The period map for cubic fourfolds

By E. Looijenga. 2007

108 citations recorded by the source corpus.

The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding. An algebraic version of our main result is an identification of the algebra of SL (6,ℂ)-invariant polynomials on the representation space Sym 3(ℂ6)* with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20. We also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a Vinberg-Dynkin diagram.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b3de767ba979f88692254d19eabb7feaeaf57095

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces (2018, methodology): “ present the recent proof of the Torelli theorem for cubic fourfolds by Huybrechts and Rennemo [HR19]. This result was originally proved by Voisin in [Voi86] and subsequently reproved by Loojienga in [Loo09]. Based on the Torelli theorem for hyperka¨hler manifolds [Ver13], Charles [Cha12] gave an el”
Hodge theory and derived categories of cubic fourfolds (2012, unknown): “Choose a 21-dimensional positive definite real subspace Then by Laza and Looijenga’s description of the image of the period map [28, Thm. 1.1], [31, Thm. 3.1], the condition (4.5) guarantees the existence of a cubic X and an isomorphism L ∼ = H 4 ( X, Z ) preserving h 2 and taking σ to a generator…”
Stability conditions on Kuznetsov components (2017, background): “Later Loojienga provided another proof in [Loo09] while describing the image of the period map.”
Cubic Fourfolds, K3 Surfaces, and Rationality Questions (2016, background): “§4.4] and the cubic fourfolds with an ordinary double point [Has00, §4.2]. The non-special cohomology lattice K⊥ d is also well-deﬁned for all (X,Kd) ∈ Cd. Laza [Laz10], building on work of Looijenga [Loo09], gives precise necessary and suﬃcient conditions for when the CM in Proposition 12 are nonem”
Remarks on degenerations of hyper-K\"ahler manifolds (2017, methodology): “We are going to use here a different degeneration which was introduced by Hassett [31], and plays an important role in [49], [53].”
The integral Hodge conjecture for two-dimensional Calabi–Yau categories (2020, methodology): “1] uses Laza and Looijenga’s description of the image of the period map for cubic fourfolds [60, 63] to show that X is deformation equivalent within the Hodge locus for v to a cubic fourfold containing a plane.”