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Projective manifolds with splitting tangent bundle. I. (English) Zbl 1065.14054
The authors investigate complex projective manifolds $$X$$ whose tangent bundle splits as a sum of vector bundles $$T_X=\bigoplus E_i$$. This is motivated by a fundamental theorem of de Rham: If for a complete Riemannian manifold $$X$$, $$T_X$$ splits into a direct sum of subbundles invariant under the holonomy, then the universal cover $$\tilde X$$ of $$X$$ splits as a product of Riemann manifolds and this splitting is compatible with that of $$T_X$$. The same result holds for complex Kähler manifolds.
As a natural generalization, one may ask whether there is such splitting of $$\tilde X$$ when the assumptions on holonomy and integrability are dropped. First results were given by A. Beauville [in: Complex analysis and algebraic geometry. A volume in memory of Michael Schneider. 61–70 (2000; Zbl 1001.32010)], who proved that if $$\dim X=2$$ or $$X$$ is Kähler-Einstein, then the problem has a positive answer, and that in general the answer is negative (for example, Hopf manifolds provide easy counterexamples). S. Druel [J. Reine Angew. Math. 522, 161–171 (2000; Zbl 0946.14005)] proved the claim in the case of projective manifolds where $$T_X$$ splits into a direct sum of line bundles $$L_i$$, under the assumption that either $$K_X$$ is nef, or all subbundles $$\bigoplus_k L_{i_k}$$ are integrable.
In the paper under review, the authors consider the case of projective threefolds where one can split off a single line bundle summand from $$T_X$$, and Fano manifolds. For threefolds, Mori theory is used to end up with a fundamental dichotomy – either $$X$$ is uniruled admitting a Mori contraction, or $$K_X$$ is nef. They also ask, if the summands are integrable, whether the decomposition of $$T_{\widetilde X}$$ is induced by the decomposition of $$T_X$$ (this situation is called “diagonal splitting”). The main result is the following
Proposition. Let $$X$$ be a smooth projective threefold with a splitting $$T_X=L\oplus V$$, where $$L$$ is a line bundle and $$\phi:X\to Y$$ is a birational contraction in the sense of Mori theory; i.e., $$-K_X$$ is $$\phi$$-ample. Then $$Y$$ is smooth, $$\phi$$ is the blow-up along a smooth curve; $$L'=\phi_*(L)$$ and $$V'=\phi_*(V)^{\vee\vee}$$ are locally free with $$L=\phi^*(L')$$ such that $$T_Y=L'\oplus V'$$. Moreover, if the universal cover $$\widetilde Y$$ of $$Y$$ splits diagonally with respect to $$T_Y=L'\oplus V'$$, then $$\widetilde X$$ splits diagonally with respect to $$T_X=L\oplus V$$.
Both cases are further investigated. In the first one, the authors prove the following
Theorem. Let $$X$$ be a smooth projective threefold with $$\kappa(X)=-\infty$$ and $$T_X=L\oplus V$$. Then $$\widetilde X$$ splits. The splitting is diagonal with respect to $$T_X=L\oplus V$$ unless the following holds: $$X$$ is a successive blow-up along smooth curves in a smooth projective threefold $$Y$$, the splitting $$T_X=L\oplus V$$ induces canonically a splitting $$T_Y=L'\oplus V'$$ and there is a $$\mathbb{P}^1$$-bundle structure $$\psi\:Y\to Z$$ such that $$L'=T_{Y/Z}$$.
Along with this theorem, several results on projective bundles in any dimension are shown.
In the second case, Beauville’s results cover the case $$c_1(X)=0$$, and $$K_X$$ ample. The authors concentrate to the case that $$K_X$$ is nef, big (the case of $$\kappa(X)=1,2$$ is announced to be the subject of the forthcoming second part) and not ample.
Theorem. Let $$X$$ be a smooth projective threefold with $$K_X$$ big and nef and let $$\phi:X\to Y$$ be the canonical model.
(1) The universal cover $$\widetilde Y$$ of $$Y$$ is of the form $$\widetilde Y\simeq\Delta\times S$$ with $$\Delta\subset C$$ the unit disc and $$S$$ a surface with only rational double points as singularities.
(2) If $$g:\widehat S\to S$$ denotes the minimal resolution, then the universal cover is of the form $$\widetilde X\simeq\Delta\times\widehat S$$ and $$\phi\simeq\text{id}_\Delta\times g$$. This decomposition is compatible with $$T_X=L\oplus V$$.
For arbitrary dimension, the authors study the Fano case, where it is sufficient (since $$\pi_1(X)=0$$) to consider 2-factor splittings.
Theorem. Let $$X$$ be a Fano $$n$$-fold. Assume that $$n\leq5$$ or that every contraction of an extremal ray contracts a rational curve such that for its normalisation $$f: \mathbb{P}^1\to X$$ one has $$f^*(T_X)={\mathcal O}(2)\oplus\bigoplus{\mathcal O}(a_i),\;a_i\leq1$$. If $$T_X=E_1\oplus E_2$$, then $$X\simeq Z_1\oplus Z_2$$ diagonally.

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14E30 Minimal model program (Mori theory, extremal rays) 14J30 $$3$$-folds
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