The de Rham homotopy theory of complex algebraic varieties II

We show that the local system of homotopy groups, associated with a topologically locally trivial family of smooth pointed varieties, underlies a good variation of mixed Hodge structure. In particular we show that there is a limit mixed Hodge structure on homotopy associated with a degeneration of such varieties.Supported in part by the National Science Foundation grant DMS-8401175.     

Mixed Hodge complexes on algebraic varieties

Using the theory of mixed Hodge Modules, we introduce the notion of mixed Hodge complex on an algebraic variety, and establish the relation between the filtered complex of Du Bois and the corresponding complex of mixed Hodge Modules. Some application to the Du Bois singularity is given. Received: 20 February 1999     

Hodge Theory and Geometry

Hodge theory for a smooth algebraic curve includes both theHodge structure (period matrix) on cohomology and the use ofthat Hodge structure to study the geometry of the curve, viathe Jacobian variety. Hodge extended the theory of the periodmatrix to smooth algebraic varieties of any dimension, definingin general a Hodge structure on the cohomology of the variety.He gave a few applications to the geometry of the variety, butthese did not attain the richness of the Jacobian variety. Inrecent years, Hodge theory has been successfully extended toarbitrary varieties, and to families of varieties. In this expositorypaper, some of these developments are reviewed, with specialemphasis on instances where these extensions can be used tostudy the geometry – especially the algebraic cycles –on the variety. 2000 Mathematics Subject Classification 14CDFJ.     

Regularity results for real analytic homotopies

Summary In this paper, we study two main features of the homotopy curves which we follow when we use the homotopy method for solving the zeros of analytic maps. First, we prove that near the solution the curve behaves nicely. Secondly, we prove that the set of starting points which give smooth homotopy curves is open and dense. The second property is of particular importance in computer implementation.This research was supported by the National Science Foundation under Grants MCS 78-02420 (Li) and MCS-7818858 (Mallet-Paret) and MCS-7818221 (Yorke), by the Army Research office under Grant DAAG-29-80-C-0040 (Li and Yorke)     

Exterior products of forms and the cohomology ring of a complex space

In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology.     

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.     

Mixed Hodge structures and equivariant sheaves on the projective plane

We describe an equivalence of categories between the category of mixed Hodge structures and a category of equivariant vector bundles on a toric model of the complex projective plane which verify some semistability condition. We then apply this correspondence to define an invariant which generalizes the notion of R ‐split mixed Hodge structure and give calculations for the first group of cohomology of possibly non smooth or non‐complete curves of genus 0 and 1. Finally, we describe some extension groups of mixed Hodge structures in terms of equivariant extensions of coherent sheaves. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Hodge Structures Associated to SU(p, 1)

Let A be an abelian variety over ? 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.     

Compact Kähler manifolds with elliptic homotopy type

Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. For surfaces there are only two possibilities, namely h1,1?2 with hp,q=0 for p≠q. For threefolds, there are three possibilities, namely h1,1?3 with hp,q=0 for p≠q. This characterization in terms of the Hodge diamonds is applied to explicitly classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type.     

Intermediate Jacobians and Hodge structures of moduli spaces

The mixed Hodge structure on the low degree cohomology of the moduli space of vector bundles on a curve is studied. Analysis of the third cohomology yields a new proof of a Torelli theorem.     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

The locus of Hodge classes in an admissible variation of mixed Hodge structure

We generalize the theorem of E. Cattani, P. Deligne, and A. Kaplan to admissible variations of mixed Hodge structure.     

Quartic double solids with ordinary singularities

We study the mixed Hodge structure on the third homology group of a threefold which is the double cover of projective three-space ramified over a quartic surface with a double conic. We deal with the Torelli problem for such threefolds.     

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].     

Weight filtration of the limit mixed Hodge structure at infinity for tame polynomials

We give three new proofs of a theorem of C. Sabbah asserting that the weight filtration of the limit mixed Hodge structure at infinity of cohomologically tame polynomials coincides with the monodromy filtration up to a certain shift depending on the unipotent or non-unipotent monodromy part.     

Motivic integral of K3 surfaces over a non-archimedean field

We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) [34] of a K3 surface over C((t)) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces.