The WYSIWYG compactification

Abstract

We show that the partial compactification of a stratum of Abelian differentials previously considered by Mirzakhani and Wright is not an algebraic variety. Despite this, we use a combination of algebro-geometric and other methods to provide a short, unconditional proof of Mirzakhani and Wright's formula for the tangent space to the boundary of a GL(2,R) orbit closure, and give new results on the structure of the boundary.

arXiv commentMinor revisions, primarily to sections 7 and 8

References in corpus (13)

• Connected components of the moduli spaces of Abelian differentials with prescribed singularities
• Compactification of strata of abelian differentials
• Semisimplicity and rigidity of the Kontsevich-Zorich cocycle
• Splitting mixed Hodge structures over affine invariant manifolds
• Topologies and structures of the Cremona groups
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• Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle
• The moduli space of multi-scale differentials
• Full rank affine invariant submanifolds
• The Deligne-Mumford and the Incidence Variety Compactifications of the Strata of $\Omega {\mathcal{M}}_g$
• Différentielles à singularités prescrites
• GL(2,R) Orbit Closures in Hyperelliptic Components of Strata

Cited by in corpus (2)

• Equations of linear subvarieties of strata of differentials
• Quasidiagonals in Strata of Translation Surfaces