Mixed Hodge Structure Research Articles

Chain integral solutions to tautological systems

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system τ, arising from the universal family of Calabi–Yau hypersurfaces Y_a in a G-variety X of dimension n. First, we construct a natural topological correspondence between relative cycles in H_n (X−Y_a, ∪ D−Y_a) bounded by the union of G-invariant divisors ∪D in X to the solution sheaf of τ, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in Y_a, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of H_n (X−Y_a,∪ D−Y_a). In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann–Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.

Semistable reduction of a normal crossing $$\mathbb {Q}$$ Q -divisor

In a previous work, we have introduced the notion of embedded \(\mathbb {Q}\)-resolution, which allows the final ambient space to contain abelian quotient singularities, and A’Campo’s formula was calculated in this setting. Here, we study the semistable reduction associated with an embedded \(\mathbb {Q}\)-resolution and compute the mixed Hodge structure on the cohomology of the Milnor fiber in the isolated case using a generalization of Steenbrink’s spectral sequence. Examples of Yomdin-Le surface singularities are presented as an application.

An explicit description of the relative $${SL_{4}(\mathbb{C})}$$ S L 4 ( C ) -character variety of the projective line

In this paper, we give an explicit description of a certain character variety of dimension 2, and we construct a compactification of the character variety. We show that the compactification is a del Pezzo surface. One motivation of the explicit description is investigation of the mixed Hodge structure of the character variety. For the character variety, we can verify some conjectures on the mixed Hodge structures and the configurations of the boundaries of compactifications.

Reidemeister torsion, peripheral complex and Alexander polynomials of hypersurface complements

Let $f:\CN \rightarrow \C $ be a polynomial, which is transversal (or regular) at infinity. Let $\U=\CN\setminus f^{-1}(0)$ be the corresponding affine hypersurface complement. By using the peripheral complex associated to $f$, we give several estimates for the (infinite cyclic) Alexander polynomials of $\U$ induced by $f$, and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of $\U$.

Mixed Hodge complexes and higher extensions of mixed Hodge modules on algebraic varieties

In [1, 2] A. Beilinson provides a description of the bounded derived category of polarizable mixed Hodge structures in terms of the more flexible triangulated category of polarizable mixed Hodge complexes. In this work we provide a partial generalization of this result to higher dimension.

Purity for graded potentials and quantum cluster positivity

Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of $f$ on proper components of the critical locus of $f$, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

On the geometry of regular maps from a quasi-projective surface to a curve

By exploring the consequences of the triviality of the monodromy group for a class of surfaces of which the mixed Hodge structure is pure, we extend results of Miyanishi and Sugie, Dimca, Zaidenberg and Kaliman.

Mixed Hodge complexes on algebraic varieties and t-structures

The aim of this paper is to provide some complements on the categories of mixed Q-Hodge complexes introduced by M. Saito in [17]. In particular we prove that the category of pure diagrams has a t-structure that lifts the perverse t-structure and derive some consequences of this fact. We also compare the category of mixed Hodge complexes defined by M. Saito with a more naive version. These complements are used in [12] to provide a partial generalization to higher dimension of a result of A. Beĭlinson [1,2] which shows that the derived category of polarizable mixed Hodge structures is equivalent to the more flexible triangulated category of polarizable mixed Hodge complexes.

On the cohomology groups of local systems over Hilbert modular varieties via Higgs bundles

Let $X$ be a Hilbert modular variety and $\Bbb{V}$ a non-trivial local system over $X$ with infinite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology group $H^k(X,\Bbb{V})$ using the method of Higgs bundles. Among other results we prove the Eichler-Shimura isomorphism, give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over $\Bbb{R}$. These results are analogous to the work of Y. Matsushima and G. Shimura in the cocompact case and complement the results of E. Frietag for constant coefficients.

Asymptotics of degenerations of mixed Hodge structures

We construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate [6] between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric introduced by Hain [12], analogs of the norm estimates of [19] and the asymptotics of the naive limit Hodge filtration considered in [21].

Filtration perverse et théorie de Hodge

The compatibility of the perverse filtration with Hodge theory with coefficients in an admissible variation of a mixed Hodge structure on the complement of a normally crossing divisor is established using a logarithmic complex, with a view to obtaining a new proof of the decomposition theorem.

Variations of Mixed Hodge Structure and Semipositivity Theorems

We discuss variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove a generalization of the Fujita–Kawamata semipositivity theorem.

Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy

We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is applied to the category of mixed Hodge diagrams of differential graded algebras. Using Sullivan's minimal models, we prove a multiplicative version of Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain functoriality for the mixed Hodge structures on the rational homotopy type of complex algebraic varieties. In this context, the mixed Hodge structures on homotopy groups obtained by Morgan's theory follow from the derived functor of the indecomposables of mixed Hodge diagrams.

The geometry on smooth toroidal compactifications of Siegel varieties

We study smooth toroidal compactifications of Siegel varieties thoroughly from the viewpoints of Hodge theory and K\"ahler-Einstein metric. We observe that any cusp of a Siegel space can be identified as a set of certain weight one polarized mixed Hodge structures. We then study the infinity boundary divisors of toroidal compactifications, and obtain a global volume form formula of an arbitrary smooth Siegel variety $\scr{A}_{g,\Gamma} (g>1)$ with a smooth toroidal compactification $\overline{\scr{A}}_{g,\Gamma}$ such that $D_\infty:=\overline{\scr{A}}_{g,\Gamma}\setminus\scr{A}_{g,\Gamma}$ is normal crossing. We use this volume form formula to show that the unique group-invariant K\"ahler-Einstein metric on $\scr{A}_{g,\Gamma}$ endows some restraint combinatorial conditions for all smooth toroidal compactifications of $\scr{A}_{g,\Gamma}$. Again using the volume form formula, we study the asymptotic behavior of logarithmical canonical line bundle on any smooth toroidal compactification of $\scr{A}_{g,\Gamma}$ carefully and we obtain that the logarithmical canonical bundle degenerate sharply even though it is big and numerically effective.

Some remarks on limit mixed Hodge structures and spectrum

Abstract In this note we clarify some subtle points on the limit mixed Hodge structures and on the spectrum. These are more or less well-known to the specialists, but do not seem to be stated explicitly in the literature. However, as they do not seem to be obvious to the beginners, we consider them to be worth writing down explicitly. The general constructions are exemplified by considering the isolated (weighted) homogeneous singularities in detail.

T-branes and geometry

T-branes are a non-abelian generalization of intersecting branes in which the matrix of normal deformations is nilpotent along some subspace. In this paper we study the geometric remnant of this open string data for six-dimensional F-theory vacua. We show that in the dual M-theory / IIA compactification on a smooth Calabi-Yau threefold X, the geometric remnant of T-brane data translates to periods of the three-form potential valued in the intermediate Jacobian of X. Starting from a smoothing of a singular Calabi-Yau, we show how to track this data in singular limits using the theory of limiting mixed Hodge structures, which in turn directly points to an emergent Hitchin-like system coupled to defects. We argue that the physical data of an F-theory compactification on a singular threefold involves specifying both a geometry as well as the remnant of three-form potential moduli and flux which is localized on the discriminant. We give examples of T-branes in compact F-theory models with heterotic duals, and comment on the extension of our results to four-dimensional vacua.

Some Remarks on the Semipositivity Theorems

We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the dualizing sheaves of smooth varieties. Using the weight spectral sequence of mixed Hodge modules, we then reduce the semipositivity theorem for the higher direct images of dualizing sheaves to the smooth case where the assertion is well known. This may be used to simplify some constructions in a recent paper of Y. Kawamata. We also give a simple proof of the semipositivity theorem for admissible variations of mixed Hodge structure in [FF] by using the theories of Cattani, Kaplan, Schmid, Steenbrink, and Zucker. This generalizes Kawamata's classical result in the pure case.

Nefness: Generalization to the lc case

This note is devoted to a proof of the b-nefness of the moduli part in the canonical bundle formula for an lc-trivial fibration that is lc and not klt over the generic point of the base. This result is proved in [3, §8] and [4] by using the theory of variation of mixed Hodge structure. Here we present a proof that makes use only of the theory of variation of Hodge structure and follows Ambro's proof of [2, Theorem 0.2].

Number of Jordan blocks of the maximal size for local monodromies

AbstractWe prove formulas for the number of Jordan blocks of the maximal size for local monodromies of one-parameter degenerations of complex algebraic varieties where the bound of the size comes from the monodromy theorem. In the case when the general fibers are smooth and compact, the proof calculates some part of the weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In the singular case, we can prove a similar formula for the monodromy on the cohomology with compact supports, but not on the usual cohomology. We also show that the number can really depend on the position of singular points in the embedded resolution, even in the isolated singularity case, and hence there are no simple combinatorial formulas using the embedded resolution in general.

Weight filtration on the cohomology of complex analytic spaces

We extend Deligne's weight filtration to the integer cohomology of complex analytic spaces (endowed with an equivalence class of compactifications). In general, the weight filtration that we obtain is not part of a mixed Hodge structure. Our purely geometric proof is based on cubical descent for resolution of singularities and Poincar\'e-Verdier duality. Using similar techniques, we introduce the singularity filtration on the cohomology of compactificable analytic spaces. This is a new and natural analytic invariant which does not depend on the equivalence class of compactifications and is related to the weight filtration.