A note on Lang's conjecture for quotients of bounded domains

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains. Comment: 10 pages, no figures, comments are welcome. v3: following suggestions made by the referee, the exposition has been improved all along the paper, we added a variant of Theorem A which includes manifolds whose universal cover admits a bounded psh function which is strictly psh just at one point, and we added a section of examples. Final version, to appear on \'Epijournal G\'eom. Alg\'ebrique

ON PROJECTIVE MANIFOLDS WITH PSEUDO-EFFECTIVE TANGENT BUNDLE

In this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

Strictly nef vector bundles and characterizations of ℙ n

In this note, we give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.

NOTES ON VANISHING CYCLES AND APPLICATIONS

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

Deformations of the tangent bundle of projective manifolds

We investigate when the tangent bundle of a projective manifold has a nontrivial first-order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.