## Breadcrumb

# Geometry and Physics Seminar: Emanuele Macri

## Topic

Curves on Irreducible Holomorphic Symplectic Varieties

## Details

The goal of the talk is to present derived category techniques to study holomorphic symplectic varieties. In particular, we study and answer the following questions:

(1) the Hassett-Tschinkel Conjecture on the structure of the Mori cone of curves;

(2) the Bogomolov-Tyurin-Hassett-Tschinkel-Huybrechts-Sawon Conjecture on the existence of Lagrangian fibrations;

(3) the Kawamata-Morrison Cone Conjecture.

Irreducible Holomorphic Symplectic varieties (IHS for short) are simply connected projective manifolds endowed with a unique (up to scalars) holomorphic symplectic form; K3 surfaces are the lowest dimensional example. In this talk we concentrate on IHS of K3^[n]-type, namely IHS deformation equivalent to the punctual Hilbert scheme on a K3 surface. After giving a short introduction to the basics of IHS theory, we will present recent joint work with Arend Bayer on how to prove (1), (2), and (3) for moduli spaces of sheaves on K3 surfaces, by using derived categories and Bridgeland stability. If time permits, I will also sketch how to extend these results to all IHS of K3^[n]-type, as recently proven by Bayer-Hassett-Tschinkel, Mongardi, Matsushita, Markman-Yoshioka, and Amerik-Verbitsky.

(1) the Hassett-Tschinkel Conjecture on the structure of the Mori cone of curves;

(2) the Bogomolov-Tyurin-Hassett-Tschinkel-Huybrechts-Sawon Conjecture on the existence of Lagrangian fibrations;

(3) the Kawamata-Morrison Cone Conjecture.

Irreducible Holomorphic Symplectic varieties (IHS for short) are simply connected projective manifolds endowed with a unique (up to scalars) holomorphic symplectic form; K3 surfaces are the lowest dimensional example. In this talk we concentrate on IHS of K3^[n]-type, namely IHS deformation equivalent to the punctual Hilbert scheme on a K3 surface. After giving a short introduction to the basics of IHS theory, we will present recent joint work with Arend Bayer on how to prove (1), (2), and (3) for moduli spaces of sheaves on K3 surfaces, by using derived categories and Bridgeland stability. If time permits, I will also sketch how to extend these results to all IHS of K3^[n]-type, as recently proven by Bayer-Hassett-Tschinkel, Mongardi, Matsushita, Markman-Yoshioka, and Amerik-Verbitsky.

## Additional Information

Location: ESB 4127
This seminar is broadcast live to UAlberta CAB Room 449

Emanuele Macri (Ohio State University)

Emanuele Macri (Ohio State University)

This is a Past Event

Event Type

**Scientific, Seminar**

Date

**March 24, 2014**

Time

5:00pm
- 7:00pm

Location

University of British Columbia