Lionel Darondeau

Title (talk 1):

Jet techniques for hyperbolicity problems

Abstract:

Hyperbolicity is the study of the geometry of holomorphic entire curves $f\colon\mathbb{C} \to X$, with values in a given complex manifold $X$. In this introductory first talk, we will give some definitions and provide historical examples motivating the study of the hyperbolicity of complements $\mathbb{P}^n\setminus X_d$ of projective hypersurfaces $X_d$ having sufficiently high degree $d\gg n$. Then, we will introduce the formalism of jets, that can be viewed as a coordinate free description of the differential equations that entire curves may satisfy, and explain a successful general strategy due to Bloch, Demailly, Siu, that relies in an essential way on the relation between entire curves and jet differentials vanishing on an ample divisor.

Title (talk 2):

Torus action and Segre classes in the context of the Green-Griffiths conjecture

Abstract:

The goal of this second talk is to study the existence of global jet differentials. Thanks to the algebraic Morse inequalities, the problem reduces to the computation of a certain Chern number on the Demailly tower of projectivized jet bundles. We will describe the significant simplification due to Gergely Bérczi consisting in integrating along the fibers of this tower by mean of an iterated residue formula. Instead of the original argument coming from equivariant geometry (already explained by Bérczi himself at IMPANGA in 2010), we will explain our alternative proof of such a formula: essentially replacing the weights of the torus action by Chern roots, we will give a natural interpretation of the fiber integration formula in terms of Segre classes.