We construct correspondences in logarithmic Hodge theory over a perfect field of arbitrary characteristic. These are represented by classes in the cohomology of sheaves of differential forms with log poles and, notably, log zeroes on cartesian products of varieties. From one perspective this generalizes work of Chatzistamatiou and Rülling, who developed (non-logarithmic) Hodge correspondences over perfect fields of arbitrary characteristic; from another we provide partial generalizations of more recent work of Binda, Park and Østvær on logarithmic Hodge correspondences by relaxing finiteness and strictness conditions on the correspondences considered.
We show that the description of Deligne--Beilinson cohomology is improved by using log Hodge theory. We consider the log relative version of it, and also present a fundamental conjecture in log Hodge theory.
In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. We also give criteria for a variety to be a Hodge theoretic K(pi,1), which roughly means that the cohomology of variations of mixed Hodge structure can be determined from the group.
We give a definition and study the basic properties of the irregular Hodge filtration on the exponentially twisted de Rham cohomology of a smooth quasi-projective complex variety.
We use Higgs cohomology to determine the Hodge numbers of the first intersection cohomology group of a local system V arising from the third direct image of a family of Calabi-Yau 3-folds over a smooth, quasi-projective curve. We give applications to Rhode's families of Calabi-Yau 3-folds without MUM.
We consider a hypersurface in $\mathbb{C}^n$ with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of the Jordan blocks in the Milnor monodromy.
We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural \textit{limiting} mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotnet orbit in several variables.
We establish a blow-up formula for Hodge cohomology of locally free sheaves on smooth proper varieties over an algebraically closed field of positive characteristic. For this, we introduce a notion of relative Hodge sheaves and study their behavior under blow-ups along smooth centers. In particular, as an application, we study the blow-up invariance of the $E_2$-degeneracy of the Hochschild--Kostant--Rosenberg spectral sequence for smooth proper varieties.
It is known that the Picard group of a complex manifold can be expressed as a Deligne cohomology group. One may wonder if the same holds for the Picard group of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not true, as already remarked by M. Saito. We explain how one has to modify the latter, show that the Picard group can be expressed by absolute Hodge cohomology, too, and introduce an intermediate object between Picard group and usual Deligne-Beilinson cohomology group. Similarly as in the case of Deligne cohomology one can relate line bundles with a regular connec…
Let $X$ be a compact Riemann surface, $Σ$ a finite set of points and $M = X\setminus Σ$. We study the $L^2$ cohomology of a polarized complex variation of Hodge structure on a Galois covering of the Riemann surface of finite type $M$. In this article we treat the case when the covering comes from a branched covering of $X$, and where $M$ is endowed with a metric asymptotic to a Poincaré metric. We prove that after tensorisation with the algebra of affiliated operators, the $L^2$ cohomology admits a pure Hodge structure.
We interpret syntomic cohomology of Nekovář-Nizioł as a $p$-adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne-Beilinson cohomology as an absolute Hodge cohomology by Beilinson and generalizes the results of Bannai and Chiarellotto, Ciccioni, Mazzari in the good reduction case, and of Yamada in the semistable reduction case. This interpretation yields a simple construction of the syntomic descent spectral sequence and its degeneration for proper and smooth varieties. We introduce syntomic coefficients and show that in dimension zero they form a full triangulated…
We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian.
With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory and polarization, Deligne's Mixed Hodge Structure on cohomology of complex algebraic varieties is described.
We prove the vanishing modulo torsion of the higher direct images of the sheaf of Witt vectors (and the Witt canonical sheaf) for a purely inseparable projective alteration between normal finite quotients over a perfect field. For this, we show that the relative Hodge-Witt cohomology admits an action of correspondences. As an application we define Witt-rational singularities which form a broader class than rational singularities. In particular, finite quotients have Witt-rational singularities. In addition, we prove that the torsion part of the Witt vector cohomology of a smooth, proper scheme…
For a smooth and proper scheme over an artinian local ring with ordinary reduction over the perfect residue field we prove - under some general assumptions - that the relative de Rham-Witt spectral sequence degenerates and the relative crystalline cohomology, equipped with its display structure arising from the Nygaard complexes, has a Hodge-Witt decomposition into a direct sum of (suitably Tate-Twisted) multiplicative displays. As examples our main results include the cases of abelian schemes, complete intersections, varieties of K3 type and some Calabi-Yau $n$-folds.
We prove two results regarding Hodge structures appearing in the cohomology of complex tori. First, we prove that if a polarizable Hodge structure appears in the cohomology of a complex torus $T$, it appears in the cohomology of an abelian variety isomorphic to a subquotient of $T$. Second, we prove a universality result for the Kuga-Satake construction applied to Hodge structures of $K3$ type that might not be polarized.
The nonabelian Hodge correspondence for vector bundles over noncompact curves is adequately described by implementing a weighted filtration on the objects involved. In order to establish a full correspondence between a Dolbeault and a de Rham space for a general complex reductive group $G$, we introduce torsors given by parahoric group schemes in the sense of Bruhat--Tits. Combined with existing results on the Riemann--Hilbert correspondence for logarithmic parahoric connections, this gives a full nonabelian Hodge correspondence from Higgs bundles to fundamental group representations over a no…
Let A be an abelian variety over C such that the semisimple part of the Hodge group of A is a product of copies of SU(p,1) for some p>1. We show that any effective Tate twist of a Hodge structure occurring in the cohomology of A is isomorphic to a Hodge structure in the cohomology of some abelian variety.
The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold.
We prove a Hodge-type decomposition for the de-Rham cohomology of $ p$-adically uniformized varieties by the product of Drinfeld's symmetric spaces. It is based on work of Schneider, Stuhler, Iovita and Spiess on the cohomology of Drinfeld's symmetric space.
We show that certain abelian varieties A have the property that for every Hodge structure V in the cohomology of A, every effective Tate twist of V occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV.
We use $L^2$-Higgs cohomology to determine the Hodge numbers of the parabolic cohomology $H^1(\bar S, j_*\V)$, where the local system $\V$ arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve $\bar S$. The method gives a way to predict the presence of algebraic 2-cycles in the total space of the family and is applied to some examples.
We prove the Hodge symmetry type result on the Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in the direct sum of holomorphic line bundles. Consequently, we show the vanishing and non-vanishing of Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in holomorphic line bundles.
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. 2020 AMS Classification: 32J27.
We present a detailed overview of the construction of the A_inf-cohomology theory from the preprint "Integral p-adic Hodge theory", joint with B. Bhatt and P. Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham--Witt complexes.
We study the $L^2$--cohomology of certain local systems on non-compact arithmetic ball quotients $X=Γ\backslash \B_n$, in particular vanishing and non--vanishing results. We also give generalizations to higher dimensional ball quotients and study the mixed Hodge structure on the sheaf cohomology of a local system with the $L^2$-cohomology contributing to the lowest weight part.
For complex parallelisable manifolds $Γ\backslash G$, with $G$ a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de-Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.
We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown-Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson's unstable splitting. As a consequence, we obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology for complex manifolds.
