Half twists and the cohomology of hypersurfaces

Abstract

A Hodge structure V of weight k on which a CM field acts defines, under certain conditions, a Hodge structure of weight k − 1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which defines an action of a cyclotomic field on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of the hypersurfaces. We use our results to prove the existence of a Kuga-Satake correspondence for certain cubic 4-folds.

Families of Motives and the Mumford–Tate Conjecture

“…See also [101], Section 5. Our formulation of this construction is the one we introduced in [55], Section 7; see also [56], Section 3.…”

Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

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Let M d,n be the moduli stack of hypersurfaces X ⊂ P n of degree d ≥ n + 1, and let M (1) d,n be the sub-stack, parameterizing hypersurfaces obtained as a d-fold cyclic covering of P n−1 ramified over a hypersurface of degree d. Iterating this construction, one obtains M (ν) d,n . Bert van Geemen found an error in the first version of Section 8, and he pointed out the relation between our Sections 7 and 8 the Sections 3 and 6 of [13]. However, for all d ≥ n + 1 the sub-stack M (2) d,n deforms.

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“…There it is shown that M (3) 5,4 has a natural structure of a ball quotient (see Remark 8.7) and that ς(M (3) 5,4 ) ≤ 2. As a byproduct of the calculation of variations of Hodge structures for families in M (2) d,n (Section 7, see also [13], Section 3) we show that for a universal family g : Z → S in M (3) 5,4 the set of CM-points is dense in S, i.e. the set of points s ∈ S where the fibre g −1 (s) has complex multiplication (see Section 8).…”

A family of cubic fourfolds with finite-dimensional motive

“…This subsection presents the first main ingredient of this note: the van Geemen-Izadi construction of an algebraic Kuga-Satake correspondence for the cubic fourfolds under consideration. Theorem 2.4 (van Geemen-Izadi [19]). Let X ⊂ P 5 be a smooth cubic fourfold, defined by an equation…”

“…Unlike the Fermat cubic, the cubics as in theorem 3.1 are not obviously dominated by a product of curves, so we need some more indirect reasoning. In a nutshell, the idea of the proof of theorem 3.1 is as follows: thanks to the work of van Geemen-Izadi [19], there exists a Kuga-Satake correspondence for these special cubic fourfolds. This implies that the homological motive of X is a direct summand of the motive of an abelian variety.…”