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 $\mathrm{SL}(2,\mathbb{R})$-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.

v2: Final version, appeared in Journal of the European Mathematical Society. Various small corrections, additions, and clarifications

References in corpus (3)

• The moduli space of multi-scale differentials
• The Chern classes and the Euler characteristic of the moduli spaces of abelian differentials
• Canonical extensions of local systems

Cited by in corpus (1)

• Reconstructing orbit closures from their boundaries