A survey of Torelli and monodromy results for holomorphic-symplectic varieties

By E. Markman. 2011

282 citations recorded by the source corpus.

We survey recent results about the Torelli question for holomorphicsymplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup WExc, of the isometry group of the weight 2 Hodge structure, generated by reflection with respect to exceptional divisors. A description of the birational Kahler cone as a fundamental domain for the WExc action on the positive cone. A proof of a weak version of Morrison’s movable cone conjecture. A description of the moduli spaces of polarized holomorphic symplectic varieties as monodromy quotients of period domains of type IV.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d372ec2e7cee29c83c954b266e5bc0e1132a7bd6

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Citations in the corpus

Symplectic involutions on deformations of K3[2] (2011, methodology): “According to a global Torelli theorem, proved by Verbitsky in the case of hyperkähler manifolds considered in this paper, two marked hyperkähler manifolds having the same period are birational; see also [10, 12, 22].”
Ergodic complex structures on hyperkähler manifolds (2013, unknown): “romy transform mapsfor the Gauss-Manin local systems obtained from all deformations of (M,I) over a complex base ([V2, Deﬁnition 7.1]). This is how this group was originally deﬁned by Markman ([Ma1], [Ma2]). Remark 2.21: A caution: usually “the global Torelli theorem” is understood as a theorem abou”
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:”
Classification of automorphisms on a deformation family of hyper-Kähler four-folds by $p$-elementary lattices (2014, unknown): “We recall some well known facts from [20] and [24].”