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.
We mention that the log twisted differential considered in this paper is closely related to the strata compactification studied by Gu ' e r ' e ( @cite_16 ), whose construction also relies on the theory of stable log maps, but is motivated from the viewpoint of Gromov-Witten theory.
@cite_16
2320222014
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. 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. We give an approach for relative and degenerate Gromov--Witten invariants, inspired by that of Jun Li but replacing predeformable maps by transversal maps to a twisted target. The main advantage is a significant simplification in the definition of the obstruction theory. We reprove in our language the degeneration formula, extending it to the orbifold case. We set up an algebraic framework for the study of pseudoholomorphic discs bounding nonorientable Lagrangians, as well as equivariant extensions of such structures arising from a torus action. First, we define unital cyclic twisted @math algebras and prove some basic results about them, including a homological perturbation lemma which allows one to construct minimal models of such algebras. We then construct an equivariant extension of @math algebras which are invariant under a torus action on the underlying complex. Finally, we construct a homotopy retraction of the Cartan-Weil complex to equivariant cohomology, which allows us to construct minimal models for equivariant cyclic twisted @math algebras. In a forthcoming paper we will use these results to define and obtain fixed-point expressions for the open Gromov-Witten theory of @math , as well as its equivariant extension.