The boundary of linear subvarieties

Abstract

We describe the boundary of linear subvarieties in the moduli space of multi-scale differentials. Linear subvarieties are algebraic subvarieties of strata of (possibly) meromorphic differentials that in local period coordinates are given by linear equations. The main example of such are affine invariant submanifolds, that is, closures of SL(2, R)-orbits. We prove that the boundary of any linear subvariety is again given by linear equations in generalized period coordinates of the boundary. Our main technical tool is an asymptotic analysis of periods near the boundary of the moduli space of multi-scale differentials which yields further techniques and results of independent interest.

The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials

“…Proof of Proposition 3.4. This is the main result of [Ben20] or the restatement in [BDG22, Proposition 3.3] and this together with the Proposition 3.2 implies the dimension statement.…”

The closure of double ramification loci via strata of exact differentials

“…Thus by [Ben20,Prop. 10.1] the intersection of DR c g ( µ) with the open boundary stratum corresponding to Γ is a linear subvariety given by Ann ⊕ 0 i=−L(Γ) Im gi .…”

“…Since exact differentials are described by the vanishing of all absolute periods, this requires an analysis of periods near the boundary of ΞM g,n (µ). The results obtained in [Ben20] and further refined in [BDG20] describe the boundary of subvarieties of strata of meromorphic differentials given by linear equations on periods in ΞM g,n (µ), and can thus be applied to the case of exact differentials. The idea of studying Hurwitz spaces via exact differentials has also appeared in [Sav17,Sec.…”

“…Furthermore, being an exact differential is equivalent to the vanishing of all absolute periods, a condition that is linear in period coordinates of the stratum. We can then describe the boundary of DR c g ( µ) in terms of twisted rational functions using the main result from [Ben20] and some further consequences from [BDG20]. A twisted rational function is a twistable rational function where we additionally mark all the critical points of the rational function, and such that the associated exact differential is a twisted differential in the sense of [BCGGM18].…”

Linear submanifolds and strata of k-differentials : invariants and applications

“…A well-behaved compactification of PΩM g,n (µ), the so called moduli space of multi-scale differentials PΞM g,n (µ), has been constructed by Bainbridge-Chen-Gendron-Grushevsky-Möller [BCGGM19b]. By a recent result of Benirschke [Ben20], the boundary of a GL 2 (R) + -orbit closure in the moduli space of multi-scale differentials is locally given by R-linear equations, but few non-trivial examples of such boundaries appear in the literature. In this article, which is still work in progress, we will highlight some aspects of the boundary of the closure of the gothic locus PΞG := PΩG ⊆ PΞM 4,6 (0 3 , 2 3 ).…”