Introduction. There has been a recent explosion in the number of mathematical publications due to the discovery of a certain duality between some families of Calabi-Yau threefolds made by a group of theoretical physicists (see [11,26] for references). Roughly speaking this duality, called mirror symmetry, pairs two families F and F ∗ of Calabi-Yau threefolds in such a way that the following properties are satisfied:

Abstract: Quasisymmetric homeomorphisms of the circle, that arise in the Teichmüller theory of Riemann surfaces as boundary values of quasiconfomal diffeomorphisms of the disk, have fractal graphs in general and are consequently not so amenable to usual analytical or calculus procedures. In this paper we make use of the remarkable fact this group QS(S1) acts by substitution (i.e., pre-composition) as a family of bounded symplectic operators on the Hilbert space H=“H1/2 ” (comprising functions mod constants on S1 possessing a square-integrable half-order derivative). Conversely, and that is also important for our work, quasisymmetric homeomorphisms are actually characterized amongst homeomorphisms of S1 by the property of preserving the space H. Interpreting H via boundary values as the square-integrable first cohomology of the disk with the cup product symplectic structure, and complex structure provided by the Hodge star, we obtain a universal form of the classical period mapping extending the map of [12] [13] from Diff(S 1)/Mobius(S 1) to all of QS(S1)/Mobius(S 1) – namely to the entire universal Teichmüller space, T(1). The target space for the period map is the universal Siegel space of period matrices; that is the space of all the complex structures on H that are compatible with the canonical symplectic structure.

The Abel–Jacobi map of the family of elliptic quintics lying on a general cubic threefold is studied. It is proved that it factors through a moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c1 = 0, c2 = 2, whose general point represents a vector bundle obtained by Serre’s construction from an elliptic quintic. The elliptic quintics mapped to a point of the moduli space vary in a 5-dimensional projective space inside the Hilbert scheme of curves, and the map from the moduli space to the intermediate Jacobian is étale. As auxiliary results, the irreducibility of families of elliptic normal quintics and of rational normal quartics on a general cubic threefold is proved. This implies the uniqueness of the moduli component under consideration. The techniques of Clemens–Griffiths and Welters are used for the calculation of the infinitesimal Abel–Jacobi map.

Abstract. Koiso was first to introduce the Weil-Petersson metric in higher dimension. Tian showed that a moduli space of Calabi-Yau n-manifolds comes naturally with Weil-Petersson metric. In this paper we focus on determining for which degenerations the central fibre is at finite distance with respect to the Weil-Petersson metric. First we obtain a simple condition on the limiting mixed Hodge structure which is a necessary and sufficient condition for finite Weil-Petersson distance to the central fibre. This issue has been raised in the Physics literature but not extensively analyzed there. Then we combine the result with the canonical mixed Hodge structure of the central fibre and obtain a simple cohomological necessary and sufficient condition for the central fibre to be at finite distance. As a corollary, we prove that a central fibre with simple nodes is at finite distance.

Abstract. The principle “ambient cohomology of a Kaehler manifold annihilates obstructions ” has been known and exploited since pioneering work of Kodaira. This paper extends and unifies many known results in two contexts, abstract deformations of compact Kaehler manifolds and deformations of submanifolds within a given deformation of the ambient manifold. 1.

This is a survey of history, methods and developments in the theory of cycle spaces of flag domains, and new results on double fibration transforms and their applications.

Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over Q carries a natural Higgs bundle structure ¯ ∂ + θ which is invariant under the C ∗ action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov-Witten potential of the underlying family of Calabi–Yau threefolds. 1

. We consider a mixed Hodge module M on a normal surface singularity (X; x) and a holomorphic function germ f : (X; x) ! (C; 0). For the case that M has an abelian local monodromy group, we give a formula for the spectral pairs of f with values in M. This result is applied to generalize the Sebastiani-Thom formula and to describe the behaviour of spectral pairs in series of singularities. Contents 1. Introduction 150 2. Mixed Hodge Modules and Spectral Pairs 151 3. The General Setup 154 4. The Definition of Spp \Gamma 154 5. The Main Result 157 5.1. The proof of Theorem 5.1 158 6. Examples 162 6.1. Abelian coverings 162 6.2. The case of the trivial mixed Hodge module 164 7. Topological Series of Curve Singularities 165 7.1. Geometric meaning 166 7.2. Topologically trivial series 166 7.3. Intrinsic invariants 167 8. Topological Series of Plane Singularities with Coefficients in a Mixed Hodge Module 167 8.1. Limit mixed Hodge structures 168 8.2. Intrinsic meaning 168 8.3. Further compu...

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Abstract. Assuming suitable convergence properties for the Gromov-Witten potential of a Calabi-Yau manifold X one may construct a polarized variation of Hodge structure over the complexified Kähler cone of X. In this paper we show that, in the case of fourfolds, there is a correspondence between “quantum potentials ” and polarized variations of Hodge structures that degenerate to a maximally unipotent boundary point. Under this correspondence, the WDVV equations are seen to be equivalent to the Griffiths ’ trasversality property of a variation of Hodge structure. 1.