Kummer Rigidity for Irreducible Holomorphic Symplectic Manifolds
Date:
A talk in Online Seminar on Algebraic and Complex Dynamics.
Abstract
In the preliminary part, we will go through the notion of irreducible holomorphic symplectic (IHS) manifolds, which is one of the high-dimensional generalizations of K3 surfaces. The action of holomorphic automorphisms on IHS manifolds is well-known, in terms of cohomology actions and characteristic currents (i.e., Green currents) that the automorphism induces. We will briefly go through the known theory, together with an easy, computable example originating from Arnold’s Cat map.
The main talk will discuss studying holomorphic automorphisms on IHS manifolds that have the volume-class Green measures. The tools are analogous to those for the K3 surfaces, as seen in recent research by Cantat, Dupont, Filip, and Tosatti. We will see which part of the arguments for the K3 surface can be generalized easily, and which part faces some difficulties. I will present the current status of overcoming each, and for which assumptions it is known that such automorphisms originate from a toral affine map (i.e., is ‘Kummer’).