# Collapsing geometry of hyperkaehler four-manifolds

##
Ruobing Zhang

created by daniele on 20 Sep 2021

modified on 08 Nov 2021

16 nov 2021
-- 14:30

DIMaI, Firenze (online)

Seminari di Geometria Differenziale e Analisi Complessa del Dipartimento di Matematica e Informatica "Ulisse Dini" dell'Università di Firenze

**Abstract.**

This talk focuses on the recent resolution of the following three well-known conjectures in the study of Ricci-flat four manifolds (joint with Song Sun).

(1) Any volume collapsed limit of unit-diameter hyperkaehler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3-torus by an involution, a singular special Kaehler metric on the 2-sphere, or the unit interval.
(2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature must have one of the following asymptotic model geometries: ALE, ALF, ALG, ALH, ALG** and ALH**.
(3) Any gravitational instanton is holomorphic to an open dense subset of some compact algebraic surface.

With the above classification results, we obtain a rather complete picture of the collapsing geometry of hyperkaehler four manifolds.