MAPPING CLASS GROUP AND A GLOBAL TORELLI THEOREM FOR HYPERKAHLER MANIFOLDS

作者：Verbitsky, Misha^*; Markman, Eyal

来源：Duke Mathematical Journal, 2013, 162(15): 2929-2986.

DOI：10.1215/00127094-2382680

• IRREDUCIBLE SYMPLECTIC-MANIFOLDS
• COMPLEX ANALYTIC STRUCTURES
• K3 SURFACE
• MODULI SPACES
• KAHLER-MANIFOLDS
• COUNTER-EXAMPLE
• CY MANIFOLDS
• COMPACT
• SUBVARIETIES
• DEFORMATIONS

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hyperkahler manifold M, showing that it is commensurable to an arithmetic lattice in SO(3, b(2) - 3). A Teichmuller space of M is a space of complex structures on M up to isotopies. We define a birational Teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds. We show that the period map gives the isomorphism between connected components of the birational Teichmuller space and the corresponding period space SO(b(2) - 3, 3)/SO(2) x SO(b(2) - 3, 1). We use this result to obtain a Torelli theorem identifying each connected component of the birational moduli space with a quotient of a period space by an arithmetic group. When M is a Hilbert scheme of n points on a K3 surface, with n 1 a prime power, our Torelli theorem implies the usual Hodge-theoretic birational Torelli theorem (for other examples of hyperkahler manifolds, the Hodge-theoretic Torelli theorem is known to be false).

• 出版日期2013-12-1