Quid on the Three Faces of Hyperbolicity: Kobayashi, Brody and Kähler Hyperbolicity

par M. Abdelouahab Khelifati (Institute de Mathematiques de Toulouse)

mercredi 18 nov. 2026, 18:00 → 19:30 Europe/Paris

Salle Pellos (207) (1R2)

Hyperbolicity is a central theme in complex geometry, with several apparently different ways of expressing that a complex manifold contains no “flat” complex directions. Starting from the Poincaré metric on the unit disc, we will introduce the Kobayashi metric, which measures the intrinsic size of holomorphic discs in a complex manifold. We will then turn to entire curves, namely holomorphic maps $\mathbb{C}\longrightarrow X$, and explain Brody’s theorem: for compact complex manifolds, the absence of non-constant entire curves is equivalent to Kobayashi hyperbolicity.

We will finally discuss Gromov’s notion of Kähler hyperbolicity. For a compact Kähler manifold, it is defined by requiring that the lift of a Kähler form to the universal cover admit a bounded primitive. We will explain the geometric meaning of this condition through an isoperimetric inequality and describe its relation with Kobayashi/Brody hyperbolicity in the compact setting.

The aim of the talk is to present these three notions as different perspectives on the same guiding question: when does the complex geometry of a manifold force it to behave like a negatively curved space?