Orbifold Aspects of Certain Occult Period Maps

Abstract

We first characterize the automorphism groups of Hodge structures of cubic threefolds and cubic fourfolds. Then we determine for some complex projective manifolds of small dimension (cubic surfaces, cubic threefolds, and non-hyperelliptic curves of genus 3 or 4), the action of their automorphism groups on Hodge structures of associated cyclic covers, and thus confirm conjectures made by Kudla and Rapoport in [KR12].

Automorphisms and periods of cubic fourfolds

“…This was subsequently verified by the first author [32] and Looijenga [36]. More recently, the second author [55] proved a stronger version of the Torelli theorem: the automorphisms of cubic fourfolds are detected by (polarized) Hodge isometries.…”

“…This is equivalent to the strong global Torelli theorem, i.e., the statement that any isomorphism between the polarized Hodge structures of two smooth cubic fourfolds is induced by a unique isomorphism between the two cubic fourfolds. Using the fact that automorphisms of cubic fourfolds X are induced by linear transformations of the ambient projective space P 5 , and that Aut(X ) acts faithfully on the middle cohomology H 4 (X , Z) (e.g., [28,Proposition 2.16]), the second author [55] has verified the Strong Global Torelli Theorem. Proposition 2.4 [55] Let X 1 and X 2 be two smooth cubic fourfolds.…”

“…Using the fact that automorphisms of cubic fourfolds X are induced by linear transformations of the ambient projective space P 5 , and that Aut(X ) acts faithfully on the middle cohomology H 4 (X , Z) (e.g., [28,Proposition 2.16]), the second author [55] has verified the Strong Global Torelli Theorem. Proposition 2.4 [55] Let X 1 and X 2 be two smooth cubic fourfolds. Assume that there is an isomorphism ϕ :…”

Complete moduli of cubic threefolds and their intermediate Jacobians

“…It is more natural to study cubic threefolds via the period map used by Clemens and Griffiths, associating to a cubic threefold its intermediate Jacobian, which is a principally polarized abelian fivefold. Recall that they prove the Torelli theorem for cubic threefolds: the map from the moduli space of cubic threefolds to the moduli space of principally polarized abelian varieties of dimension 5 is an injective immersion (the local and global Torelli Theorems hold and the automorphism groups coincide, see [24, (0.11); 33, Theorem 9.8(b); 70, Theorem 5.1]). It is thus natural to compactify this map, that is, to study degenerations of intermediate Jacobians.…”

Cubic Fourfolds with an Involution

“…One can further adapt these techniques to the case of automorphisms of a cubic fourfold X. The Strong Torelli theorem ([Voi86] [Zhe19]) asserts that automorphisms of X are equivalent to (polarized) Hodge isometries of the middle cohomology H 4 (X, Z). Such an isometry in turn determines the lattice of algebraic primitive cycles A(X) prim := H 2,2 (X, C) ∩ H 4 (X, Z) prim contained in the cubic X.…”