Logarithmic Surfaces and Hyperbolicity

In 1981 J.Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: Theorem: In a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2, any Brody curve is algebraically degenerate. We also deal with the case of arbitrary logarithmic Kodaira dimension. As a corollary, we get hyperbolicity for such logarithmic surfaces not containing non-hyperbolic algebraic curves and having hyperbolically stratified boundary divisors. In particular we get the "best possible" result on algebraic degeneracy of Brody curves in the complex plane minus a curve consisting of three components, thus improving results of Dethloff-Schumacher-Wong from 1995.Comment: 34 pages. Final version, to appear in Annales Fourie

A characterization of finite quotients of Abelian varieties

In this paper we prove a characterization of quotients of Abelian varieties by the actions of finite groups that are free in codimension-one via some vanishing conditions on the orbifold Chern classes. The characterization is given among a class of varieties with mild singularities that are more general than quotient singularities, namely among the class of klt varieties. Furthermore we show that over a projective klt variety, any semistable reflexive sheaf with vanishing orbifold Chern classes can be obtained as the invariant part of a locally-free sheaf on a finite Galois cover whose associated vector bundle is flat.Comment: Added more details for the arguments in the final section. To appear in International Mathematics Research Notice

Logarithmic Jet Bundles and Applications

We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a metric proof for the logarithmic version of Lang's conjecture concerning the hyperbolicity of complements of divisors in a semi-abelian variety as well as for the corresponding big Picard theorem.Comment: 49 pages, Late