an:05881837
Zbl 1218.14030
Campana, Fr??d??ric; Peternell, Thomas
Geometric stability of the cotangent bundle and the universal cover of a projective manifold
EN
Bull. Soc. Math. Fr. 139, No. 1, 41-74 (2011).
00277421
2011
j
14J40 32Q26 32J27 14E30
uniruledness; Kodaira dimension; pseudo-effective line bundle; rational ample class; semi-stable bundle
The 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 [\textit{N. I. Shepherd-Barron}, Ast??risque. 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 K??hler 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 \textit{C. Simpson} on jumping loci of numerically trivial line bundles [Ann. Sci. ??c. Norm. Sup??r. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)] and on \textit{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.
Eberhard Oeljeklaus (Bremen)
Zbl 0809.14034; Zbl 0798.14005; Zbl 0513.14019