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.
* Incidence variety compactification. The problem of the compactification of the strata is extensively studied from different approaches in a joint work of Bainbridge, Chen, Gendron, Grushevsky, and Moeller (see @cite_13 and @cite_26 ). Their compactification (called incidence variety compactification ) is slightly different from the one that we use here. We will recall their definitions in since we will make use of some of their results.
@cite_26 @cite_13
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The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of the odd minimal stratum in genus three. 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. It was shown by Connes, Douglas, Schwarz [hep-th 9711162] that one can compactify M(atrix) theory on a non-commutative torus To. We prove that compactifications on Morita equivalent tori are in some sense physically equivalent. This statement can be considered as a generalization of non-classical SL(2,Z)N duality conjectured by Connes, Douglas and Schwarz for compactifications on two-dimensional non-commutative tori. Abstract The effective action for type II string theory compactified on a six-torus is N = 8 supergravity, which is known to have an E7 duality symmetry. We show that this is broken by quantum effects to a discrete subgroup, E 7 ( Z ) , which contains both the T-duality group O(6, 6; Z ) and the S-duality group SL(2; Z ). We present evidence for the conjecture that E 7 ( Z ) is an exact ‘U-duality’ symmetry of type II string theory. This conjecture requires certain extreme black hole states to be identified with massive modes of the fundamental string. The gauge bosons from the Ramond-Ramond sector couple not to string excitations but to solitons. We discuss similar issues in the context of toroidal string compactifications to other dimensions, compactifications of the type II string on K3 × T2 and compactifications of 11-dimensional supermembrane theory.