Hodge Structure Research Articles

The mirror Clemens–Schmid sequence

We introduce a four-term long exact sequence that relates the cohomology of a smooth variety admitting a projective morphism onto a projective base to the cohomology of the open set obtained by removing the preimage of a general linear section. We show that this sequence respects the perverse Leray filtration and induces exact sequences of mixed Hodge structures on its graded pieces. We conjecture that this exact sequence should be thought of as mirror to the Clemens–Schmid sequence, which describes the cohomology of degenerations. We exhibit this mirror relationship explicitly for all Type II and many Type III degenerations of K3 surfaces. In three dimensions, we show that for Tyurin degenerations of Calabi–Yau threefolds our conjecture is a consequence of existing mirror conjectures, and we explicitly verify our conjecture for a more complicated degeneration of the quintic threefold.

Non-Density of the Exceptional Components of the Noether–Lefschetz Locus

Abstract We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^{3}$ acquires extra classes. In particular we show that the so-called exceptional components of the Noether–Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether–Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective three-fold and $L$ a very ample line bundle. Both results are applications of the Zilber–Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.

End of the world brane networks for infinite distance limits in CY moduli space

Dynamical Cobordism provides a powerful method to probe infinite distance limits in moduli/field spaces parameterized by scalars constrained by generic potentials, employing configurations of codimension-1 end of the world (ETW) branes. These branes, characterized in terms of critical exponents, mark codimension-1 boundaries in the spacetime in correspondence of finite spacetime distance singularities at which the scalars diverge. Using these tools, we explore the network of infinite distance singularities in the complex structure moduli space of Calabi-Yau fourfolds compactifications in M-theory with a four-form flux turned on, which is described in terms of normal intersecting divisors classified by asymptotic Hodge theory. We provide spacetime realizations for these loci in terms of networks of intersecting codimension-1 ETW branes classified by specific critical exponents which encapsulate the relevant information of the asymptotic Hodge structure characterizing the corresponding divisors.

On the transcendental lattices of Hyper-Kähler manifolds

We introduce the notion of a Hyper-Kähler manifold X induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of Hyper-Kähler manifolds studying those that are induced by a K3 or abelian surface (that is, induced by the Hodge structure of their transcendental lattice), giving lattice-theoretic criteria to decide whether or not they are birational to a moduli space of sheaves over said surface. We highlight the different behaviors we find for the particular class of Hyper-Kähler manifolds of O'Grady type.

Mixed Hodge structures and Siegel operators

Abstract In this paper, we study mixed Hodge structures on the cohomology of locally symmetric varieties and give an application to modular forms. After proving vanishing of some Hodge numbers, we focus on the weight filtration on the last Hodge subspace of the middle degree cohomology. We prove that the weight filtration coincides with the corank filtration on the space of modular forms of canonical weight defined by the Siegel operators, and calculate the graded quotients. As an application, we deduce surjectivity of the total Siegel operators in many cases, and identify an obstruction space in the remaining case.

Vanishing of local cohomology with applications to Hodge theory

Let H=((H,F•),L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U. By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M∈HM(U). Let X be a compactification of U, and j:U↪X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j+M∈DbMHM(X). In particular, we study the graded pieces of the de Rham complex GrpFDR(j+M)∈Dcohb(X), and the Hodge structure of Hi(U,L) for i in sufficiently low degrees.

A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.

Cohomological $\chi$-independence for Higgs bundles and Gopakumar–Vafa invariants

The aim of this paper is two-fold. Firstly, we prove Toda’s \chi -independence conjecture for Gopakumar–Vafa invariants of arbitrary local curves. Secondly, following Davison’s work, we introduce the BPS cohomology for moduli spaces of Higgs bundles of rank r and Euler characteristic \chi which are not necessary coprime, and show that it does not depend on \chi . This result extends the Hausel–Thaddeus conjecture on the \chi -independence of E-polynomials proved by Mellit, Groechenig–Wyss–Ziegler and Yu in two ways: We obtain an isomorphism of mixed Hodge modules on the Hitchin base rather than an equality of E-polynomials, and we do not need the coprime assumption. The proof of these results is based on a description of the moduli stack of one-dimensional coherent sheaves on a local curve as a global critical locus which is obtained in the companion paper by the first author and Naruki Masuda.

Period Integrals of Hypersurfaces via Tropical Geometry

Abstract Let $\left \{ Z_{t} \right \}_{t}$ be a one-parameter family of complex hypersurfaces of dimension $d \geq 1$ in a toric variety. We compute asymptotics of period integrals for $\left \{ Z_{t} \right \}_{t}$ by applying the method of Abouzaid–Ganatra–Iritani–Sheridan, which uses tropical geometry. As integrands, we consider Poincaré residues of meromorphic $(d+1)$-forms on the ambient toric variety, which have poles along the hypersurface $Z_{t}$. The cycles over which we integrate them are spheres and tori, which correspond to tropical $(0, d)$-cycles and $(d, 0)$-cycles on the tropicalization of $\left \{ Z_{t} \right \}_{t}$, respectively. In the case of $d=1$, we explicitly write down the polarized logarithmic Hodge structure of Kato–Usui at the limit as a corollary. Throughout this article, we impose the assumption that the tropicalization is dual to a unimodular triangulation of the Newton polytope.

Motivic Geometry of two-Loop Feynman Integrals

Abstract We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch (Double box motive. SIGMA 2021;17,048) that in the well-known double-box example, there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the non-planar two-loop tardigrade graph is that of a K3 surface. In an appendix by Eric Pichon-Pharabod, we argue via high-precision numerical computations that the Picard number of this K3 surface is generically 11 and we compute the expected lattice polarization. Lastly, we show that generic members of the ice cream cone family of graph hypersurfaces correspond to the pairs of sunset Calabi–Yau varieties.

Plectic Jacobians

ABSTRACT Looking for a geometric framework to study plectic Heegner points, we define a collection of abelian varieties – called plectic Jacobians—using the middle-degree cohomology of quaternionic Shimura varieties (QSVs). The construction is inspired by the definition of Griffiths’ intermediate Jacobians and rests on Nekovář–Scholl’s notion of plectic Hodge structures. Moreover, we construct exotic Abel–Jacobi maps sending certain zero cycles on QSVs to plectic Jacobians.

On the intermediate Jacobian of M5-branes

We study Euclidean M5-branes wrapping vertical divisors in elliptic Calabi-Yau fourfold compactifications of M/F-theory that admit a Sen limit. We construct these Calabi-Yau fourfolds as elliptic fibrations over coordinate flip O3/O7 orientifolds of toric hypersurface Calabi-Yau threefolds. We devise a method to analyze the Hodge structure (and hence the dimension of the intermediate Jacobian) of vertical divisors in these fourfolds, using only the data available from a type IIB compactification on the O3/O7 Calabi-Yau orientifold. Our method utilizes simple combinatorial formulae (that we prove) for the equivariant Hodge numbers of the Calabi-Yau orientifolds and their prime toric divisors, along with a formula for the Euler characteristic of vertical divisors in the corresponding elliptic Calabi-Yau fourfold. Our formula for the Euler characteristic includes a conjectured correction term that accounts for the contributions of pointlike terminal ℤ2 singularities corresponding to perturbative O3-planes. We check our conjecture in a number of explicit examples and find perfect agreement with the results of direct computations.

𝑉-filtrations and minimal exponents for local complete intersections

Abstract We define and study a notion of minimal exponent for a local complete intersection subscheme 𝑍 of a smooth complex algebraic variety 𝑋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 𝑉-filtration associated to 𝑍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r ⁢ ( O X ) \mathcal{H}^{r}_{Z}(\mathcal{O}_{X}) , where 𝑟 is the codimension of 𝑍 in 𝑋. We also study its relation to the Bernstein–Sato polynomial of 𝑍. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension 1 case. A key ingredient for our main result is a description of the Kashiwara–Malgrange 𝑉-filtration associated to any ideal ( f 1 , … , f r ) (f_{1},\ldots,f_{r}) in terms of the microlocal 𝑉-filtration associated to the hypersurface defined by ∑ i = 1 r f i ⁢ y i \sum_{i=1}^{r}f_{i}y_{i} .

Mixed Hodge structures on character varieties of nilpotent groups

AbstractLet $$\textsf{Hom}^{0}(\Gamma ,G)$$ Hom 0 ( Γ , G ) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group $$\Gamma $$ Γ into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety $$\textsf{Hom}^{0}(\Gamma ,G)$$ Hom 0 ( Γ , G ) and on the character variety $$\textsf{Hom}^{0}(\Gamma ,G)/\!\!/G$$ Hom 0 ( Γ , G ) / / G . We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.

Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let U U be a smooth connected complex algebraic variety and let f : U → C ∗ f\colon U\to \mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C ∗ \mathbb {C}^* by f f gives rise to an infinite cyclic cover U f U^f of U U . The action of the deck group Z \mathbb {Z} on U f U^f induces a Q [ t ± 1 ] \mathbb {Q}[t^{\pm 1}] -module structure on H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) . We show that the torsion parts A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) of the Alexander modules H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) carry canonical Q \mathbb {Q} -mixed Hodge structures. We also prove that the covering map U f → U U^f \to U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f : U → C ∗ f\colon U\to \mathbb {C}^* is proper, we prove the semisimplicity and purity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , and we compare our mixed Hodge structure on A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) with the limit mixed Hodge structure on the generic fiber of f f .

The second integral cohomology of moduli spaces of sheaves on K3 and Abelian surfaces

In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If S is a projective K3 surface, v a Mukai vector and H a polarization on S that is general with respect to v, we show that H2(Mv,Z) is a free Z-module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which H2(Mv,Z) is Hodge isometric to the Hodge sublattice v⊥ of the Mukai lattice of S. Similar results are proved for Abelian surfaces.

The four operations on perverse motives

Let k be a field of characteristic zero with a fixed embedding \sigma:k\hookrightarrow \mathbf{C} into the field of complex numbers. Given a k -variety X , we use the triangulated category of étale motives with rational coefficients on X to construct an abelian category \mathscr{M}(X) of perverse mixed motives. We show that over \mathop{\mathrm{Spec}}(k) the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories \mathrm{D}^{\mathrm{b}}(\mathscr{M}(X)) are equipped with the four operations of Grothendieck (for morphisms of quasi-projective k -varieties) as well as nearby and vanishing cycles functors and a formalism of weights. In particular, as an application, we show that many classical constructions done with perverse sheaves, such as intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recovers and strengthens work by Zucker, Saito, Arapura and de Cataldo–Migliorini and provides an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures.

Seiberg–Witten differentials on the Hitchin base

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss–Manin) derivative of the Seiberg–Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration.

A note on varieties of weak CM-type

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level-n subspace of Hn(X;Q) is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire H⁎(X;Q) is of CM-type. We study in particular abelian varieties when the dimension of the level-n subspace is two or four, and K3×T2. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974.

Hodge modules and cobordism classes

We show that the cobordism class of a polarization of Hodge module defines a natural transformation from the Grothendieck group of Hodge modules to the cobordism group of self-dual bounded complexes with real coefficients and constructible cohomology sheaves in a compatible way with push-forward by proper morphisms. This implies a new proof of the well-definedness of the natural transformation from the Grothendieck group of varieties over a given variety to the above cobordism group (with real coefficients). As a corollary, we get a slight extension of a conjecture of Brasselet, Schürmann and Yokura, showing that in the {\mathbb Q} -homologically isolated singularity case, the homology L -class which is the specialization of the Hirzebruch class coincides with the intersection complex L -class defined by Goresky, MacPherson, and others if and only if the sum of the reduced modified Euler–Hodge signatures of the stalks of the shifted intersection complex vanishes. Here Hodge signature uses a polarization of Hodge structure, and it does not seem easy to define it by a purely topological method.