Mixed Hodge Structure Research Articles

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.

Iterated integrals and higher order automorphic forms

Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's theory, we prove a structure theorem for automorphic forms of all orders. This allows us to define an analogue of a mixed Hodge structure on a space of higher order automorphic forms.

Hodge structures for orbifold cohomology

We construct a polarized Hodge structure on the primitive part of Chen and Ruan’s orbifold cohomology H o r b k ( X ) H_{orb}^k(X) for projective S L SL -orbifolds X X satisfying a “Hard Lefschetz Condition”. Furthermore, the total cohomology ⨁ p , q H o r b p , q ( X ) \bigoplus _{p,q}H_{orb}^{p,q}(X) forms a mixed Hodge structure that is polarized by every element of the Kähler cone of X X . Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of X X . This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of X X , in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

Polylogarithms

The k -th polylogarithm function is defined on |z|<1 by Li_k(z) = \sum_{n\geq 1} \frac{z^n}{n^k}. In the past 25 years or so, polylogarithms have appeared in many different areas of Mathematics. The following list is taken, for the most part, from [Oe]: volumes of polytopes in spherical and hyperbolic geometry, volumes of hyperbolic manifolds of dimension 3, combinatorial description of characteristic classes, special values of zeta functions, geometry of configurations of points in \mathbb P^1 , cohomology of GL_n(\mathbb C) , calculation of Green's functions associated to perturbation expansions in quantum field theory, Chen iterated integrals, regulators in algebraic K -theory, differential equations with nilpotent monodromy, Hilbert's problem on cutting and pasting, nilpotent completion of \pi_1(\mathbb P^1-\{0,1,\infty\}) , Bethe's Ansatz in thermodynamics, and combinatorial problems in quantum field theory. Of course, these problems are not all unrelated. One common thread is that values of polylogarithms appear naturally as periods of certain mixed Hodge structures associated to mixed Tate motives over cyclotomic fields. How these periods are related to special values of L -functions is a part of the Beilinson conjectures, which were discussed in a previous Arbeitsgemeinschaft. Since that time, the general picture has clarified. A number of talks are devoted to aspects of this more general philosophy (talk 2-5, 9, 10 13-17). The p -adic aspects of the theory have been studied and will be explained in the eleventh talk. In addition, a vast generalization, multiple polylogarithms of the form Li_{s_1,\dotsc,s_k} := \sum_{n_1>\ldots > n_k \geq 1} \frac{z_1^{n_1}\cdots z_k^{n_k}}{n_1^{s_1}\cdots n_k^{s_k}}, have come to play a role as presented in the fourth and twelfth talk. The connection between volumes of hyperbolic manifolds and polylogarithms is described in the sixth talk. The seventh talk introduces higher torsion and explains why polylogarithms occur in that setting. Polylogarithms play a role in physics. The eighth talk explains how zeta values, polylogarithms, and multiple polylogarithms appear in calculations of perturbative expansions in quantum field theory. The final talks 13-17 return to the relationship with special values of L -functions. The thirteenth and fourteenth talk are devoted to the Zagier conjectures for number fields. The fifteenth talk concerns the elliptic polylogarithm sheaves and Zagier's conjecture for elliptic motives. Finally, talk 16 and 17 describe how polylogarithms relate to Euler systems and to the Bloch-Kato conjectures on special values of L -functions. The Arbeitsgemeinschaft was organized by Spencer Bloch (University of Chicago), Guido Kings (Universität Regensburg) and Jörg Wildeshaus (Université Paris 13). It was held October 3rd – October 9th, 2004 with 46 participants.

String-Theorie und Geometrie

Of the three organizers of the conference N. Hitchin is a mathematician, A. Kapustin and W. Nahm are physicists. Ideas from physics, in particular from string theory and related areas had a profound influence on algebraic and differential geometry. Conversely, many developments in string theory use very recent mathematical results. There were 19 talks at the workshop, 12 by mathematicians and 7 by physicists, but in many cases an outside observer would have had difficulties to sort them out. Quantum field theory often allows continuous interpolations between geometrical structures of topologically different manifolds. A particularly important example is mirror symmetry, where the complexified Kähler cone of one manifold corresponds to the moduli space of complex structures of the mirror manifold. Several talks explored geometrical and number theoretical aspects of this connection. Mirror symmetry can be understood in terms of torus fibrations and an isomorphism between the quantum field theories given by maps to a torus and to its dual (T-duality). The B-field of string theory can be described in terms of non-commutative geometry on the tori, and its effect on T-duality has been investigated. The quantum field theories which provide these dualities are supersymmetric conformally invariant quantum field theories in two dimensions. In mathematics, large parts of their structures have been axiomatised in the language of vertex operator algebras, but much remains to be done. A promising mathematical approach is provided by the chiral de Rham complex. This cohomology of this complex seems to be invariant under mirror symmetry, but its physical meaning is not yet clear. This led to many discussions between mathematicians and physicists. Indeed, the free time for discussions was at least as important as the lectures for learning the language of researchers with a different background and for making use of their ideas. An area in which the common understanding is well advance is topological field theory. Here one has good axioms and many of the analytic problems of conformally invariant quantum field theories can be ignored. One can solve rather complex problems and one can make contact with new geometrical ideas, in particular generalised complex structures. The latter allow to interpolate between complex and symplectic geometry, which should become important for both physics and mathematics. To understand all of conformal field theory at a similar mathematical depth will take more time, but already now relations to mixed Hodge structures have given much insight. Eventually many so far intractable problems should be solvable by the use of quantum field theoretical ideas, for example the construction of the non-singular Ricci flat metrics on K3. Perturbative string theory can be described in terms of conformal field theory, but the complete theory has more excitations than strings. Among them the best understood are the D-branes, many features of which can be understood in terms of boundary states in conformal field theory. Some of these branes are related to complex submanifolds, others to certain Lagrangian submanifolds, and their description involves twisted K-theory and derived categories. The development of a common language between mathematicians and physicists is well under way, and the workshop has made a contribution which all participants found very stimulating. In Europe such meetings between mathematicians and string theorists are rare, so the workshop was particularly useful. Much of the time was reserved for informal discussions. On one evening there was a discussion on the relation between classical and quantum geometry. Weather predictions for the free afternoon had been bad, but finally the weather was splendid and the supply of Black Forest cake matched the demand.

Rational cohomology of the moduli space of genus 4 curves

We compute the mixed Hodge structure on the rational cohomology of the moduli space of smooth genus 4 curves. Specifically, we prove that its Poincare–Serre polynomial is 1 + t2u2 + t4u4 + t5u6. We show this by producing a stratification of the space, such that all strata are geometric quotients of complements of discriminants.

The Leray spectral sequence is motivic

Our goal is to prove that the Leray spectral sequence associated to a map of algebraic varieties is motivic in the following sense: If the singular cohomology groups of the category of quasiprojective varieties defined over a subfield of C can be canonically endowed with some additional structure, such as its mixed Hodge structure or its motive (in Nori's sense or as a product of realizations). Then the Leray spectral sequence for a projective map in this category is automatically compatible with this structure. As byproduct, we get a relatively cheap construction of a mixed Hodge structure on the cohomology of variation of Hodge structure of geometric origin.

Filtrations on Chow Groups and Transcendence Degree

For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending on the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of transcendence degree one. This may be viewed as a refinement of results of Nori, Schoen, and Green–Griffiths–Paranjape. For higher graded pieces we have a similar assertion assuming a conjecture of Beilinson and Grothendieck’s generalized Hodge conjecture.

Cycles over fields of transcendence degree 1

where (a) the group of cycles Z(V ) is the free abelian group on scheme-theoretic points of V of codimension p and (b) rational equivalence R(V ) is the subgroup generated by cycles of the form divW(f ), where W is a subvariety of V of codimension p − 1 and f is a nonzero rational function on it. There is a natural cycle class map clp : CH (V ) → H(V ), where the latter denotes the singular cohomology group H2p(V(C),Z) with the (mixed) Hodge structure given by Deligne (see [D]). The kernel of clp is denoted by F 1 CH(V ). There is an Abel–Jacobi map (see [G]) p : F 1 CH(V ) → IJp(H2p−1(V )),

Higher direct images of log canonical divisors

In this paper, we investigate higher direct images of log canonical divisors. After we reformulate Kollar's torsion-free theorem, we treat the relationship between higher direct images of log canonical divisors and the canonical extensions of Hodge filtration of gradedly polarized variations of mixed Hodge structures. As a corollary, we obtain a logarithmic version of Fujita--Kawamata's semi-positivity theorem. The final section is an appendix, which is a result of Morihiko Saito.

On the Reductive Borel-Serre Compactification, III: Mixed Hodge Structures

We establish a procedure for constructing compatible mixed Hodge structures for the cohomology of various topological compactifications of locally symmetric varieties, notably ones that are not algebraic varieties. This is carried out in full for the case of the reductive Borel-Serre compactification, and conditionally for the excentric compactifications. Introduction. In (BS), Borel and Serre gave a construction that compactified the arithmetic quotient X of a symmetric space (also certain more general locally homogeneous spaces) to a manifold-with-corners X. With its faces fibered by nilman- ifolds, it was both an easy and a useful little construction to collapse each fiber to a point, producing what is now called the reductive Borel-Serre compactification X red

Cycle map for strictly decomposable cycles

We introduce a class of cycles, called nondegenerate, strictly decomposable cycles, and show that the image of each cycle in this class under the refined cycle map to an extension group in the derived category of arithmetic mixed Hodge structures does not vanish. This class contains certain cycles in the kernel of the Abel-Jacobi map. The construction gives a refinement of Nori's argument in the case of a self-product of a curve. As an application, we show that a higher cycle which is not annihilated by the reduced higher Abel-Jacobi map produces uncountably many indecomposable higher cycles on the product with a variety having a nonzero global 1-form.

The Brauer group and the second Abel-Jacobi map for 0-cycles on algebraic varieties

This paper focuses on the connection between the Brauer group and the $0$-cycles of an algebraic variety. We give an alternative construction of the second $l$-adic Abel-Jacobi map for such cycles, linked to the algebraic geometry of Severi-Brauer varieties on $X$. This allows us then to relate this Abel-Jacobi map to the standard pairing between $0$-cycles and Brauer groups (see [M], [L]), completing results from [M] in this direction. Second, for surfaces, it allows us to present this map according to the more geometrical approach devised by M. Green in the framework of (arithmetic) mixed Hodge structures (see [G]). Needless to say, this paper owes much to the work of U. Jannsen and, especially, to his recently published older letter [J4] to B. Gross.

Tt* geometry, Frobenius manifolds, their connections, and the construction for singularities

The base space of a semiuniversal unfolding of a hypersurface singularity carries a rich geometry. By work of K. Saito and M. Saito is can be equipped with the structure of a Frobenius manifold. By work of Cecotti and Vafa it can be equipped with tt* geometry if the singularity is quasihomogeneous. tt* geometry generalizes the notion of variation of Hodge structures. In the second part of this paper (chapters 6-8) Frobenius manifolds and tt* geometry are constructed for any hypersurface singularity, using essentially oscillating integrals; and the intimate relationship between polarized mixed Hodge structures and this tt* geometry is worked out. In the first part (chapters 2-5) tt* geometry and Frobenius manifolds and their relations are studied in general. To both of them flat connections with poles are associated, with distinctive common and different properties. A frame for a simultaneous construction is given.

Singularities of variations of mixed Hodge structure

We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$, if and only if certain grading of the complexified weight filtration, depending smoothly on the Hodge filtration, extends across the puncture. In particular, the result exactly supplements Schmid's Theorem for pure structures, which holds for the graded variation, and gives a Hodge-theoretic condition for the relative monodromy weight filtration to exist.

The mixed Hodge structure on the fundamental group of a complement of hyperplanes

The complement of an arrangement of hyperplanes is a good example of a mixed Hodge structure on the fundamental group of an algebraic variety. We compute its isomorphism class using iterated integrals and then show that the cross ratio of a arrangement is a combinatorial and projective invariant. So cross ratio equivalent arrangements are isomorphic. Moreover, an isomorphism of polarized mixed Hodge structures on the fundamental group induces cross ratio equivalent arrangements.

Periods of limit mixed Hodge structures

The first goal of this paper is to explain some important results of Wilfred Schmid from his fundamental paper [30] in which he proves very general results which govern the behaviour of the periods of a of smooth projective variety Xt as it degenerates to a singular variety. As has been known since classical times, the periods of a smooth projective variety sometimes contain significant information about the geometry of the variety, such as in the case of curves where the periods determine the curve. Likewise, information about the asymptotic behaviour of the periods of a variety as it degenerates sometimes contain significant information about the degeneration and the singular fiber. For example, the Hodge norm estimates, which are established in [30] and [3], describe the asymptotics of the Hodge norm of a cohomology class as the variety degenerates in terms of its monodromy. They are an essential ingredient in the study of the L2 cohomology of smooth varieties with coefficients in a variation of Hodge structure [37, 4]. A second goal is to give some idea of how geometric and arithmetic information can be extracted from the limit periods, both in the geometric case and in the case of the limits of the mixed Hodge structures on fundamental groups of curves.

The mixed Hodge structure on the fundamental group of hyperelliptic curves and higher cycles

In this paper we give a geometrical interpretation of an extension of mixed Hodge structures (MHS) obtained from the canonical MHS on the group ring of the fundamental group of a hyperelliptic curve modulo the fourth power of its augmentation ideal. We show that the class of this extension coincides with the regulator image of a canonical higher cycle in a hyperelliptic jacobian. This higher cycle was introduced and studied by Collino.

Structures de Hodge mixtes et faisceaux réflexifs semistables

Let X be a smooth projective variety over an algebraically closed field of characteristic 0. We prove that the category of μ-semistable reflexive sheaves of slope μ equivariant for the action of some group on X is Abelian. The same claim for X= P 2 C and a stronger semistability condition gives us a geometric proof of the fact that the category of mixed Hodge structures is Abelian. To cite this article: O. Penacchio, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 475–480.

Variance of the spectral numbers and Newton polygons

Using the theory of the mixed Hodge structure one can define a notion of spectrum of a singularity or of a polynomial. Recently Claus Hertling proposed a conjecture about the variance of the spectrum of a singularity. Alexandru Dimca proposed a similar conjecture on polynomials. Here, we prove these two conjectures in the case of dimension 2 and when the singularity or the polynomial is Newton non-degenerated and commode.