A global Torelli theorem for hyperkahler manifolds

By Misha Verbitsky. 2009

240 citations recorded by the source corpus.

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 hypekahler manifold $M$, showing that it is commensurable to an arithmetic subgroup 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, and show that the period map gives an isomorphism of the birational Teichmuller space and the corresponding period space $SO(b_2-3, 3)/SO(2)\times SO(b_2 -3, 1)$. We use this result to obtain a Torelli theorem identifying any connected component of birational moduli space with a quotient of a period space by an arithmetic subgroup. 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).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a00506d033dd2631d02166934dfa31c5822332d8

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Citations in the corpus

On two rationality conjectures for cubic fourfolds (2014, methodology): “Thanks to the global Torelli theorem for hyperkähler manifolds [10, 15, 19] we can now prove a more complete result:”
Lagrangian Fibrations of Holomorphic-Symplectic Varieties of K3[n]-Type (2013, unknown): “(The Global Torelli Theorem [Ver1, Hu3] ) The period map P 0 ∶ M 0Λ → Ω Λ is surjective.”
Fano varieties of cubic fourfolds containing a plane (2009, unknown): “d be deduced from [11, Thm. 1.0.3]. This would probably allow to avoid completely the argument in Section 3.2. Moreover, [12] in the last part of the previous argument could be replaced by the recent [25]. We stick here to our elementary approach. As an application of the remarks at the beginning of”
The global moduli theory of symplectic varieties (2018, background): “ is irreducible with respect to their decomposition theorem. arXiv:1812.09748v1 [math.AG] 23 Dec 2018 2 BENJAMIN BAKKER AND CHRISTIAN LEHN very rich. In particular, Verbitsky’s global Torelli theorem [Ver13] gives a precise description of the global deformations of a symplectic manifold in terms of ”
Lectures on K3 Surfaces (2016, methodology): “It is inspired by the approach used in higher dimensions, see the Bourbaki talk [257] on Verbitsky’s paper [609].”
Symplectic involutions on deformations of K3[2] (2011, unknown): “ σX is a symplectic 2-form. The period map is surjective and it is a local isomorphism, moreover a global Torelli theorem was proven by Verbitsky in our and in several other cases, see [12], [10] and [22]. Whenever this theorem holds it states that two marked Hyperkahler manifolds having the same pe”