Hodge Structure Research Articles

Mixed $\ell$-adic complexes for schemes over number fields

If X is a variety over a number field, Annette Huber has defined a category of “horizontal” (or “almost everywhere unramified”) \ell -adic complexes and \ell -adic perverse sheaves on X . For such objects, the notion of weights makes sense (in the sense of Deligne), just as in the case of varieties over finite fields. However, contrary to what happens in that last case, mixed perverse sheaves (or mixed locally constant sheaves) on X do not have a weight filtration in general, even when X is a point. The goal of this paper is to show how to avoid this problem by working directly in the derived category of the abelian category of perverse sheaves that do admit a weight filtration. As an application, the method of the author to calculate the intermediate extension of a pure perverse sheaf using weight truncations apply over any finitely generated field, and not just over a finite field.

Rich representations and superrigidity

Abstract We investigate and compare applications of the Zilber–Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu’s immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.

A cone conjecture for log Calabi-Yau surfaces

Abstract We consider log Calabi-Yau surfaces $(Y, D)$ with singular boundary. In each deformation type, there is a distinguished surface $(Y_e,D_e)$ such that the mixed Hodge structure on $H_2(Y \setminus D)$ is split. We prove that (1) the action of the automorphism group of $(Y_e,D_e)$ on its nef effective cone admits a rational polyhedral fundamental domain; and (2) the action of the monodromy group on the nef effective cone of a very general surface in the deformation type admits a rational polyhedral fundamental domain. These statements can be viewed as versions of the Morrison cone conjecture for log Calabi–Yau surfaces. In addition, if the number of components of D is no greater than six, we show that the nef cone of $Y_e$ is rational polyhedral and describe it explicitly. This provides infinite series of new examples of Mori Dream Spaces.

Notes on Characterizations of 2d Rational SCFTs: Algebraicity, Mirror Symmetry, and Complex Multiplication

AbstractS. Gukov and C. Vafa proposed a characterization of rational superconformal field theories (SCFTs) in dimensions with Ricci‐flat Kähler target spaces in terms of the Hodge structure of the target space, extending an earlier observation by G. Moore. The idea is refined, and a conjectural statement on necessary and sufficient conditions for such SCFTs to be rational is obtained, which is indeed proven to be true in the case the target space is . In the refined statement, the algebraicity of the geometric data of the target space turns out to be essential, and the Strominger–Yau–Zaslow fibration in the mirror correspondence also plays a vital role.

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.

Ax–Schanuel for variations of mixed Hodge structures

Ax–Schanuel for variations of mixed Hodge structures

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 .