Let ${\cal L}$ be a variation of Hodge structures on the complement $X^{*}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{*} = X - Y \to X $ denotes the open embedding. The purpose of this paper is to describe the weight filtration $W$ on a combinatorial logarithmic complex computing the (higher) direct image ${\bf j}_{*}{\cal L} $, underlying a mixed Hodge complex when $X$ is proper, proving in this way the results in the note [14] generalizing the constant coefficients case. When a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)$, the weight filtration on the complex of nearby cocycles $Ψ_f ({\cal L})$ on $Y$ can be described by these logarithmic techniques and a comparison theorem shows that the filtration coincides with the weight defined by the logarithm of the monodromy which provides the link with various results on the subject.
Let ${\cal L}$ be a local system on the complement $X^{\star}$ of a normal crossing divisor (NCD) $ Y$ in a smooth analytic variety $X$ and let $ j: X^{\star} = X - Y \to X $ denotes the open embedding. The purpose of this paper is to describe a weight filtration $W$ on the direct image ${\bf j}_{\star}{\cal L} $ and in case a morphism $f: X \to D$ to a complex disc is given with $Y = f^{-1}(0)$, the weight filtration on the complex of nearby cocycles $Ψ_f ({\cal L})$ on $Y$. A comparison theorem shows that the filtration coincides with the weight defined by the logarithm of the monodromy and …
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.
We promote Beilinson's triangulated equivalence between the bounded derived category of rational polarizable mixed Hodge structures and the derived category of rational polarizable mixed Hodge complexes to an equivalence of symmetric monoidal quasi-categories. We use this equivalence to construct a presheaf of commutative differential graded algebras in the ind-completion of the category of rational mixed Hodge structures which computes Deligne's mixed Hodge structure on the rational Betti cohomology of $\mathbf{C}$-schemes of finite type. This leads to a preshea--in the quasi-categorical sens…
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…
We compute the Hodge-Deligne polynomials of the moduli spaces of representations of the fundamental group of a complex surface into SL(2,C), for the case of small genus g, and allowing the holonomy around a fixed point to be any matrix of SL(2,C), that is Id, -Id, diagonalisable, or of either of the two Jordan types. For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the Hodge-Deligne polynomial under fibrations.
We investigate the interplay between the monodromy and the Deligne mixed Hodge structure on the Milnor fiber of a homogeneous polynomial. In the case of hyperplane arrangement Milnor fibers, we obtain a new result on the possible weights. For line arrangements, we prove in a new way the fact due to Budur and Saito that the spectrum is determined by the weak combinatorial data, and show that such a result fails for the Hodge-Deligne polynomials.
According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three different splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we…
Let $X$ be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products $Sym^{n}X$ when the cohomology of $X$ is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials $μ_{X^{n}}^{S_{n}}\left(t,u,v\right)$, codifying the permutation action of $S_{n}$ as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products $μ_{Sym^{n}X}\left(t,u,v\right)$. These formulas are then applied to the case of linear alg…
We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are defined over number fields (assuming the initial family is), as expected from the Hodge conjecture. We also show that the Hodge conjecture for (weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly) absolute Hodge classes on varieties defined over number fields.
In a beautiful paper Deligne and Illusie proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. In a recent paper Arinkin, Căldăraru and the author of this paper gave a geometric interpretation of the problem of Deligne-Illusie showing that the triviality of a certain line bundle on a derived scheme implies the the Deligne-Illusie result. In the present paper we generalize these ideas to logarithmic schemes and using the theory of twisted derived intersection of logarithmic schemes we obtain the Hodge theorem for the logarithmic de Rham complex…
Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime p>3. Consider its perfectoid cover S(p^\infty) and the Hodge-Tate period map introduced by A. Caraiani and P. Scholze. We compare the pull-back to S(p^\infty) of the Ekedahl-Oort stratification on the mod p special fiber of S and the pull back to S(p^\infty) of the fine Deligne-Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge-Tate period map. An application to the non-emptiness of Ekedhal-Oort strata is provided.
Motivated by a connection between the topology of (generalized) configuration spaces and chromatic polynomials, we show that generating functions of Hodge-Deligne polynomials of quasiprojective varieties and colorings of acyclic directed graphs with the complete graph $K_n$ as the underlying undirected graph. In order to do this, we combine an interpretation of Crew-Spirkl for plethysms involving chromatic symmetric polynomials using colorings of directed acyclic graphs to plethystic exponentials often appearing Hodge-Deligne polynomials of varieties. Applying this to a recent result of Floren…
We define notions of higher order spectra of a complex quasi-projective manifold with an action of a finite group $G$ and with a $G$-equivariant automorphism of finite order, some of their refinements and give Macdonald type equations for them.
The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligne and Illusie, using positive characteristic methods. We argue that the central algebraic object of their proof can be understood geometrically as a line bundle on a derived scheme. In this interpretation, the Deligne-Illusie result can be seen as a proof that this line bundle is trivial under certain assumptions. We give a criterion for the triviality of this line bundle in a more general context. The proof uses techniques from derived algebraic geometry, specifically arguments which show the formality of certain derive…
The goal of this paper is to develop the theory of Deligne-Beilinson cohomology of affine groups with a mixed Hodge structure. The motivation comes from Hodge theory and the study of motives, where such groups appear. Several of Francis Brown's period computations (arXiv:1407.5167) are interpreted as elements of the DB cohomology of the relative unipotent completion of $SL_2(Z)$ and their cup products. The results in this paper are used in arXiv:1403.6443 where they are used to prove that Pollack's quadratic relations are motivic.
Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.
For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of abelian Hurwitz-Hodge integrals as a sum over Feynman graphs, where the weight of each graph is expressed in terms of descendant integrals over moduli spaces of stable curves and representations of G. This expression will play a crucial role in the proof of the remodeling conjecture (arXiv:0709.1453, arXiv:0807.0597) for affine toric Calabi-Yau 3-orbifolds …
We introduce the notion of mixed Hodge complex on an algebraic variety, improving Du Bois' filtered complex, and relate Deligne's theory of mixed Hodge structure with the theory of mixed Hodge module. This was supposed to be true, but is quite nontrivial, because the theory of mixed Hodge module does not work on a simplicial scheme. Some application to the Du Bois singularity is given.
Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A. Weil. The conjecture (and one-motives) are motivated by means of Jacobians, generalized Jacobians of Rosenlicht, and Serre's generalized Albanese varieties. We discuss the connections with the Hodge and the generalized Hodge conjecture. We end with some applications to number theory by providing partial answers to questions of Serre, Katz and Jannsen.
If $X$ is a smooth projective variety over ${\mathbb R}$, the Hodge ${\mathcal D}$-conjecture of Beilinson asserts the surjectivity of the regulator map to Deligne cohomology with real coefficients. It is known to be false in general but is true in some special cases like Abelian surfaces and $K3$-surfaces - and still expected to be true when the variety is defined over a number field. We prove an analogue of this for Abelian surfaces at a non-Archimedean place where the surface has bad reduction. Here the Deligne cohomology is replaced by a certain Chow group of the special fibre. The case of…
We use a version of the method of Deligne-Illusie to prove that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates for a large class of associative, not necessariyl commutative DG algebras. This proves, under some assumption, a conjecture by Kontsevich and Soibelman made in math.RA/0606241. The approach is similar to my earlier paper math.AG/0511665, but the proof is more straightforward, and the underlying algebraic topology notions are explicitly described. The paper is independent of math.AG/0511665 and in a sense, supercedes it.
This note is inspired by the work of Deligne on the local behavior of Hodge structures at infinity. We study limit mixed Hodge structures of degenerating families of compact hyperkähler manifolds. We show that when the monodromy action on $H^2$ has maximal index of unipotency, the limit mixed Hodge structures on all cohomology groups are of Hodge-Tate type.
We show that the pairing on de Rham realizations of 1-motives in "Theorie di Hodge III", IHES 44, can be defined over any base scheme and we prove that it gives rise to a perfect duality if one is working with a 1-motive and its Cartier dual. Furthermore we study universal extensions of 1-motives and their relation with $\natural$-extensions.
The Getzler's formula relates the S_n-equivariant Hodge-Deligne polynomial of the space of ordered tuples of distinct points on a given variety X with the Hodge-Deligne polynomial of X. We obtain the analogue of this formula for the case when X has a nontrivial automorphism group. Collecting together all strata of $\mathcal{M}_2$ with different automorphism groups, we derive a formula for the S_n-equivariant Euler characteristic of $\mathcal{M}_{2,n}$.
In this article we describe three constructions of complex variations of Hodge structure, proving the existence of interesting opposite filtrations that generalize a construction of Deligne. We also analyze the relation between deformations of Frobenius modules and certain maximally degenerate variations of Hodge structures. Finally, under a certain generation hypothesis, we show how to construct a Frobenius manifold starting from a deformation of a Frobenius module.
Over a subfield of the field of complex numbers, the Hodge realization of a geometrical motive is defined and represented as the cohomology of a mixed Hodge DG-complex in the sense of Deligne. Both filtrations are represented by truncation functors, on a Bondarko weight complex for the weight filtration and on the De Rham motivic complex for the Hodge one. The Deligne-Beilinson realization is also constructed. This preprint partially replaces the former "Réalisation des complexes motiviques de Voevodsky".
We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new Hodge-theoretic results on the cohomology of projective varieties which extend naturally the classical theory and provide new applications.
In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of "rational Tate classes" on varieties over finite fields having the properties that the algebraic classes would have if the Hodge and Tate conjectures were known. v3. Submitted version.
In this paper, we shall generalize the theory of mixed Hodge structures due to Deligne and obtain a subcategory GMHS in the category of mixed Hodge structures such that we have Ext_{GMHS}^2(Q,-)\not=0 in general.
We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered lo…
Given a family of smooth complex projective varieties, the Hodge conjecture predicts the algebraicity of the locus of Hodge classes. This was proven unconditionnally by Cattani, Deligne and Kaplan in 1995. In a similar way, conjectures on algebraic cycles have led Green and Griffiths to conjecture the algebraicity of the zero locus of normal functions. This corresponds to a mixed version of the theorem of Cattani, Deligne and Kaplan. This result has been proven recently by Brosnan-Pearlstein, Kato-Nakayama-Usui, and Schnell building on work of M. Saito. We will present some of the ideas around…
We generalize the logarithmic decomposition theorem of Deligne-Illusie to a filtered version. There are two applications. The easier one provides a mod $p$ proof for a vanishing theorem in characteristic zero. The deeper one gives rise to a positive characteristic analogue of a theorem of Deligne on the mixed Hodge structure attached to complex algebraic varieties.
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…
For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking each arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of an…
Let $n$ be a positive even integer and $d$ a positive integer . To every complete family $Z$ of n dimensional degree d hypersurfaces in the projective space with isolated A-D-E singularities we construct according to an idea of Carlson-Toledo a Deligne-Mumford stack $\bar Z$ whose moduli space is $Z$ such that the monodromy representation extends. We study the corresponding periods mapping and establish an infinitesimal Torelli theorem along the isosingular strata of lZ$ under transversality assumptions. We apply this result to prove Steiness of the universal covering space of $\bar Z$.
In its simplest form the Decomposition Theorem asserts that the rational intersection cohomology of a complex projective variety occurs as a summand of the cohomology of any resolution. This deep theorem has found important applications in algebraic geometry, representation theory, number theory and combinatorics. It was originally proved in 1981 by Beilinson, Bernstein, Deligne and Gabber as a consequence of Deligne's proof of the Weil conjectures. A different proof was given by Saito in 1988, as a consequence of his theory of mixed Hodge modules. More recently, de Cataldo and Migliorini foun…
We are showing that the Deligne--Beilinson cohomology sheaves ${\cal H}^{q+1}({\bf Z}(q)_{\cal D})$ are torsion free by assuming Kato's conjectures hold true for function fields. This result is `effective' for $q=2$; in this case, by dealing with `arithmetic properties' of the presheaves of mixed Hodge structures defined by singular cohomology, we are able to give a cohomological characterization of the Albanese kernel for surfaces with $p_g=0$.
This is partly a survey article on nonabelian Hodge theory, but we also give proofs of results that have only been announced elsewhere. In the introduction we discuss a wide range of recent work on this subject and give some references. In the body of the paper, we discuss Corlette's nonabelian Hodge theorem, Hitchin's quaternionic structure on the moduli space of representations of $π_1(X)$ (for a compact Kähler manifold $X$), and Deligne's construction of the resulting twistor space. We then mention the interpretation of Deligne's space of $λ$-connections as the Hodge filtration of the nonab…
For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in arXiv:1203.2338 a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential d+df, extending a definition of Deligne in the case of curves. In this article, we show the degeneration at E_1 of the spectral sequence attached to the irregular Hodge filtration, by using the method of arXiv:0804.4328. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on its proof of the corresponding E_1 degeneration, by reduction to characteris…
We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and $\A^1$-homotopy invariance. Moreover, we obtain transfer maps along projective morphisms.
At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are products of values of the gamma function at rational arguments, with exponents determined by the Hodge decomposition. We prove an alternating variant of this conjecture for smooth projective varieties acted upon by an automorphism of finite order, thus improving previous results of Maillot and Rössler. The proof relies on a product formula for periods of regular singular connections due to Saito and Terasoma.
Let $\mathcal{X}_ΓG:=\mathrm{Hom}(Γ,G)/\!/G$ be the $G$-character variety of $Γ$, where $G$ is a complex reductive group and $Γ$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of $\mathcal{X}_ΓG$, and present some applications, focusing on the cases when $Γ$ is a free or free abelian group. Detailed constructions and proofs of the main results will appear elsewhere.
Given an integral p-adic variety, we observe that if the integral Hodge--de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p-adic Hodge theory, we show that such a nice behavior is guaranteed if the p-adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne--Illusie and Fontaine--Messing. Lastly, we discuss an example illustrating the necessity of t…
We show that the limit, by rescaling, of the `new supersymmetric index' attached to the Fourier-Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the same object. We also extend the definition by Deligne of a Hodge filtration on the de Rham cohomology of a exponentially twisted polarized variation of complex Hodge structure and prove a E_1 degeneration property for it.
The category of rational mixed Hodge-Tate structures is a mixed Tate category. So thanks to the Tannakian formalism, it is equivalent to the category of finite dimensional graded comodules over a graded commutative Hopf algebra H over Q. Since the category has homological dimension 1, the Hopf algebra H is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the direct sum of the groups C/Q(n) over n>0. However this isomorphism is not natural in any sense, e.g. does not work in families. We give a natural explicit construction of …
