Polarized variation of Hodge structures of Calabi–Yau type and characteristic subvarieties over bounded symmetric domains

In this paper, we extend the construction of the canonical polarized variation of Hodge structures over tube domain considered by Gross (Math Res Lett 1:1–9, 1994) to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of Hodge structures, which we call characteristic subvarieties. We prove that the characteristic subvariety of the canonical polarized variations of Hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by Mok (Ann Math 125(1):105–152, 1987). We verified the generating property of Gross for all irreducible bounded symmetric domains, which was predicted in Gross (Math Res Lett 1:1–9, 1994).     

A note on the Bogomolov-type smoothness on deformations of the regular parts of isolated singularities

We apply the Tian-Todorov method, proving the Bogomolov
smoothness theorem (for deformations of compact Kähler manifolds) to deformations of the regular part of a Stein space with a finite number of isolated singular points. By the argument based on the Hodge structure on a strongly pseudo-convex Kähler domain or on a punctured Kähler space, we obtain an unobstructed subspace of the infinitesimal deformation space.

     

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.     

Hodge structures for orbifold cohomology

We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology for projective -orbifolds satisfying a ``Hard Lefschetz Condition'. Furthermore, the total cohomology forms a mixed Hodge structure that is polarized by every element of the Kähler cone of . Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of .

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 , in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

     

The second cohomology with symmplectic coefficients of the moduli space of smooth projcetive curves

Each finite dimensional irreducible rational representation V of the symplectic group Sp_2g(Q) determines a generically defined local system V over the moduli space M_g of genus g smooth projective curves. We study H² (M_g; V) and the mixed Hodge structure on it. Specifically, we prove that if g 6, then the natural map IH²(M~_g; V) H²(M_g; V) is an isomorphism where M~_{_g} is tfhe Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H²(M_g; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H²(M_g; V) for 3 g < 6. Results of this article can be applied in the study of relations in the Torelli group T_g.     

Monodromy of Variations of Hodge Structure

We present a survey of the properties of the monodromy of local systems on quasi-projective varieties which underlie a variation of Hodge structure. In the last section, a less widely known version of a Noether–Lefschetz-type theorem is discussed.     

Schubert varieties as variations of Hodge structure

We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS ‘span’ the space of all infinitesimal VHS; and (3) show that the cohomology classes dual to the Schubert VHS form a basis of the invariant characteristic cohomology associated with the infinitesimal period relation (a.k.a. Griffiths’ transversality).     

Addendum to: Hodge Classes on Self-Products of a Variety with an Automorphism

There are infinitely many fundamentally distinct families of polarized Abelian fourfolds of Weil type with multiplication from the cyclotomic field of cube roots of unity. The Hodge conjecture is shown to hold at a sufficiently general fiber in any of these families.     

Moduli of polarized Hodge structures

Around 1970 Griffiths introduced the moduli of polarized Hodge structures/ the period domain D and described a dream to enlarge D to a moduli space of degenerating polarized Hodge structures. Since in general D is not a Hermitian symmetric domain, he asked for the existence of a certain automorphic cohomology theory for D, generalizing the usual notion of automorphic forms on symmetric Hermitian domains. Since then there have been many efforts in the first part of Griffith's dream but the second part still lives in darkness. The objective of the present text is two-folded. First, we give an exposition of the subject. Second, we give another formulation of the Griffiths problem, based on the classical Weierstrass uniformization theorem.     

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.     

The de rham homotopy theory of complex algebraic varieties I

In this paper we use Chen's iterated integrals to put a mixed Hodge structure on the homotopy Lie algebra of an arbitrary complex algebraic variety, generalizing work of Deligne and Morgan. Similar techniques are used to put a mixed Hodge structure on other topological invariants associated with varieties that are accessible to rational homotopy theory such as the cohomology of the free loopspace of a simply connected variety.Supported in part by the National Science Foundation through grants MCS-8201642, DMS-8401175 and MCS-8108814(A04).     

On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in {\mathbb{P}^3}

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi–Yau threefolds coming from eight planes in ${\mathbb{P}^3}$ does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.     

The Hodge Structure on a Filtered Boolean Algebra

Let (B _n) be the order complex of the Boolean algebra and let B(n, k) be the part of (B _n) where all chains have a gap at most k between each set. We give an action of the symmetric group S _l on the l-chains that gives B(n, k) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The S _n action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary.We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods.Finally we present some data on the homology of B(n, k).     

Transformée de Mellin des intégrales-fibres associées aux singularités isolées d'intersection complète quasihomogènes. (Mellin Transform of Fibre Integrals Associated to Isolated Complete Intersection Singularities)

The Mellin transform of the fibre integral is calculated for certain quasihomogeneous isolated complete intersection singularities (above all, unimodal singularities of the list by Giusti and Wall). We show the symmetry property of the Gauss–Manin spectra (Theorem 3.1) and shed light on the lattice structure of the poles of the Mellin transform that are expressed by means of some topological data of the singularities (Theorem 4.3, Theorem 5.2). As an application of these results, we express the Hodge number of the fibre in terms of the Gauss–Manin spectra.     

Some aspects of the Hodge conjecture

I will discuss positive and negative results on the Hodge conjecture. The negative aspects come on one side from the study of the Hodge conjecture for integral Hodge classes, and on the other side from the study of possible extensions of the conjecture to the general K?hler setting. The positive aspects come from algebraic geometry. They concern the structure of the so-called locus of Hodge classes, and of the Hodge loci. This article is based on the 1st Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on November 25 and 26, 2006.     

Limits of Hodge Structure in Several Variables

We define the notion of a morphism of generalized semi-stable type, which is a generalization of the notion of a semistable degeneration over a curve. We partially generalize Steenbrink's results on the limit of Hodge structures to the case of such a morphism. As an application we prove the E1-degeneration of the relative Hodge–De Rham spectral sequence for this case.     

A formula of two-partition Hodge integrals

Motivated by the Mariño-Vafa formula of Hodge integrals and physicists' predictions on local Gromov-Witten invariants of toric Fano surfaces in a Calabi-Yau threefold, the third author conjectured a formula of certain Hodge integrals in terms of certain Chern-Simons invariants of the Hopf link. We prove this formula by virtual localization on moduli spaces of relative stable morphisms.

     

On K1 and K2 of Algebraic Surfaces

In this paper we study higher Chow groups of smooth, projective surfaces over a field k of characteristic zero, using some new Hodge theoretic methods which we develop for this purpose. In particular we investigate the subgroup of CH ^r+1 (X,r) with r = 1,2 consisting of cycles that are supported over a normal crossing divisor Z on X. In this case, the Hodge theory of the complement forms an interesting variation of mixed Hodge structures in any geometric deformation of the situation. Our main result is a structure theorem in the case where X is a very general hypersurface of degree d in projective 3-space for d sufficiently large and Z is a union of very general hypersurface sections of X. In this case we show that the subgroup of CH ^r+1 (X,r) we consider is generated by obvious cycles only arising from rational functions on X with poles along Z. This can be seen as a generalization of the Noether–Lefschetz theorem for r = 0. In the case r = 1 there is a similar generalization by Müller-Stach, but our result is more precise than it, since it is geometric and not only cohomological. The case r = 2 is entirely new and original in this paper. For small d, we construct some explicit examples for r = 1 and 2 where the corresponding higher Chow groups are indecomposable, i.e. not the image of certain products of lower order groups. In an appendix Alberto Collino constructs even more indecomposable examples in CH ³ (X,2) which move in a one-dimensional family on the surface X.Contribution to appendix.     

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].