Emanuele Macri - Hyper-Kähler manifolds and Lagrangian fibrations

Abstract

A Hyper-Kähler manifold is a complex Kähler manifold that is simply connected, compact, and has a unique holomorphic symplectic form, up to constants. This important class of manifolds has been studied in the past in many contexts, from an arithmetic, algebraic, geometric point of view, and in applications to physics and dynamics.
The theory in dimension two, namely K3 surfaces, is well understood. The aim of this talk is to give an introduction to the theory of hyper-Kähler manifolds in higher dimension, from a point of view of their classification; in particular, about existence of Lagrangian fibrations. We will present some results in dimension four, obtained in collaboration with Olivier Debarre, Daniel Huybrechts and Claire Voisin.

Talk

$X$ will be compact Complex manifold. We assume that our Manifold can be embedded into a projective space ${\mathbb{P}}_{\mathbb{C}}^N$.

Say $X$ is irreducible, holomorphic symplectic. We call those Kähler mfds.

Question: Can we classify Kähler manifolds? Can we construct those mfds? We answer those questions using the structure of Lagrangian fibrations.

We look at a special class of manifolds in three dimensions, the K3. An example is a quartic surfacce embedded in ${\mathbb{P}}_{\mathbb{C}}^3$.

An important theorem is due to Kodaina 1964: All k3 surfaces are deformation equivalent. The proof uses deformation theory.

We give an example of a legrangian fibration which was given by Beauville-Fujiki

Keywords

• SYZ conjecture
• Abelian variety
• Compactification

Literature

• O’Grady, Debaure, Huybrechts, Macri, Voisin
• Kim, Laza, Martin

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