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Article Dans Une Revue International Mathematics Research Notices Année : 2019

On the Period Map for Polarized Hyperkähler Fourfolds

Résumé

Abstract We study smooth projective hyperkähler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy’s Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW (Eisenbud–Popescu–Walter) sextics. In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperkähler fourfolds with Picard number 1 or 2.
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hal-03805869 , version 1 (17-02-2024)

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Olivier Debarre, Emanuele Macrì. On the Period Map for Polarized Hyperkähler Fourfolds. International Mathematics Research Notices, 2019, 2019 (22), pp.6887-6923. ⟨10.1093/imrn/rnx333⟩. ⟨hal-03805869⟩
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