks are exhaustive and mutually exclusive. This proves the corollary.
==============================================================================
APPENDIX A. RESOLVING A RATIONAL MAP FROM A SURFACE
==============================================================================
Elimination of indeterminacy by point blowups is classical (Beauville, Complex
Algebraic Surfaces). We give the precise statement used in Section 3.3, together
with enough of the argument to make the finite termination explicit.
LEMMA A.1. A rational map R: P^1 x P^1 --> P^N is holomorphic outside a finite
set B. A finite sequence of point blowups over B produces a smooth projective
surface mu: S' -> P^1 x P^1 on which R o mu extends holomorphically. The fibers
of mu are connected, and its exceptional components are smooth curves.
Proof. Step 1: B is finite. Clear denominators to write R as a tuple of
bihomogeneous polynomials of one common bidegree, with no common factor. The
highest and lowest degrees in each variable group are additive in products, so a
common factor may be taken bihomogeneous. In any product affine chart the
dehomogenized tuple h_0(x, y), ..., h_N(x, y) then has no common nonconstant
factor (bihomogenizing such a factor would give one for the original tuple). In
the polynomial ring C(x)[y] the tuple therefore has gcd one; a univariate
Bezout identity with denominators cleared puts a nonzero polynomial in x into the
ideal (h_0, ..., h_N). Interchanging the variables gives a nonzero polynomial in
y. The common zero set in the chart is finite, and there are four charts.
Step 2: each base point has an ideal of finite colength. At a common zero, in
centered formal coordinates, the ideal generated by the local tuple contains a
power of x and a power of y, hence has finite colength in C[[x, y]].
Step 3: one blowup strictly decreases colength. Let D = C[[x, y]], m = (x, y),
and let I subset D be the ideal of a local tuple with finite colength; write
k = min ord(f_nu) >= 1. Blow up the origin. In the chart (x, z) |-> (x, xz),
divide the pulled-back tuple by x^k; the quotients are holomorphic, and the
other chart gives the same map because the two tuples differ by a unit on the
overlap. On the exceptional line the remaining base points are the common zeros
of the degree-k initial forms; at least one such form is nonzero, so there are
finitely many, and a linear change of coordinates places a chosen one at
(x, z) = (0, 0). Put
J = ( x^{-k} f_nu(x, xz) )_nu subset C[[x, z]], D' = C[[x, z]] / J.
A power x^l belongs to I; substituting and dividing puts x^{l-k} in J, so the
class of x in D' is nilpotent. Modulo x, a nonzero degree-k initial form gives a
nonzero polynomial in z of degree at most k, whose germ at the new base point is
z^q times a unit with 1 <= q <= k. Hence D'/(x) is spanned by 1, z, ..., z^{k-1}.
Lifting successively modulo powers of the nilpotent x shows D' is spanned by
finitely many classes x^a z^b with 0 <= b < k.
Now the C-linear map
m^k / I -> D', h |-> x^{-k} h(x, xz)
is well defined: a monomial x^a y^b of total degree at least k goes to
x^{a+b-k} z^b with nonnegative exponents, higher orders go to higher orders, and
if h = sum_nu q_nu f_nu lies in I then its transform sum_nu q_nu(x, xz) x^{-k}
f_nu(x, xz) lies in J. Each spanning monomial x^a z^b (b < k) is the image of
x^{a+k-b} y^b in m^k, so the map is surjective. Therefore
dim D' <= dim (m^k / I) = dim(D/I) - k(k+1)/2 < dim(D/I).
So the new base ideals have finite colength, and that positive integer drops
strictly along every surviving branch.
Step 4: the process terminates. There are finitely many initial base points and
finitely many children at each step; the maximum initial colength bounds the
length of each branch by the inequality above. The whole tree is finite, so
finitely many point blowups remove all base points, and the divided tuples define
the holomorphic extension R o mu.
Step 5: geometry of the blowups. A point blowup is proper, has a projective line
over its center, and is an isomorphism elsewhere. The strict transform of a
smooth curve stays smooth, and a new exceptional line meets the strict transforms
of the components through the center. Induction gives smooth exceptional
components and connected fibers for the composite mu. This proves the lemma.
==============================================================================
APPENDIX B. TWO-POINT VARIATION FROM RATIONAL CONNECTEDNESS
==============================================================================
This appendix proves Lemma 5.1. We use only three tools: the polynomial
description of maps from P^1, the fact that a proper closed analytic subset of a
connected complex manifold is covered by countably many lower-dimensional
submanifolds, and the holomorphic constant-rank theorem. For background on
rationally connected varieties, see Kollar-Miyaoka-Mori (1992) and Kollar (1999).
We first record the covering observation.
LEMMA B.1. Let A be a proper closed analytic subset of a connected
second-countable complex manifold M. Then A is covered by countably many
complex submanifolds of dimension strictly smaller than dim M. (These
submanifolds need not be contained in A.)
Proof. Cover A by countably many connected coordinate balls on which A is
defined by holomorphic functions, and in each ball pick a defining function f
that is not identically zero (a proper analytic subset has empty interior). At
a zero of f of finite order r >= 1, some derivative of order r-1 vanishes there
and has nonzero differential; that point therefore lies on the regular zero
locus of one partial derivative of f. There are countably many such derivatives
and balls, and their regular zero loci (connected components separated) give
the covering.
Proof of Lemma 5.1. Embed X in P^N. Every holomorphic map C = P^1 -> P^N is
given by a tuple of homogeneous polynomials of some degree d >= 0 with no
common zero: the pullback of O(1) is O_C(d), and its coordinate sections give
such a tuple. For fixed d these tuples form an open subset M_d of a
finite-dimensional vector space (openness uses compactness of C), and there is
a universal holomorphic evaluation map C x M_d -> P^N.
We cover the tuples whose maps land in X by countably many manifold families,
each consisting entirely of such tuples. Begin with the connected components of
M_d. For a connected submanifold M, either every tuple in M maps C into X
(then accept M), or there are tau in M and c in C with G(c, tau) not in X. In
the second case the set { eta in M : G(c, eta) in X } is a proper closed
analytic subset of M containing every desired tuple, and Lemma B.1 replaces it
by countably many lower-dimensional submanifolds; repeat the test on each. All
desired tuples survive, and extra points introduced by the covering are simply
tested again. Each non-accepted family strictly lowers the dimension, so the
process has finite depth with countably many branches and yields countably many
accepted families covering all desired tuples. On an accepted family the
universal evaluation factors holomorphically through the embedded submanifold X.
Do this for every d. The resulting countably many two-point evaluation maps to
X x X have images containing general pairs, because a rational curve through two
distinct points can be reparametrized to take those points at the fixed values
a, b. By the convention defining rational connectedness, these images together
contain a nonempty Zariski open subset of X x X, which is also open in the usual
topology and has positive measure in local coordinates.
Suppose for contradiction that every one of these evaluation maps has complex
rank everywhere less than 2 dim X. Then the image of each is measure zero; here
is the direct argument. For a holomorphic phi: M -> X x X with the stated rank
bound, take M connected. On the open set where the rank is maximal, the
constant-rank theorem puts the image locally in a submanifold of smaller
dimension, and countably many charts give a measure-zero image; if the maximal
rank is zero this already handles all of M. Otherwise the complement of that open
set is a proper closed analytic subset, covered by countably many lower-
dimensional submanifolds by Lemma B.1, and we induct on dim M, the strict rank
bound persisting on each restriction. The zero-dimensional case is immediate. A
countable union of measure-zero images cannot contain the positive-measure open
set, so some family has a point where two-point evaluation is a submersion.
Restrict its parameter space to the nonempty open set where the differential is
surjective, and call the result T.
Finally, a tangent vector to the parameter space differentiates the holomorphic
family to a global section of g^*T_X. Surjectivity of the two-point evaluation
differential provides, for every w in T_{g(b)} X, such a section with values 0
at a and w at b. Projecting with P_i gives the last assertion of Lemma 5.1.
==============================================================================
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END OF PAPER
==============================================================================