Hyperbolicity

In a broad sense, complex hyperbolicity aims at understanding the geometry of entire curves in a given complex manifold. We say roughly that a complex manifold is “complex hyperbolic” if it does not contain any such entire curves. The study of this notion has attracted the interest of many complex geometers during the last century (among many others, we can mention Picard, Bloch, H. Cartan…). Nowadays, it continues to be a very active field of research, and the last few years have lead to quite spectacular developments. In particular, a lot of work has been devoted to the Green-Griffiths-Lang conjecture, which would relate the notion of complex hyperbolicity to more algebro-geometric or arithmetic properties of complex manifolds.

In this lecture, we will start by introducing the basic notions of complex hyperbolicity: we will see that there are several non-equivalent ways of quantifying the absence of entire curves. An important result we will prove is that they all amount to the same in the compact case. We will then state the Green-Griffiths-Lang conjecture, and describe some of the tools that were introduced to approach it: our presentation will mainly revolve around metric methods and jet differential techniques. All along the way, we will present several illustrating examples, as well as recent important applications of the techniques we will discuss.

The Green–Griffiths–Lang Conjecture relates some transcendental and algebraic properties of complex projective varieties. In particular, such a variety X should be complex hyperbolic if and only if all its subvarieties are of general type. Campana has introduced a notion of special type that allows to give a clear conjectural picture of the situation. The conjecture is widely open in higher dimensions.

In the special case where X is a smooth projective hypersurface, there is deep connections  between this conjecture and a famous conjecture of Kobayashi, assertingthat if X is generic of large degree, it should be hyperbolic (the optimal degree should be linear in the dimension). This conjecture has recently been established by Brotbek for very high degrees. We will (quickly) review some other important results of the literature supporting the conjectures.

1) GGL => Kobayashi for very generic hypersurfaces (Clemens, Ein, Voisin, Pacienza)

2) general type <=> existence of differential equations satisfied by all entire curves (Demailly, Campana–Paun)

3) GGL for generic projective hypersurfaces of high degrees (Diverio–Merker–Rousseau)

4) GGL for generic projective hypersurfaces via jets => Kobayashi for very generic hypersurfaces (Riedl–Yang)

I will discuss joint work with Bram Petri in which we use the Selberg trace formula and numerical optimization to prove new upper bounds on four geometric invariants associated to hyperbolic surfaces: the systole, the kissing number, the first positive eigenvalue of the Laplacian, and the multiplicity of that eigenvalue. Our method is inspired by work of Cohn and Elkies on the density of sphere packings in Euclidean spaces. We combine this approach with other techniques to prove that the Klein quartic maximizes the multiplicity of the first eigenvalue among closed hyperbolic surfaces of genus 3.

In Teichmüller theory, dynamical hyperbolicity results are typically phrased in terms of the Hodge norm, and geometric hyperbolicity results in terms of the Teichmüller metric. I’ll explain what these objects are, give examples of known results, and discuss recent joint work with Jeremy Kahn that gives fresh hope for a more unified perspective.