FRG workshop, 2011

E. Markman

Title: Morrison's movable cone conjecture for irreducible holomorphic symplectic varieties


Let X be a smooth projective variety with a numerically trivial canonical line-bundle. The ample and movable cones of X can have infinitely many linear as well as circular boundary faces, and are often quite complicated. Morrison's cone conjectures state, roughly, that the ample cone is simple modulo the action of the automorphism group of X, and the movable cone is simple modulo the action of the group Bir(X) of birational automorphisms of X. A version of the movable cone conjecture in the title is derived as a corollary of the Global Torelli Theorem for irreducible holomorphic symplectic manifolds. As a consequence it is shown that for each non-zero integer d there are only finitely many Bir(X)-orbits of complete linear systems, which contain a reduced and irreducible divisor of Beauville-Bogomolov degree d. A similar finiteness result holds in degree zero as well.