Geometric stability of the cotangent bundle and the universal cover of a projective manifold.
(Stabilité géométrique du fibré cotangent et du recouvrement universel d’une variété projective.)

*(English. French summary)*Zbl 1218.14030The focus of the article lies on birational positivity properties of the cotangent bundle \(\Omega^1_X\) of a connected projective algebraic manifold \(X\). A key role for the considerations is taken by a general theorem about the stability of tensor products of torsion free sheaves on \(X\) which reads in the notation of the paper as follows: Let \(\alpha\) be a rational ample class in the closed cone of classes of movable curves on \(X\) and let \({\mathcal{E}}\) and \({\mathcal{F}}\) be \(\alpha\)-semi-stable torsion free sheaves on \(X\). Then \({\mathcal E}\otimes{\mathcal F}\)/torsion is again \(\alpha\)-semi-stable.

The algebraic proof of the authors is complemented by an analytic proof, given by M. Toma. The first applications deal with uniruledness criteria related to properties of \(\Omega^1_X\) and generalizations of a theorem of Y. Miyaoka which says that \(X\) is uniruled if \(\Omega^1_X\) is not generically nef. For a proof, see [N. I. Shepherd-Barron, Astérisque. 211, 103–114 (1992; Zbl 0809.14034)].

The authors prove that the torsion free coherent quotients \(\mathcal S\) of \((\Omega^1_X)^{\otimes m}\) have pseudo-effective determinants (i.e., \(c_1(\det\mathcal S)\) lies in the closure of the Kähler cone) if \(X\) is not uniruled. This result serves in the following as a tool for the proof of several interesting results and as a basis for the statement of conjectures about the geometric stability of \(\Omega^1_X\). Among others, it is shown that the Kodaira dimension \(\kappa(X)\) equals \(\kappa^+(X):=\max\{\kappa(\det\mathcal {I})\,|\, \mathcal {I}\) a saturated coherent subsheaf of \( \Omega^p_X, 1\leq p\leq\dim X\}\) if \(\kappa(X)\geq \dim X-3\) or if \(\kappa^+(X)=\dim X\). Moreover, \(\kappa(X)\) can be estimated from below by any numerically trivial line bundle \(L\in\) Pic\(^0(X)\) in the sense that \(\kappa(X,mK_X\otimes L)\leq\kappa(X)\), and if \(h^0(mK_X\otimes L)\not=0\) for some \(m\in\mathbb N\), then \(\kappa(X)>0\) or \(L\) is a torsion element.

The proof of this statement is based on the results of C. Simpson on jumping loci of numerically trivial line bundles [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361–401 (1993; Zbl 0798.14005)] and on E. Viehweg’s cyclic covers [Adv. Stud. Pure Math. 1, 329–353 (1983; Zbl 0513.14019)]. Other applications are related to the geometry of the universal cover \(\tilde X\) of \(X\), e.g., if there is no positive dimensional compact subvariety through a general point of \(\tilde X\) and if \(\chi(X,{\mathcal{O}}_X)\not=0\), then \(X\) is of general type.

The algebraic proof of the authors is complemented by an analytic proof, given by M. Toma. The first applications deal with uniruledness criteria related to properties of \(\Omega^1_X\) and generalizations of a theorem of Y. Miyaoka which says that \(X\) is uniruled if \(\Omega^1_X\) is not generically nef. For a proof, see [N. I. Shepherd-Barron, Astérisque. 211, 103–114 (1992; Zbl 0809.14034)].

The authors prove that the torsion free coherent quotients \(\mathcal S\) of \((\Omega^1_X)^{\otimes m}\) have pseudo-effective determinants (i.e., \(c_1(\det\mathcal S)\) lies in the closure of the Kähler cone) if \(X\) is not uniruled. This result serves in the following as a tool for the proof of several interesting results and as a basis for the statement of conjectures about the geometric stability of \(\Omega^1_X\). Among others, it is shown that the Kodaira dimension \(\kappa(X)\) equals \(\kappa^+(X):=\max\{\kappa(\det\mathcal {I})\,|\, \mathcal {I}\) a saturated coherent subsheaf of \( \Omega^p_X, 1\leq p\leq\dim X\}\) if \(\kappa(X)\geq \dim X-3\) or if \(\kappa^+(X)=\dim X\). Moreover, \(\kappa(X)\) can be estimated from below by any numerically trivial line bundle \(L\in\) Pic\(^0(X)\) in the sense that \(\kappa(X,mK_X\otimes L)\leq\kappa(X)\), and if \(h^0(mK_X\otimes L)\not=0\) for some \(m\in\mathbb N\), then \(\kappa(X)>0\) or \(L\) is a torsion element.

The proof of this statement is based on the results of C. Simpson on jumping loci of numerically trivial line bundles [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361–401 (1993; Zbl 0798.14005)] and on E. Viehweg’s cyclic covers [Adv. Stud. Pure Math. 1, 329–353 (1983; Zbl 0513.14019)]. Other applications are related to the geometry of the universal cover \(\tilde X\) of \(X\), e.g., if there is no positive dimensional compact subvariety through a general point of \(\tilde X\) and if \(\chi(X,{\mathcal{O}}_X)\not=0\), then \(X\) is of general type.

Reviewer: Eberhard Oeljeklaus (Bremen)

##### MSC:

14J40 | \(n\)-folds (\(n>4\)) |

32Q26 | Notions of stability for complex manifolds |

32J27 | Compact Kähler manifolds: generalizations, classification |

14E30 | Minimal model program (Mori theory, extremal rays) |