Geometry of theta divisors --- a survey
arXiv:1204.2734
Abstract
We survey the geometry of the theta divisor and discuss various loci of principally polarized abelian varieties (ppav) defined by imposing conditions on its singularities. The loci defined in this way include the (generalized) Andreotti-Mayer loci, but also other geometrically interesting cycles such as the locus of intermediate Jacobians of cubic threefolds. We shall discuss questions concerning the dimension of these cycles as well as the computation of their class in the Chow or the cohomology ring. In addition we consider the class of their closure in suitable toroidal compactifications and describe degeneration techniques which have proven useful. For this we include a discussion of the construction of the universal family of ppav with a level structure and its possible extensions to toroidal compactifications. The paper contains numerous open questions and conjectures.
arXiv commentFinal version; remark 6.5 on the non-basic codimension 5 stratum missed in our previous work added
References in corpus (11)
- Perfect forms and the moduli space of abelian varieties
- Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces
- The Chow Ring of the Moduli Space of Abelian Threefolds
- Andreotti-Mayer loci and the Schottky problem
- Extending Torelli map to toroidal compactifications of Siegel space
- The Moduli Space of Abelian Varieties and the Singularities of the Theta Divisor
- The class of the locus of intermediate Jacobians of cubic threefolds
- Perfect forms and the cohomology of modular groups
- Singularities of theta divisors and the geometry of A_5
- Cohomology of the second Voronoi compactification of A_4
- Extended Torelli map to the Igusa blowup in genus 6, 7, and 8