Projective manifolds with splitting tangent bundle. I.

*(English)*Zbl 1065.14054The 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.

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.

Reviewer: Olaf Teschke (Berlin)