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We show that the vanishing of the $(g+1)$-st power of the theta divisor on the universal abelian variety $mathcal{X}_g$ implies, by pulling back along a collection of Abel--Jacobi maps, the vanishing results in the tautological ring of $mathcal{M}_{g,n}$ of Looijenga, Ionel, Graber--Vakil, and Faber--Pandharipande. We also show that Pixtons double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem~$star$ of Graber and Vakil, and we provide an explicit algorithm for expressing any tautological class on $overline{mathcal{M}}_{g,n}$ of sufficiently high codimension as a boundary class.
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We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. We use this to obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus, and apply this bound in examples, and to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.
Relations among tautological classes on the moduli space of stable curves are obtained via the study of Wittens r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl2 is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Wittens r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r=4 relations are used to bound the Betti numbers of the tautological ring of the moduli of nonsingular curves. At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Wittens r-spin class. In the Appendix (with F. Janda), a conjecture relating the r=0 limit of Wittens r-spin class to the class of the moduli space of holomorphic differentials is presented.
The moduli space of stable bundles of rank 2 and degree 1 on a Riemann surface has rational cohomology generated by the so-called universal classes. The work of Baranovsky, King-Newstead, Siebert-Tian and Zagier provided a complete set of relations between these classes, expressed in terms of a recursion in the genus. This paper accomplishes the same thing for the non-compact moduli spaces of Higgs bundles, in the sense of Hitchin and Simpson. There are many more independent relations than for stable bundles, but in a sense the answer is simpler, since the formulas are completely explicit, not recursive. The results of Kirwan on equivariant cohomology for holomorphic circle actions are of key importance. Together, Parts I and II describe the cohomology rings of spaces of rank 2 Higgs bundles at essentially the same level of detail as is known for stable bundles.
Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $ell$-adic tautological systems for the case where the ground field is of characteristic $p$.
Let C be a complex curve of genus g, let J(C) be its Jacobian and let R(C) be its tautological ring, that is, the group of algebraic cycles modulo algebraic equivalence. We study the algebraic structure of R(C). In particular, we give a detailed description of all the possibilities that may occur for g<9: we construct convenient basis and we determine the matrices representing the Fourier transform and both intersection and Pontryagin products explicitly. In particular, we estimate the dimension of R(C).
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