Abstract. In this paper we prove a conjecture of Hershel Farkas [8] that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian.We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.

We complete the program indicated by the Ansatz of D'Hoker and Phong in genus 4 by proving the uniqueness of the restriction to Jacobians of the weight 8 Siegel cusp forms satisfying the Ansatz. We prove dim[ 4 (1, 2), 8] 0 = 2 and dim[ 4 (1, 2), 8] = 7. In each genus, we classify the linear relations among the selfdual lattices of rank 16. We extend the program to genus 5 by constructing the unique linear combination of theta series that satisfies the Ansatz.

2011It also contains Z andH := H/Z is a group of order 12 288 = 2 12 · 3 which acts faithfully on X and which leaves the holomorphic 3-form invariant. In [FS2] has been proved:Theorem. Let G be any subgroup ofH. Then there exists a desingularization (in the category of complex spaces) of X /G that is a weak Calabi-Yau threefold.

Abstract. We show that a generic principally polarized abelian variety is uniquely determined by its gradient theta-hyperplanes, the non-projectivized version of those studied in [Ca01], [CS00], [CS02], which in a sense are a generalization to ppavs of bitangents of plane curves. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli space of ppavs. Definitions and notationsWe denote by H g the Siegel upper half-space -the space of complex symmetric g × g matrices with positive definite imaginary part. An element τ ∈ H g is called a period matrix, and defines the complex abelian variety X τ := C g /Z g + τ Z g . The group Γ g := Sp(2g, Z) acts on H g by automorphisms: for γ := a b c d ∈ Sp(2g, Z) the action is γτ := (aτ + b)(cτ + d) −1 . The quotient of H g by the action of the symplectic group is the moduli space of principally polarized abelian varieties (ppavs): A g := H g /Sp(2g, Z). A ppav is called irreducible if it is not a direct product of two lower-dimensional ppavs, i.e. if its period matrix τ is not conjugate by the action of Γ g to a matrix that splits as τ 1 ⊕ τ 2 for two lower-dimensional period matrices. For us the case g = 1 is special and in the following we will always assume g > 1.We define the level subgroups of the symplectic group to beThe corresponding level moduli spaces of ppavs are denoted A n g and A n,2n g , respectively.

Abstract. Combining certain identities for modular forms due to Igusa with Schottky-Jung relations, we study the cosmological constant for the recently proposed ansatz for the chiral superstring measure in genus 5. The vanishing of this cosmological constant turns out to be equivalent to the long-conjectured vanishing of a certain explicit modular form of genus 5 on the moduli of curves M 5 , and we disprove this conjecture, thus showing that the cosmological constant for the proposed ansatz does not vanish identically. We exhibit an easy modification of the genus 5 ansatz satisfying factorization constraints and yielding a vanishing cosmological constant. We also give an expression for the cosmological constant for the proposed ansatz that should hold for any genus if certain generalized Schottky-Jung identities hold.