Fano manifolds and stability of tangent bundles. (English) Zbl 1472.14044
Let \(X\) be a Fano manifold, that is a complex projective manifold such that the anticanonical divisor \(-K_X\) is ample. If the Picard number of \(X\) is one, a widely believed folklore conjecture claims that the tangent bundle \(T_X\) is semistable (in the sense of Mumford-Takemoto). In this paper the author gives a series of counterexamples to this conjecture!
The counterexamples are obtained by a family of horospherical varieties classified by B. Pasquier [Math. Ann. 344, No. 4, 963–987 (2009; Zbl 1173.14028)]: for these manifolds the action of the group \(\mbox{Aut}^0(X)\) on \(X\) has two orbits, the open orbit \(X^0\) and a closed orbit \(Z\). Moreover the action on the blow-up \(\mbox{Bl}_Z X\) again has two orbits, the open orbit \(X^0\) and the exceptional divisor \(E\). The manifold \(\mbox{Bl}_Z X\) admits a smooth fibration onto a lower-dimensional manifold \(Y\), the push-forward of the relative tangent bundle to \(X\) defines an algebraically integrable foliation \(\mathcal F \subset T_X\). The author shows that this foliation is canonical in the sense that it is the unique algebraically integrable foliation on \(X\) that is \(\mbox{Aut}^0(X)\)-invariant. General arguments show that the stability of \(T_X\) can be verified by computing the slope of the foliation \(\mathcal F\). It turns out that for infinitely many manifolds in Pasquier’s list, the subsheaf \(\mathcal F \subset T_X\) destabilises the tangent bundle. The reviewer recommends to any complex geometer to read this beautiful paper.
The counterexamples are obtained by a family of horospherical varieties classified by B. Pasquier [Math. Ann. 344, No. 4, 963–987 (2009; Zbl 1173.14028)]: for these manifolds the action of the group \(\mbox{Aut}^0(X)\) on \(X\) has two orbits, the open orbit \(X^0\) and a closed orbit \(Z\). Moreover the action on the blow-up \(\mbox{Bl}_Z X\) again has two orbits, the open orbit \(X^0\) and the exceptional divisor \(E\). The manifold \(\mbox{Bl}_Z X\) admits a smooth fibration onto a lower-dimensional manifold \(Y\), the push-forward of the relative tangent bundle to \(X\) defines an algebraically integrable foliation \(\mathcal F \subset T_X\). The author shows that this foliation is canonical in the sense that it is the unique algebraically integrable foliation on \(X\) that is \(\mbox{Aut}^0(X)\)-invariant. General arguments show that the stability of \(T_X\) can be verified by computing the slope of the foliation \(\mathcal F\). It turns out that for infinitely many manifolds in Pasquier’s list, the subsheaf \(\mathcal F \subset T_X\) destabilises the tangent bundle. The reviewer recommends to any complex geometer to read this beautiful paper.
Reviewer: Andreas Höring (Nice)
MSC:
| 14J45 | Fano varieties |
| 32Q26 | Notions of stability for complex manifolds |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
Citations:
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