Integrability of split tangent bundles on rationally connected manifolds

1. Trivialize the complementary bundle on a factor curve

Take $F$ satisfying $F|_\Delta=g$ and $dF(T^{(i)})\subset F^*E_i$, where $T^{(i)}=\operatorname{pr}_i^*T_C$. The map may have critical points: a tangent box does not supply coordinates or a local product decomposition of $X$.

For $i\ne j$, $\xi\in E_j$ and $e\in E_i$, the formula $\nabla^i_\xi e=P_i[\xi,e]$ is tensorial in the direction and satisfies the Leibniz rule in $e$. This partial connection requires no integrability of either bundle. The tangency of $F$ pulls it back to a holomorphic connection on $F^*E_i$ relative to $\operatorname{pr}_i$. The construction does not invert $dF$ and remains defined at critical points.

On each $\mathbb P^1$ fiber its curvature is a holomorphic two-form and vanishes. Simple connectedness makes parallel transport path-independent. Transport starting on the diagonal varies holomorphically and gives the isomorphisms of Proposition 3.1:

Flatness here concerns the restriction to the curve fibers; no flat connection on all of $X$ has been produced.

Proposition 2.1 · p. 3 · Proposition 3.1; equation (3.2) · p. 10

2. Produce sections vanishing at one point and spanning at the other

Fix distinct $a,b\in C$. Lemma 4.1 supplies a nonempty family $G\colon C\times T\to X$ for which $(\mathrm{ev}_a,\mathrm{ev}_b)\colon T\to X\times X$ is a submersion. Appendix B constructs countably many families using polynomial presentations in projective space. Their two-point images contain general pairs. If all images had deficient rank, their union would have measure zero, so some family has full rank.

For $g=G(\cdot,\tau)$, differentiating in a parameter direction gives a global section of $g^*T_X$. Surjectivity of two-point evaluation, followed by projection with $P_i$, shows that global sections of $g^*E_i$ vanishing at $a$ have values spanning $E_{i,g(b)}$. The tangent boxes of the separate curves need not themselves be chosen as a holomorphic family.

Lemma 4.1 · p. 11 · Appendix B; Lemma 4.1 (proof) · pp. 14–15

3. Kill both brackets by reducing their values to constant sections

The obstruction to integrability is the holomorphic bundle morphism

On $L_i=\{s_j=b\}\subset S$, the map $\Theta_i$ identifies the input bundle with $g^*E_i$, while $\Theta_j$ identifies the target with $\mathcal O_C\otimes E_{j,g(b)}$. For global sections $u,v$ of $g^*E_i$ vanishing at $a$,

is constant. It vanishes at $a$ and therefore at $b$. At $(b,b)$, one has $F(b,b)=g(b)$ and the transport maps are the identity, so $B_i(g(b))(u(b)\wedge v(b))=0$. The spanning property gives $B_i(g(b))=0$. The image of $\mathrm{ev}_b$ is nonempty and open, and the identity principle on connected $X$ makes $B_i$ vanish everywhere. Applying the argument for both $i=1,2$ proves integrability of both specified subbundles.

Theorem 1.1 (proof) · p. 11

1. Solve a formal Cauchy problem using the projectors

With $u=s,v=t-s$ near the diagonal, the required tangencies are equivalent to

In $\widehat F=\sum f_k(u)v^k$, comparison of coefficients determines $(k+1)f_{k+1}$ from previous coefficients and their $u$ derivatives. Uniqueness glues the local solutions across coordinate changes. No convergence in the normal direction is asserted at this stage.

Lemma 2.2; equation (2.2) · pp. 4–5

2. Turn positivity of the diagonal’s normal bundle into a section bound

For a projective embedding $X\subset\mathbb P^N$, let $L$ be the formal pullback of the hyperplane bundle, with $L|_\Delta\simeq\mathcal O_C(d)$. Since $I/I^2\simeq\mathcal O_C(-2)$, the filtration by normal vanishing order of $M_m=L\otimes\widehat{\mathcal O}_S(m,m)$ yields

Pairs of ordinary polynomial sections form a space of dimension $2(m+1)^2$. This forces a nonzero relation $AS_0+BS_\nu=0$ between formal homogeneous coordinates, expressing $S_\nu/S_0=-A/B$ as a rational function. Choose a point of the diagonal where the leading denominator coefficients are nonzero. Canceling common powers of the normal parameter then gives a holomorphic map near that point with the same formal expansion. The manuscript does not invoke an external G3 theorem for this step.

Lemma 2.3; equations (2.3)–(2.4) · pp. 6–7 · Remark 2.4 · p. 8

3. The two tangencies force zero derivative on exceptional curves

Resolve the resulting rational map by finitely many point blowups $\mu\colon S'\to S$, obtaining $F'\colon S'\to X$. Appendix A proves finite termination through a decreasing local base-ideal colength. The tangencies give global identities

At a general point of an exceptional curve $H$, write $s\circ\mu-s(p)=r^\alpha h$ and $t\circ\mu-t(p)=r^\beta\widetilde h$. Cancel the common powers and restrict to $r=0$. Both $P_1dF'|_{T_H}$ and $P_2dF'|_{T_H}$ vanish, so $dF'|_{T_H}=0$. Every exceptional component maps to a point. Connectedness of the fibers of $\mu$ allows $F'$ to descend to $S$, and removability at isolated points makes the descended map holomorphic. The identity principle extends its diagonal value and tangencies globally.

Lemma A.1 · pp. 13–14 · Lemma 2.5; Proposition 2.1 (proof) · pp. 8–9

1. Apply the product theorem and realize the factors as projective slices

Höring’s Theorem 1.4 starts with a specified tangent splitting on a rationally connected projective manifold and requires at least one integrable summand. It produces $X\simeq X_1\times X_2$ compatible with both summands. The manuscript’s Theorem 1.1 supplies the integrability hypothesis. The factors are smooth and occur as closed complex submanifolds of $X$ by taking slices of the product. Restriction of a projective embedding of $X$ proves that each factor is projective.

Compatibility here concerns the specified holomorphic tangent subbundles; it does not assert an orthogonal product for a metric. The product of $X$ itself should also be distinguished from one obtained only after passing to a finite or universal cover.

[Hor07, Theorem 1.4, p. 2] · Corollary 1.2 · p. 2

1. Establish the hypotheses on the general fiber

Using Definition 2.9 and Theorem 2.10 in Höring’s published version, choose $Z$ over a regular value in the fibration locus. It is a smooth connected compact submanifold, inherits a Kähler form, and is rationally connected. Campana’s result recalled immediately after Definition 2.8 makes $Z$ projective.

Proposition 3.16 says that the ambient projectors $P_i$ preserve $T_Z$. Their restrictions are complementary holomorphic idempotents of constant rank on connected $Z$. Their images are therefore subbundles $W_i$ and $T_Z=W_1\oplus W_2$. If both ranks are positive, then $\dim Z\ge2$ and Theorem 1.1 applies. Otherwise the two summands are $0,T_Z$ and integrability already holds.

[Hor08, Definition 2.9; Theorem 2.10, print p. 1070 / PDF p. 5] · [Hor08, remark after Definition 2.8, print p. 1070 / PDF p. 5] · [Hor08, Proposition 3.16, print p. 1073 / PDF p. 8] · Theorem 1.1 · p. 1

2. Pass from integrability on general fibers to two fibrations

The integrability hypothesis of Höring’s Theorem 1.3 now holds. For both $i=1,2$, it gives an almost-holomorphic fibration with $T_{F_i}=W_i|_{F_i}$. The published convention permits the identity map, and its proof handles $W_i=0$ by $\operatorname{id}_X$. This case is treated separately from Theorem 1.1, which requires positive ranks. Integrability of $W_i$ does not imply integrability of the ambient $V_i$ in general.

[Hor08, Theorems 1.3–1.4, print p. 1068 / PDF p. 3] · [Hor08, Theorem 1.3 (proof), print p. 1074 / PDF p. 9]

3. Obtain all three alternatives under the rank-two hypothesis

If $\operatorname{rank}V_1=2$, the rank of $W_1$ is $2,1$, or $0$. Under the same integrability premise, Höring’s Theorem 1.4 gives, respectively, an analytic fiber bundle with $T_{X/Y}=V_1$, an equidimensional holomorphic map with rational-curve general fiber, or $T_Z\subset V_2|_Z$. The ranks are mutually exclusive and exhaustive, proving the “exactly one” assertion in Corollary 5.1.

[Hor08, Theorems 1.3–1.4, print p. 1068 / PDF p. 3] · Corollary 5.1 · pp. 12–13