Extremal Rays and Automorphisms of Holomorphic Symplectic Varieties

For last fifteen years, numerous authors have studied the birational geometry of projective irreducible holomorphic symplectic varieties X, seeking to relate extremal contractions X → X ′ to properties of the Hodge structures on H(X,Z) and H2(X,Z), regarded as lattices under the Beauville-Bogomolov form. Significant contributions have been made by Huybrechts, Markman, O’Grady, Verbitsky, and many others [Huy99], [Mar08], [O’G99], [Ver13], see also [Huy11]. The introduction of Bridgeland stability conditions by Bayer and Macr̀ı provided a conceptual framework for understanding birational contractions and their centers [BM14a, BM14b]. In particular, one obtains a transparent classification of extremal birational contractions, up to the action of monodromy, for varieties of K3 type [BHT13]. Here we elaborate the Bayer-Macr̀ı machinery through concrete examples and applications. We start by stating the key theorem and organizing the resulting extremal rays in lattice-theoretic terms; see Sections 2 and 3. We describe exceptional loci in small-dimensional cases in Sections 4 and 5. Finding concrete examples for each ray in the classification can be computationally involved; we provide a general mechanism for writing down Hilbert schemes with prescribed contractions in Section 6. Then we turn to applications. Section 7 addresses a question of Oguiso and Sarti on automorphisms of Hilbert schemes. Finally, we show that the ample cone of a polarized variety (X, h) of K3 type cannot be read off from the Hodge structure on H(X,Z) in Section 8; this resolves a question of Huybrechts.