## University of Sydney Algebra Seminar

# Justin Sawon (Colorado State University)

## Friday 17th July, 12.05-12.55pm, Carslaw 375

### Applications of twisted Fourier-Mukai transforms

A Fourier-Mukai transform is an equivalence between the derived categories of coherent sheaves on different spaces X and Y. Mukai originally studied such transforms for abelian varieties and K3 surfaces. Geometrically, a Fourier-Mukai transform may induce a birational map between moduli spaces of stable sheaves on X and Y, which is sometimes even an isomorphism.

Often there is a gerbey obstruction to the existence of a Fourier-Mukai transform; Caldararu overcame this obstacle by incorporating the gerbe into the transform by way of twisted sheaves. In this talk, I will describe two applications of twisted Fourier-Mukai transforms to holomorphic symplectic geometry. The first proves the existence of fibrations on Hilbert schemes Hilb^nS of certain K3 surfaces S. The second gives a relative Fourier-Mukai transform between families of abelian varieties - the twist arises because these families may be non-trivial `torsors'.