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.