Anibal Aravena: The Fano visitor problem for K3 surfaces

A smooth projective variety  is called a Fano visitor if its bounded derived category of coherent sheaves can be embedded into the derived category of a smooth Fano manifold. Motivated by homological mirror symmetry, Bondal posed the question if every smooth projective variety is a Fano visitor. In this talk, I will try to answer this question  for a general K3 surface.  Our proof is a consequence of several results concerning a sequence of flips associated to the K3 surface. Its construction combines the work of Bayer and Macrì on the description of the birational geometry of a moduli space of sheaves on a K3 surface through Bridgeland stability conditions, and the study of the fixed locus of antisymplectic involutions on hyperkähler manifolds by Saccà, Macrì, O'Grady, and Flapan.