PhD research topic - Simone Diverio
Complex-analytic, differential and algebraic geometry
Kobayashi hyperbolicity of algebraic varieties, positivity in Kähler geometry, special metrics on holomorphic vector bundles and consequences, Calabi-Yau manifolds.
My research area is, broadly speaking, complex analytic and algebraic geometry. This is the study of complex spaces or complex algebraic varieties from various points of view, in particular their algebro-geometric and differential-geometric properties.
One of the guiding lines of my research are Lang's celebrated conjectures which tries to relate the property of being Kobayashi hyperbolic (which is a complex analytic property) with the positivity properties of the canonical bundle (which are algebro-geometric in nature).
In doing so, several techniques and tools are needed; they essentially are borrowed from: birational geometry of complex algebraic or compact Kähler manifolds, positivity theory of complex vector bundles (both in the differential-geometric and in the algebro-geometric context), (semi)stability of complex vector bundles, characteristic classes, complex Monge-Ampère equations, L2 methods in several complex variables, and so on...
More specifically, begin Kobayashi hyperbolic for a compact complex manifold is characterized by the fact that there are no non constant holomorphic maps from the complex plane to the manifold. In particular, a projective algebraic manifold which is hyperbolic does not posses any rational or elliptic curve, nor, more generally, any non constant holomorphic image of a positive dimensional complex torus.
Instead, being of general type is a particular positivity property of the canonical bundle: it means that some big multiple of it is linearly equivalent to an ample plus an effective divisor (or, from a differential geometric point of view, that it carries a possibly singular hermitian metric whose curvature current is strictly positive). Easy examples of projective manifold of general type are given by smooth projective hypersurfaces of high degree.
Lang's conjectures predict that a complex projective manifold is hyperbolic if and only if it is of general type as well as all its subvarities.
 Jean-Pierre Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. Algebraic geometry—Santa Cruz 1995, 285–360, Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997.
 Serge Lang, Hyperbolic and Diophantine analysis. Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159–205.
 Claire Voisin, On some problems of Kobayashi and Lang; algebraic approaches. Current developments in mathematics, 2003, 53–125, Int. Press, Somerville, MA, 2003.
 Simone Diverio and Erwan Rousseau, The exceptional set and the Green-Griffiths locus do not always coincide. Enseign. Math. 61 (2015), no. 3-4, 417–452. (The introduction only)