Hodge Structure Research Articles (Page 1)

Abstract In this paper we provide applications of general results of Baldi-Klingler-Ullmo and Khelifa-Urbanik on the geometry of the Hodge locus associated to an integral polarized variation of Hodge structures to the case of Noether-Lefschetz loci for families of smooth surfaces. In particular, we consider the family of smooth surfaces in the linear system of a sufficiently ample line bundle on a smooth projective threefold Y , in case Y is a Fano or a Calabi-Yau threefold, and we prove results on the different behaviour of the union of the general, respectively exceptional, components of its Noether-Lefschetz locus.

This article constructs Von Neumann invariants for constructible complexes and coherent 𝒟-modules on compact complex manifolds, generalizing the work of the author on coherent L 2 -cohomology. We formulate a conjectural generalization of Dingoyan’s L 2 -Mixed Hodge structures in terms of Saito’s Mixed Hodge Modules and give partial results in this direction.

Abstract We show that the derived categories of perverse Nori motives investigate and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$ -categorical lifts of the six operations. As a result, we obtain realisation functors from the category of Voevodsky étale motives to the derived categories of perverse Nori motives and mixed Hodge modules that commute with the operations. We also prove that if a motivic t-structure exists then Voevodsky étale motives and the derived category of perverse Nori motives are equivalent. Finally, we give a presentation of the indization of the derived category of perverse Nori motives as a category of modules in Voevodsky étale motives.

Abstract Let be a smooth irreducible quasi‐projective algebraic variety over a number field . Suppose is equipped with a ‐adic étale local system compatible with an admissible graded‐polarized variation of mixed Hodge structures on the complex analytification of . We prove that the ‐integral points in are covered by subpolynomially many geometrically irreducible ‐subvarieties, each lying in a fiber of the mixed period mapping arising from the variation of mixed Hodge structures. This is based on recent works by Brunebarbe–Maculan and Ellenberg–Lawrence–Venkatesh. As an application, we prove that there are subpolynomially many ‐integral Laurent polynomials with fixed reflexive Newton polyhedron and fixed non‐zero principal ‐determinant. Our results answer a question asked by Ellenberg–Lawrence–Venkatesh.

Abstract The defect of a cubic threefold $X$ with isolated singularities is a global invariant that measures the failure of $\mathbb{Q}$-factoriality. We compute the defect for such cubics in terms of topological data about the curve of lines through a singular point. We express the mixed Hodge structure on the middle cohomology of $X$ in terms of both the defect and local invariants of the singularities. We then relate the defect to various geometric properties of $X$: in particular, we show that a cubic threefold is not $\mathbb{Q}$-factorial if and only if it contains either a plane or a cubic scroll. We relate the defect to existence of compactified intermediate Jacobian fibrations with irreducible fibers associated to a cubic fourfold.

Abstract The period geometry of Calabi-Yau n-folds — characterised by their variations of Hodge structure governed by Griffiths transversality, a graded Frobenius algebra, an integral monodromy and an intriguing arithmetic structure — is analysed for applications in string compactifications and to Feynman integrals. In particular, we consider type IIB flux compactifications on Calabi-Yau three-folds and elliptically fibred four-folds. After constructing suitable three-parameter three-folds, we examine the relation between symmetries of their moduli spaces and flux configurations. Although the fixed point loci of these symmetries are projective special Kähler, we show that a simultaneous stabilisation of multiple moduli on the intersection of these loci need not be guaranteed without the existence of symmetries between them. We furthermore consider F-theory vacua along conifolds and use mirror symmetry to perform a complete analysis of the two-parameter moduli space of an elliptic Calabi-Yau four-fold fibred over ℙ3. We use the relation between Calabi-Yau period geometries in various dimensions and, in particular, the fact that the antisymmetric products of one-parameter Calabi-Yau three-fold operators yield four-fold operators to establish pairs of flux vacua on the moduli spaces of the three- and four-fold compactifications. We give a splitting of the period matrix into a semisimple and nilpotent part by utilising the Frobenius structure. This helps bringing ϵ-dimensional regulated integration by parts relations between Feynman integrals into ϵ-factorised form and solve them by iterated integrals of the periods.

The Hodge structure on the singularity category of a complex hypersurface

We show how a method to construct canonical differential equations for multi-loop Feynman integrals recently introduced by some of the authors can be extended to cases where the associated geometry is of Calabi-Yau type and even beyond. This can be achieved by supplementing the method with information from the mixed Hodge structure of the underlying geometry. We apply these ideas to specific classes of integrals whose associated geometry is a one-parameter family of Calabi-Yau varieties, and we argue that the method can always be successfully applied to those cases. Moreover, we perform an in-depth study of the properties of the resulting canonical differential equations. In particular, we show that the resulting canonical basis is equivalent to the one obtained by an alternative method recently introduced in the literature. We apply our method to non-trivial and cutting-edge examples of Feynman integrals necessary for gravitational wave scattering, further showcasing its power and flexibility.

Abstract Given a smooth projective complex curve inside a smooth projective surface, one can ask how its Hodge structure varies when the curve moves inside the surface. In this paper, we develop a general theory to study the infinitesimal version of this question in the case of ample curves. We can then apply the machinery to show that the infinitesimal variation of the Hodge structure of a general deformation of an ample curve in is an isomorphism.

Mixed Hodge modules and real groups

We discuss variations of mixed Hodge structure arising from projective morphisms of complex analytic spaces. Then we treat generalizations of Kollár’s torsion-free theorem, vanishing theorem, and so on, for reducible complex analytic spaces as an application. The results will play a crucial role in the theory of minimal models for projective morphisms between complex analytic spaces.

The focus of the workshop was on recent developments in local systems in arithmetic geometry in the broad sense, including anabelian geometry. The talks covered results in complex variations of Hodge structures, \ell -adic local systems, p -adic Hodge theory with p -adic local systems, and new insights into absolute Galois groups, with applications to rational points.

Given a Hilbert modular form for a totally real field F, and a prime p split completely in F, the f-eigenspace in p-adic de Rham cohomology has a family of partial filtrations and partial Frobenius maps, indexed by the primes of F above p. The general plectic conjectures of Nekovář and Scholl suggest a “plectic comparison isomorphism” comparing these structures to étale cohomology. We prove this conjecture in the case [F:Q]=2 under some mild assumptions; and for general F we prove a weaker statement which is strong evidence for the conjecture, showing that the plectic Hodge filtration has a canonical splitting given by intersecting with simultaneous eigenspaces for the partial Frobenii.

Previously, using the theory of delta characters for Drinfeld modules, one constructed a finite free [Formula: see text]-module [Formula: see text] with a semilinear operator on it, and hence a canonical [Formula: see text]-isocrystal [Formula: see text] was attached to any Drinfeld module [Formula: see text] that depended on the invertibility of a differential modular parameter [Formula: see text]. In this paper, we prove that [Formula: see text] is invertible for a Drinfeld module of rank [Formula: see text]. As a consequence, if [Formula: see text] does not admit a lift of Frobenius and [Formula: see text] is the fraction field of the ring of definition, we show that [Formula: see text] is isomorphic to [Formula: see text] and the isomorphism preserve the canonical Hodge filtration. On the other hand, if [Formula: see text] admits a lift of Frobenius, then [Formula: see text] is isomorphic to the subobject [Formula: see text] of [Formula: see text]. The above result can be viewed as a character theoretic interpretation of de Rham cohomology.

Verdier specialization and restrictions of Hodge modules

We build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo–Kato cohomology without recourse to log-geometry, as predicted by Fontaine, and we produce an induced Clemens–Schmid chain complex.

Let σ:X→Δ be a 1-parameter family of 2-dimensional isolated hypersurface singularities. In this paper, we show that if the Milnor number is constant, then any semistable model, obtained from σ after a sufficiently large base change must satisfy non trivial restrictions. Those restrictions are in terms of the dual complex, Hodge structure, and numerical invariants of the central fibre.

Abstract We give a cohomological and geometrical interpretation for the weighted Ehrhart theory of a full-dimensional lattice polytope $P$, with Laurent polynomial weights of geometric origin. For this purpose, we calculate the motivic Chern and Hirzebruch characteristic classes of a mixed Hodge module complex ${\mathscr{M}}$ whose underlying cohomology sheaves are constant on the ${{\mathbb{T}}}$-orbits of the toric variety $X_{P}$ associated to $P$. Besides motivic coefficients, this also applies to the intersection cohomology Hodge module. We introduce a corresponding generalized Hodge $\chi _{y}$-polynomial of the ample divisor $D_{P}$ on $X_{P}$. Motivic properties of these characteristic classes are used to express this Hodge polynomial in terms of a very general weighed lattice point counting and the corresponding weighted Ehrhart theory. We introduce, for such a mixed Hodge modules complex ${\mathscr{M}}$ on $X_{P}$, an Ehrhart polynomial $E_{P,{\mathscr{M}}}$ generalizing the Hodge polynomial of ${\mathscr{M}}$ and satisfying a reciprocity formula and a purity formula fitting with the duality for mixed Hodge modules. This Ehrhart polynomial and its properties depend only on a Laurent polynomial weight function on the faces $Q$ of $P$. In the special case of the intersection cohomology mixed Hodge module, the weight function corresponds to Stanley’s $g$-function of the polar polytope of $P$, hence it depends only on the combinatorics of $P$. In particular, we obtain a combinatorial formula for the intersection cohomology signature.

We explicitly compute canonical liftings modulo p2 in a sense of Achinger–Zdanowicz of Dwork hypersurfaces. The computation involves studying a compatibility between Hodge filtrations and a crystalline Frobenius. In particular, remarkably, we explicitly compute a partial data of the crystalline Frobenius modulo p2.

We review Hodge structures, relating filtrations, Galois Theory and Jordan-Holder structures.