We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar 'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
* Compactification via log-geometry. J 'er 'emy Gu 'er 'e constructed a moduli space of rubber'' differentials using log geometry. He proved that this space is endowed with a perfect obstruction theory. Moreover, if @math , this moduli space surjects onto the moduli space of twisted canonical divisors and the class @math is the push-forward of the virtual fundamental cycle (see @cite_15 ).
@cite_15
2320222014 2244320003 2537454422 2276886773
We give a log-geometric description of the space of twisted canonical divisors constructed by Farkas--Pandharipande. In particular, we introduce the notion of a principal rubber @math -log-canonical divisor, and we study its moduli space. It is a proper Deligne--Mumford stack admitting a perfect obstruction theory whose virtual fundamental cycle is of dimension @math . In the so-called strictly meromorphic case with @math , the moduli space is of the expected dimension and the push-forward of its virtual fundamental cycle to the moduli space of stable curves equals the weighted fundamental class of the moduli space of twisted canonical divisors. Conjecturally, it yields a formula of Pixton generalizing the double ramification cycle in the moduli space of stable curves. The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in the moduli space of Deligne-Mumford stable pointed curves which includes the space of canonical divisors as an open subset. The theory leads to geometric combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus g curves are of pure codimension g in the moduli spaces of stable pointed curves. In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors. As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases). Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected components, due to spin and hyperelliptic structures. We prove that the spin parity can be distinguished on the boundary of the log compactification. Moreover, combining the techniques of log geometry and admissible covers, we introduce log twisted hyperelliptic differentials, and prove that their moduli stack provides a toroidal compactification of the hyperelliptic loci in the open strata. Curves of genus g which admit a map to CP1 with specified ramification profile mu over 0 and nu over infinity define a double ramification cycle DR_g(mu,nu) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle DR_g(mu,nu) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DR_g(mu,nu) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain's formula in the compact type case. When mu and nu are both empty, the formula for double ramification cycles expresses the top Chern class lambda_g of the Hodge bundle of the moduli space of stable genus g curves as a push-forward of tautological classes supported on the divisor of nonseparating nodes. Applications to Hodge integral calculations are given.