Modular forms and special cubic fourfolds

We study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ADE singularities. In this paper, we show that the generating series of the degrees of such divisors is a level three modular form.     

A remark on the nonrationality problem for generic cubic fourfolds

It is proved that the nonrationality of a generic cubic fourfold follows from a conjecture on the nondecomposability in the direct sum of nontrivial polarized Hodge structures of the polarized Hodge structure on transcendental cycles on a projective surface.     

Algebraic cycles and very special cubic fourfolds

Informed by the Bloch–Beilinson conjectures, Voisin has made a conjecture about 0-cycles on self-products of Calabi–Yau varieties. In this note, we consider variant versions of Voisin’s conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.     

The period map for cubic fourfolds

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.     

On the Chow groups of certain cubic fourfolds

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.     

Computing the arithmetic genus of Hilbert modular fourfolds

The Hilbert modular fourfold determined by the totally real quartic number field is a desingularization of a natural compactification of the quotient space , where PSL acts on by fractional linear transformations via the four embeddings of into . The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight , is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results.

     

New fourfolds from F‐theory

In this paper, we apply Borcea–Voisin's construction and give new examples of fourfolds containing a del Pezzo surface of degree six, which admit an elliptic fibration on a smooth threefold. Some of these fourfolds are Calabi–Yau varieties, which are relevant for the compactification of Type IIB string theory known as F‐theory. As a by‐product, we provide a new example of a Calabi–Yau threefold with Hodge numbers .     

Smooth surfaces in smooth fourfolds with vanishing first Chern class

According to a conjecture attributed to Hartshorne and Lichtenbaum and proven by Ellingsrud and Peskine [18], the smooth rational surfaces in ${\mathbb{P}}^4$ belong to only finitely many families. We formulate and study a collection of analogous problems in which ${\mathbb{P}}^4$ is replaced by a smooth fourfold X with vanishing first integral Chern class. We embed such X into a smooth ambient variety and count families of smooth surfaces which arise in X from the ambient variety. We obtain various finiteness results in such settings. The central technique is the introduction of a new numerical invariant for smooth surfaces in smooth fourfolds with vanishing first Chern class.     

Topological properties of Hilbert schemes of almost-complex fourfolds (I)

In this article, we study topological properties of Voisin??s Hilbert schemes of an almost-complex four-manifold X. We compute in this setting their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on X, which are shown to be canonical in K?Ctheory.     

The Tate conjecture for powers of ordinary cubic fourfolds over finite fields

Recently, Levin proved the Tate conjecture for ordinary cubic fourfolds over finite fields. In this paper, we prove the Tate conjecture for self-products of ordinary cubic fourfolds. Our proof is based on properties of so-called polynomials of K3-type introduced by the author about 12 years ago.     

surfaces of genus 8 and varieties of sums of powers of cubic fourfolds

The main result of this paper is that the variety of presentations of a general cubic form in variables as a sum of cubes is isomorphic to the Fano variety of lines of a cubic -fold , in general different from .

A general surface of genus determines uniquely a pair of cubic -folds: the apolar cubic and the dual Pfaffian cubic (or for simplicity and ). As Beauville and Donagi have shown, the Fano variety of lines on the cubic is isomorphic to the Hilbert scheme of length two subschemes of . The first main result of this paper is that parametrizes the variety of presentations of the cubic form , with , as a sum of cubes, which yields an isomorphism between and . Furthermore, we show that sets up a correspondence between and . The main result follows by a deformation argument.

     

Duality and the topological filtration

Let $(M,J)$ be a Fano manifold which admits a Kähler-Einstein metric $g_{KE}$ (or a Kähler-Ricci soliton $g_{KS}$ ). Then we prove that Kähler-Ricci flow on $(M,J)$ converges to $g_{KE}$ (or $g_{KS}$ ) in $C^\infty $ in the sense of Kähler potentials modulo holomorphisms transformation as long as an initial Kähler metric of flow is very closed to $g_{KE}$ (or $g_{KS}$ ). The result improves Main Theorem in [14] in the sense of stability of Kähler-Ricci flow.     

The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements

The Hodge filtration of the module of derivations on the orbit space of a finite real reflection group acting on an ℓ-dimensional Euclidean space was introduced and studied by K. Saito [5] [6]. It is closely related to the flat structure or the Frobenius manifold structure. We will show that the Hodge filtration coincides with the filtration by the order of contacts to the reflecting hyperplanes. Moreover, a standard basis for the Hodge filtration is explicitly given.Partially supported by the Grant-in-aid for scientific research (No. 14340018 and 13874005), the Ministry of Education, Sports, Science and Technology, Japan