We derive the (g − 2)(g − 3)/2 linearly independent relations among the products of pairs in a basis of holomorphic abelian differentials in the case of compact non-hyperelliptic Riemann surfaces of genus g ≥ 4. By the Kodaira-Spencer map this leads to the modular invariant metric on the moduli space induced by the Siegel metric.

Inspired by mirror symmetry, we investigate some differential geometric aspects of the space Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory Weil-Petersson geometry stringy Kahler moduli space. A few basic examples are studied. In particular, identify our metric with Bergman Siegel modular variety in case self-product an elliptic curve.

The geometry of moduli spaces of complex abelian varieties and of compactifications of those moduli spaces has been the object of much study in the last few years. Given such a compactified moduli space it is natural to ask about the fundamental group of a resolution of singularities. This problem has been studied in some special cases, for instance in [K], [HK] and [HS]. It follows from the re...

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length was proved by Eskin and Masur to generically have quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics calle...

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length was proved by Eskin and Masur to generically have quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics calle...

A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Θ s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian...

Let h2 = Sp(4,R)/U(2) be the Siegel upperhalf space of rank 2. The quotient Sp(4,Z)\h2 has three remarkable properties: (a) it is the moduli space of principally polarized abelian surfaces, (b) it has the structure of a quasi-projective complex algebraic variety which is defined over the rational numbers Q, and (c) it has a natural compactification (the BailyBorel Satake compactification) which...

Canonical curves are characterized by the vanishing of combinatorial products of g + 1 determinants of the holomorphic abelian differentials. This also implies the characterization of canonical curves in terms of (g − 2)(g − 3)/2 theta identities and the explicit expression of the volume form on the moduli space of canonical curves induced by the Siegel metric. Together with the Mumford form, i...

Let h2 = Sp(4,R)/U(2) be the Siegel upperhalf space of rank 2. The quotient Sp(4,Z)\h2 has three remarkable properties: (a) it is the moduli space of principally polarized abelian surfaces, (b) it has the structure of a quasi-projective complex algebraic variety which is defined over the rational numbers Q, and (c) it has a natural compactification (the BailyBorel Satake compactification) which...