Ruxandra Moraru : Holomorphic symplectic manifolds

Posted by Laura Schaposnik , part of the Departmental Colloquium.

At

April 26, 2024, 3 p.m.

In

636 SEO

Abstract

A holomorphic symplectic manifold is a complex manifold $X$ together with a closed, non-degenerate holomorphic 2-form $\Omega$. The top power of $\Omega$ gives a trivialisation of the canonical bundle so that $X$ has trivial first Chern class. In the context of K\”ahler geometry, such manifolds play a very important role due to the Bogomolov covering theorem, which states that any compact Kahler manifold with vanishing first Chern class has a covering that splits as the product of Calabi–Yau manifolds, complex tori and irreducible holomorphic symplectic manifolds. Among these, the last two are, in fact, compact holomorphic symplectic manifolds. Furthermore, irreducible holomorphic symplectic manifolds correspond to compact hyperk\”ahler manifolds in the K\”ahler setting. In general, finding compact holomorphic symplectic manifolds is very difficult. In this talk, I will present some classical constructions of holomorphic symplectic manifolds in various geometric contexts and discuss some recent developments in finding compact examples in the non-Kahler setting.

Please email schapos@uic.edu if you'd like to join for a meal.