Complex torus

Abstract: These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995 The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi A convenient framework for this is the category of “directed manifolds”, that is, the category of pairs (X, V ) whereX is a complex manifold and V a holomorphic subbundle of TX If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C → X tangent to V (Brody’s criterion) We describe a construction of projectivized kjet bundles PkV , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature More precisely, πk : PkV → X is a tower of projective bundles over X and carries a canonical line bundle OPkV (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on OPkV (−1) for k large enough The direct images (πk)⋆OPkV (m) can be viewed as bundles of algebraic differential operators of order k and degree m, acting on germs of curves and invariant under reparametrization Following an approach initiated by Green and Griffiths, we establish a basic Ahlfors-Schwarz lemma in the situation when OPkV (−1) has a (possibly singular) metric of negative curvature, and we infer that every nonconstant entire curve f : C → V tangent to V must be contained in the base locus of the metric This basic result is then used to obtain a proof of the Bloch theorem, according to which the Zariski closure of an entire curve in a complex torus is a translate of a subtorus Our hope, supported by explicit Riemann-Roch calculations and other geometric considerations, is that the Semple bundle construction should be an efficient tool to calculate the base locus Necessary or sufficient algebraic criteria for hyperbolicity are then obtained in terms of inequalities relating genera of algebraic curves drawn on the variety, and singularities of such curves We finally describe some techniques introduced by Siu in value distribution theory, based on a use of meromorphic connections These techniques have been developped later by Nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3-space; thanks to a suitable concept of “partial projective projection” and the associated Wronskian operators, substantial improvements on Nadel’s degree estimate will be achieved here 2 J-P Demailly, Kobayashi hyperbolic projective varieties and jet differentials

Abstract: There are three types of “building blocks” in the Bogomolov decomposition [B, Th.2] of compact Kahlerian manifolds with torsion c1, namely complex tori, CalabiYau varieties, and irreducible symplectic manifolds. We are interested in the last type, i.e. simply-connected compact Kahlerian manifolds carrying a holomorphic symplectic form which spans H. (The holonomy of a Ricci-flat Kahler metric is equal to Sp(r), hence these manifolds are hyperkahler [B].) The stock of available irreducible symplectic manifolds appears to be quite scarce, expecially if we think of the many examples of CalabiYau’s. Every known irreducible symplectic manifold is a deformation of one of the following varieties: the Hilbert scheme parametrizing zero-dimensional subschemes of a K3 of fixed length [B], the generalized Kummer variety parametrizing zero-dimensional subschemes of a complex torus of fixed length and whose associated 0-cycle sums up to 0 [B], the (10-dimensional) desingularization of the moduli space of rank-two semistable torsion-free sheaves on a K3 with c1 = 0, c2 = 4 constructed by the author [O1]. Briefly: all known examples are deformations of an irreducible factor in the Bogomolov decomposition of a moduli space of semistable sheaves on a surface with trivial canonical bundle or, as in the last case, of a symplectic desingularization of such a moduli space. This paper provides a new example in dimension 6: the manifold in question is an irreducible factor in the Bogomolov decomposition of a symplectic desingularization of a moduli space of sheaves on an abelian surface. To put our result in perspective we recall some results on moduli spaces of sheaves on a surface with trivial canonical bundle. Let X be such a surface and D an ample divisor on it: given a vector w ∈ H∗(X;Z), we let Mw(X,D) be the moduli space of D-semistable torsion-free sheaves F on X with Mukai vector