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1 December 2013 Mapping class group and a global Torelli theorem for hyperkähler manifolds
Misha Verbitsky
Duke Math. J. 162(15): 2929-2986 (1 December 2013). DOI: 10.1215/00127094-2382680


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 hyperkähler manifold M, showing that it is commensurable to an arithmetic lattice in SO(3,b23). A Teichmüller space of M is a space of complex structures on M up to isotopies. We define a birational Teichmüller 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 Teichmüller space and the corresponding period space SO(b23,3)/SO(2)×SO(b23,1). 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 n1 a prime power, our Torelli theorem implies the usual Hodge-theoretic birational Torelli theorem (for other examples of hyperkähler manifolds, the Hodge-theoretic Torelli theorem is known to be false).


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Misha Verbitsky. "Mapping class group and a global Torelli theorem for hyperkähler manifolds." Duke Math. J. 162 (15) 2929 - 2986, 1 December 2013.


Published: 1 December 2013
First available in Project Euclid: 28 November 2013

zbMATH: 1295.53042
MathSciNet: MR3161308
Digital Object Identifier: 10.1215/00127094-2382680

Primary: 53C26
Secondary: 32G13

Rights: Copyright © 2013 Duke University Press


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Vol.162 • No. 15 • 1 December 2013
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