Integrability of split tangent bundles on rationally connected manifolds

Proof. Höring’s [6, Theorem 1.4] applies to a smooth connected rationally connected projective complex manifold with a specified splitting of its tangent bundle and at least one integrable summand. By Theorem 1.1, these hypotheses hold. His theorem gives an isomorphism with a product that identifies the specified summands with the factor tangent bundles. Each factor is smooth. It is projective because its realization as a slice of the product is a closed complex submanifold of X, and a projective embedding of X restricts to an embedding of that slice.□

Proof. Use the same coordinate on a source chart in each factor, denoting the two coordinates by s,t. Set

After restricting the diagonal disk so that g takes values in one target coordinate chart, impose

Here the entries of the holomorphic matrix P_2 are substituted by the Taylor rule above. That rule preserves P_2^2=P_2 and P_1+P_2=\id.

Because \partial_u=\partial_s+\partial_t and \partial_v=\partial_t, (2.2) is equivalent to

Indeed, the equation puts \partial_v\wh F in the image of P_2(\wh F), while applying P_2(\wh F) to \partial_u\wh F-\partial_v\wh F gives zero. Conversely, these two statements recover (2.2). They are exactly the two formal tangency conditions.

Write \wh F=\sum_{k\ge0}f_k(u)v^k, with f_0=g in the chosen coordinates. At order zero, (2.2) gives

Thus along the diagonal the second-factor derivative is P_2(g)g', and the first-factor derivative is its complement g'-f_1=P_1(g)g'. The higher coefficients extend this forced first-order decomposition. The coefficient of v^k on the right side of (2.2) involves only f_0,\ldots,f_k and their u-derivatives. It determines (k+1)f_{k+1}, so induction gives a unique solution whose coefficients are holomorphic on the diagonal disk.

We verify that these local solutions define a formal map globally. Under a target-coordinate change z'=\psi(z), Taylor substitution gives \wh F'=\psi(\wh F), and the formal chain rule gives D\wh F'=d\psi(\wh F)D\wh F. The matrix of P_2 changes by the corresponding conjugation, so (2.2) is preserved. For a source change s'=h(s), t'=h(t), one has

The new normal parameter is v times a unit, so this substitution identifies the two completions. With

the continuous source derivations transform as

Since P_2(\wh F)\partial_v\wh F=\partial_v\wh F, these formulas imply

The initial value also transforms to the same diagonal map. Local uniqueness therefore identifies the coordinate series on every overlap. Their Taylor substitution maps agree as well, since those maps are determined by the coordinate series. This proves both global existence and uniqueness.□

Proof. Write g_0=\wh G|_\Delta. We first construct the formal pullback of \OO_{\PP^N}(1). Choose a cover of g_0(\Delta) by affine projective charts U_a with holomorphic frames e_a of this line bundle, and write e_b=t_{ab}e_a on overlaps. Restricting t_{ab} to Y and applying \wh G^\# gives formal functions \wh t_{ab} on g_0^{-1}(U_a\cap U_b). Their constant terms are t_{ab}\circ g_0, which are nowhere zero. Thus each \wh t_{ab} is a unit in the completion. The substitutions preserve the transition cocycle identities, so the relations \wh e_b=\wh t_{ab}\wh e_a glue free rank-one modules into a locally free \wh{\OO}_{S,\Delta}-module L.

If the homogeneous coordinate section x_\nu is c_{\nu,a}e_a on U_a, evaluate c_{\nu,a}|_Y by the same substitution. The resulting local sections \wh G^\#(c_{\nu,a}|_Y)\wh e_a agree on overlaps, giving global sections S_0,\ldots,S_N of this very module L. Its reduction L_0=L/\wh{\mathcal I}L is the ordinary holomorphic line bundle g_0^*\OO_{\PP^N}(1) on \Delta\simeq C. The reductions of the coordinate sections have no common zero, so after reordering, S_0|_\Delta is not identically zero. Its zero divisor has some degree d\ge0 and identifies

For example, a degree-d homogeneous polynomial with the same zero divisor gives this identification directly.

For a fixed m\ge0, let \wh{\OO}_S(m,m) denote the restriction of the ordinary line bundle to the completion, and put

Thus V_0=V, and V_k consists precisely of global formal sections vanishing to normal order at least k. Passing a section to its class modulo \wh{\mathcal I}^{k+1}\mathcal M_m has kernel V_{k+1}. It gives the leading-coefficient injection

Indeed, the leading-coefficient sheaf is

Only the constant terms of the frame transitions act on this quotient. Since \Delta has bidegree (1,1), its conormal bundle \mathcal I/\mathcal I^2 is \OO_C(-2); the displayed sheaf is therefore \OO_C(d+2m-2k), as asserted.

This filtration is separated. In a formal frame a section is a series in v, so membership in every V_k makes every coefficient zero. For all sufficiently large k, the space on the right side of (2.3) is zero. Hence, for some K, V_k=V_{k+1} for every k\ge K, and V_K=\bigcap_{k\ge K}V_k=0 by separatedness. The finite remaining filtration gives

Here d is fixed. More explicitly, the sum is (m+q+1)^2 for d=2q and (m+q+1)(m+q+2) for d=2q+1.

Restrict ordinary polynomial sections of \OO_S(m,m) to the completion. For each \nu=1,\ldots,N, consider

Its source has dimension 2(m+1)^2. Choose m large enough that this exceeds the bound in (2.4), and choose a nonzero kernel pair (A_\nu,B_\nu) for each \nu.

Fix a connected diagonal coordinate disk D on which S_0|_\Delta is nowhere zero, contained in one product affine chart. The section S_0 is then a formal frame for L along D, and S_\nu/S_0 is the substituted affine coordinate x_\nu/x_0. Necessarily B_\nu\ne0: otherwise A_\nu S_0=0 would say that the ordinary section A_\nu has zero formal expansion along D. Its convergent local Taylor series would make it zero on a neighborhood of D, and the identity principle on connected S would give A_\nu=0, a contradiction. The ratios

therefore define a rational map R=[1:q_1:\cdots:q_N].

It remains to show that these ratios are holomorphic simultaneously near one point of D. Use u=s, v=t-s and an ordinary frame for \OO_S(m,m) on the chosen product chart, and represent A_\nu,B_\nu by holomorphic functions a_\nu(u,v), b_\nu(u,v). Write their normal Taylor coefficients on D as a_{\nu,k}(u), b_{\nu,k}(u). For each \nu, the generic normal order

is finite. Otherwise Taylor uniqueness near D, followed by the identity principle on S, would make the nonzero ordinary section B_\nu vanish identically. Each leading coefficient b_{\nu,r_\nu} is a nonzero holomorphic function on the connected disk D. Its zeros are discrete, so a single point p\in D can be chosen where all the finitely many leading coefficients are nonzero.

In the formal frame S_0, the kernel relations become

By the definition of r_\nu, all coefficients of b_\nu below that order vanish on D; the relation makes the same true of a_\nu. Since a_\nu and b_\nu are ordinary holomorphic functions, this formal vanishing implies actual holomorphic divisibility near p:

Indeed, their convergent Taylor expansions in v have the first r_\nu coefficients zero. Shrink to one neighborhood of p where every \widetilde b_\nu is a holomorphic unit. Multiplication by v is injective on formal series, so the same power can be canceled in each formal relation. It follows that

is holomorphic there and has formal expansion S_\nu/S_0. Thus R is holomorphic on this common neighborhood. Its affine coordinate series agree with those of the composition of \wh G with Y\hookrightarrow\PP^N. The Taylor substitution rule then makes these formal maps agree on every holomorphic function on the projective target chart.

Finally, R(p)=g_0(p), so shrink the source neighborhood until its image lies in a target chart with holomorphic defining equations for Y. Their compositions with R have zero normal Taylor series, because their substitutions by \wh G are zero. The convergent Taylor series of these ordinary holomorphic functions make them vanish on a neighborhood of p. There R takes values in Y. In local submanifold coordinates, every holomorphic function on Y extends to the ambient chart. The ambient formal equality therefore also gives equality of the formal maps to Y, as required.□

Proof. Choose a projective embedding X\subset\PP^N. By Lemma A.1, there are a finite set B\subset S, a composition of point blowups \mu:S'\to S over B, and a holomorphic map F':S'\to\PP^N extending R\circ\mu. Since X is closed and F' maps a dense open set into X, the entire image lies in X.

We show that F' is constant on each exceptional component. Let H be such a component, lying over p\in B, and let s,t be coordinates on the two source factors near p. On the inverse image of this coordinate neighborhood, P_i dF' is a holomorphic one-form with values in F'^*T_X. The tangency assumptions imply the identities

The two wedge expressions are sections of the locally free sheaf \Omega^2_{S'}\otimes F'^*T_X. The identities first hold off the exceptional curves, where \mu is an isomorphism, and then hold everywhere by holomorphicity.

Both coordinate functions s\circ\mu-s(p) and t\circ\mu-t(p) vanish along H, since \mu(H)=\{p\}. Their generic orders along H are positive and finite. Indeed, in a normal coordinate along a connected disk of H, the first normal Taylor coefficient that is not identically zero gives that order. If no such coefficient existed, the pullback would vanish on an open neighborhood meeting the dense locus where \mu is an isomorphism, which is impossible for a coordinate function. Denote the two respective orders by \alpha,\beta. This reasoning applies also to components created by later blowups. At a general point of H, avoiding intersections with other components and zeros of the two leading coefficients, choose a local equation r=0. We can then write

with holomorphic units h,\widetilde h.

The first identity in (2.5) is therefore

The holomorphic local ring of the smooth surface is a domain, so multiplication by r^{\alpha-1} is injective on the locally free module of bundle-valued two-forms. It can therefore be canceled. At r=0, evaluate the resulting ambient two-form on a vector w tangent to H and a vector n with dr(n)=1. The result is \alpha h\,P_1dF'(w)=0, hence P_1dF'(w)=0. Canceling r^{\beta-1} in the second identity gives P_2dF'(w)=0 in the same way. Since P_1+P_2=\id, the derivative of F'|_H vanishes on a dense open subset of H, and hence everywhere. Since H is connected, F'|_H is constant.

The fibers of \mu away from B are points. Each nontrivial fiber over B is a connected union of exceptional components: the first blowup creates an exceptional line, and every later blowup over that point replaces its center by a new line meeting the strict transforms of the components through it, as in Lemma A.1. The constants on intersecting components agree, so F' is constant on every fiber.

The map \mu is a continuous surjection from a compact space to a Hausdorff space. It is therefore a quotient map, and F' descends to a continuous map F:S\to X, holomorphic off B. Near each point of B, continuity places the image in a single target coordinate chart. The coordinate functions are continuous and holomorphic away from the isolated point. One-variable removability on coordinate lines makes them separately holomorphic, and the iterated Cauchy formula makes them holomorphic. Hence F is holomorphic everywhere.□

Proof of Proposition 2.1. Apply Lemma 2.2 and then Lemma 2.3 to a projective embedding of X. The resulting rational map is holomorphic near a nonempty diagonal patch and has formal expansion \wh F there. Continuous differentiation and Taylor substitution identify the expansions of P_1(R)\partial_tR and P_2(R)\partial_sR with the zero formal tangency expressions of \wh F. These are ordinary holomorphic expressions, so Taylor uniqueness makes them zero on a neighborhood of that patch. Let B be the finite base set from Lemma A.1. The complement S\setminus B is connected; one can perturb any path in the connected real four-manifold S inside coordinate balls to avoid the finitely many points. The inverse image of X under the holomorphic map R:S\setminus B\to\PP^N is an analytic subset containing a nonempty open set, so it is all of S\setminus B. The two tangency conditions are then vanishing identities for holomorphic bundle morphisms on this connected complement, and continue there by the identity principle.

Now Lemma 2.5 gives a holomorphic extension F:S\to X. Tangency extends by holomorphicity. The maps F|_\Delta and g agree on the original nonempty diagonal patch, so they agree on all of \Delta.□

Proof. The bracket rules and P_i\xi=0 give

Thus (3.1) is tensorial in its E_j direction and obeys the connection Leibniz rule in its E_i argument. These two rules use no bracket of two E_j sections and require no integrability of either summand.

Fix i and let j\ne i. The tangency of F sends the vertical line bundle \ker(d\pr_i)=T^{(j)} into F^*E_j. The partial connection therefore pulls back to a holomorphic connection on F^*E_i relative to \pr_i. We spell out this construction because F may have critical points. Take local frames e_\lambda of E_i and \xi_\gamma of E_j, and write

for a local holomorphic vertical vector field V on S. Define

The two connection rules and the chain rule V(A\circ F)=dA(dF(V)) give the usual connection transformation under a change of the E_i frame; tensoriality in the first argument handles a change of the E_j frame. Hence these formulas agree on overlapping product neighborhoods. They define a holomorphic relative connection

where the relative differential is for \pr_i:S\to C. At a point where dF(V)=0, all h_\gamma vanish there and the formula still applies; no inverse of dF is used.

The restriction of D to any fiber C=\PP^1 is a holomorphic connection. Its curvature is a holomorphic two-form on that curve and therefore vanishes. Horizontal sections satisfy an everywhere nonsingular holomorphic linear ordinary differential equation. Local existence and uniqueness give continuation along every path, with reverse transport as its inverse. Flatness and simple connectedness of \PP^1 make this transport path-independent. Starting at the fiber’s diagonal point therefore identifies its bundle with the corresponding fiber of g^*E_i.

These identifications depend holomorphically on both variables. Fix a base point and a path in its fiber from the diagonal to a desired endpoint. Cover the path by a finite chain of coordinate disks and bundle frames, and shrink the base neighborhood so that the same chain applies to nearby fibers. In each disk the coefficient matrix of the horizontal equation is holomorphic in the base and fiber coordinates. Picard iteration converges normally on smaller disks by the exponential-series estimate, so its solution matrix is holomorphic in the base and the start and end coordinates. Composing these local transports and frame changes gives a holomorphic identification near the chosen point of S. Path independence makes the local identifications agree globally. Transport at the starting diagonal point is the identity, giving \Theta_i and its stated normalization.□

Proof of Theorem 1.1. Choose the family in Lemma 4.1. For each member g, take a tangent box F:S\to X from Proposition 2.1; the boxes need not be chosen holomorphically as the member varies. For j\ne i, consider

The free coordinate identifies L_i with C. As shown in Figure 1, its marked endpoint (b,b) lies on the diagonal. By Proposition 3.1, \Theta_i identifies F^*E_i|_{L_i} with g^*E_i, while \Theta_j identifies F^*E_j|_{L_i} with the trivial bundle having fiber E_{j,g(b)}.

Let u,v be any two global sections of g^*E_i vanishing at a. Restricting the transport maps to L_i, the tensor value

is a holomorphic section of a trivial bundle on \PP^1, hence is constant. It vanishes at the point with free coordinate a, so it also vanishes at the point with free coordinate b. At this latter point, F(b,b)=g(b) and both transport maps are the identity. Consequently

By Lemma 4.1, the values u(b) of sections vanishing at a span E_{i,g(b)}. Their pairwise wedges span its exterior square, and therefore \BB_i(g(b))=0.

This applies to both i and every member of the family. The map \ev_b is a submersion, being the composition of the two-point submersion with the second projection. Its image is a nonempty open subset of X, on which both bracket tensors vanish. The identity principle for holomorphic bundle morphisms on connected X makes them vanish everywhere. This is precisely integrability of both specified summands.□

Proof. Use the published version of [7]; its definition of an almost-holomorphic fibration includes the identity case. Choose Z over a regular value in the fibration locus of the quotient supplied by [7, Definition 2.9 and Theorem 2.10]. Then Z is an actual smooth connected compact submanifold of X. It inherits a Kähler form and is rationally connected. Campana’s theorem, recalled in the remark after [7, Definition 2.8], makes Z projective.

By [7, Proposition 3.16], the ambient complementary projections preserve T_Z. Their restrictions are complementary holomorphic idempotents. An idempotent has locally constant rank, so on connected Z their images are the holomorphic subbundles W_i, and

If both ranks are positive, then \dim Z\ge2, and Theorem 1.1 makes both W_i integrable. Otherwise they are 0 and T_Z, which are integrable as well.

This is the general-fiber integrability hypothesis of [7, Theorem 1.3], which gives the fibrations \phi_i and the stated tangent identities. Its proof includes the identity when W_i=0. If \rk V_1=2, [7, Theorem 1.4] gives the three conclusions according as \rk W_1 is 2, 1, or 0. These ranks are mutually exclusive and exhaustive, proving the stated trichotomy.□

Proof. Clear denominators to represent R by a tuple of bihomogeneous polynomials of the same bidegree, and remove all common factors. The factors can be taken bihomogeneous: the highest and lowest degrees in each variable group are additive in products.

In any product affine chart, the dehomogenized tuple h_0(x,y),\ldots,h_N(x,y) has no common nonconstant factor. Otherwise bihomogenizing such a factor would give a common factor of the original tuple. In \CC(x)[y], the tuple therefore has gcd one. A univariate Bézout identity, with denominators cleared, puts a nonzero polynomial in x in the ideal (h_0,\ldots,h_N). Interchanging the variables gives a nonzero polynomial in y. The common zero set in the affine chart is finite. At each common zero, the ideal in centered formal coordinates contains powers of both coordinates and consequently has finite colength. There are four product affine charts, so the global base set B is finite.

If B=\varnothing, the map is already holomorphic, and S'=S, \mu=\id_S give the conclusion. Assume from now on that B\ne\varnothing.

It remains to resolve such a base point. Let

and let I\subset D be the ideal of a representing holomorphic tuple f_0,\ldots,f_N at the point. Assume I is proper and has finite colength, and let k=\min_\nu\operatorname{ord}(f_\nu)\ge1. Blow up the point. In a chart (x,z)\mapsto(x,xz), divide the pulled-back tuple by x^k; the divided functions are holomorphic. The other chart gives the same map, since the two tuples differ by a unit on their overlap.

On the exceptional projective line, the remaining base points are common zeros of the degree-k initial forms. At least one such form is nonzero, so there are finitely many. Fix one. A linear change of the old coordinates lets us place it at (x,z)=(0,0) in the displayed chart. Let

A power x^\ell belongs to I; applying the substitution and division gives x^{\ell-k}\in J. Thus 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. Its germ at the new base point is z^q times a unit, with 1\le q\le k. Therefore D'/(x) is spanned by 1,z,\ldots,z^{k-1}. Successively lifting modulo the powers of the nilpotent x shows that D' is spanned by finitely many classes x^az^b, with 0\le b<k.

There is a well-defined \CC-linear map

Indeed a monomial x^ay^b of total degree at least k maps to x^{a+b-k}z^b. All exponents are nonnegative, and the resulting orders tend to infinity with the original orders, so the operation is defined on arbitrary formal series. If h=\sum_\nu q_\nu f_\nu lies in I, its transform is \sum_\nu q_\nu(x,xz)x^{-k}f_\nu(x,xz)\in J. This proves descent to the quotient; no multiplicativity is needed.

Each spanning monomial x^az^b, b<k, is the image of x^{a+k-b}y^b\in\fm^k, so (A.1) is surjective. It follows that

In particular, the new base ideals have finite colength, and that positive integer decreases strictly along every surviving branch of the blowup process.

There are finitely many initial base points and finitely many children at each step. The maximum initial colength bounds the length of every branch by (A.2). Thus the entire tree is finite, and finitely many point blowups remove all base points. The local divided tuples define the desired holomorphic map.

Finally, a point blowup is proper, has a projective line over its center, and is an isomorphism elsewhere. The strict transform of a smooth curve remains smooth, and a new exceptional line meets the strict transforms of the components through the center. Induction gives smooth exceptional components and connected fibers of the composite.□

Proof of Lemma 4.1. Embed X in \PP^N. Every holomorphic map C=\PP^1\to\PP^N is represented by a tuple of homogeneous polynomials of a degree d\ge0, with no common zero. Indeed its pullback of \OO_{\PP^N}(1) is \OO_C(d), and its homogeneous coordinate sections supply such a tuple. For fixed d, these tuples form an open subset M_d of a finite-dimensional vector space. Openness follows from compactness of C. There is a universal holomorphic evaluation map C\times M_d\to\PP^N.

We cover the tuples whose maps land in X by countably many manifold families all of whose maps land in X. Start with the connected components of M_d. For a current connected submanifold M, either every tuple maps C into X, in which case accept this family, or there are \tau\in M and c\in C with G(c,\tau)\notin X. In the latter case

is a proper closed analytic subset of M, and contains every desired tuple in M. Use the covering observation to replace it by countably many lower-dimensional submanifolds, and repeat the same test on each. All desired tuples survive. Extra points introduced by the covering cause no problem: each new manifold is tested anew. Dimension strictly decreases whenever a family is not accepted. In dimension zero the test accepts precisely the desired points. Finite depth and countable branching give countably many accepted families covering all the desired tuples. On an accepted family the universal evaluation factors holomorphically through the embedded submanifold X.

Perform this construction for every d. The resulting countably many two-point evaluation maps to X\times X have images containing general pairs: a rational curve through two distinct points can be reparametrized to take those values at the fixed distinct points a,b. By the convention in the introduction, these images contain a nonempty Zariski open subset of X\times X. It is also open in the usual topology and has positive measure in smooth local coordinates.

Suppose, for a contradiction, that every one of these evaluation maps has complex rank everywhere less than 2\dim X. The image of each is then measure zero. Here is a direct verification of that fact for a holomorphic map \phi:M\to X\times X with the stated rank bound. We may take M connected. On the open set of maximum rank, the constant-rank theorem places the image locally in a submanifold of smaller dimension than X\times X. Countably many local charts give a measure-zero image. If the maximum rank is zero, this already treats all of M. Otherwise its complement is a proper closed analytic subset. Cover that subset by countably many lower-dimensional submanifolds and apply induction on \dim M. The strict rank bound persists on every such restriction, including the extra points of a covering submanifold. The zero-dimensional case is immediate. This proves the asserted measure-zero property.

A countable union of measure-zero images cannot contain that open subset. Some family therefore has a point where two-point evaluation is submersive. Restrict its parameter space to the nonempty open set where the differential is surjective; call the resulting manifold T.

Finally, a parameter tangent vector 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,w at a,b. Projecting with P_i gives the last assertion of Lemma 4.1.□