Compactification of strata of abelian differentials

Abstract

We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.

v2: Final version, to appear in Duke Mathematical Journal. Various small corrections, additions, and clarifications

References in corpus (4)

• The moduli space of twisted canonical divisors
• Realizability of tropical canonical divisors
• Towards a tropical Hodge bundle
• On the effective cone of ${\overline{\mathcal{M}}}_{g,n}$

Cited by in corpus (36)

• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
• The moduli space of multi-scale differentials
• A transfer principle: from periods to isoperiodic foliations
• Full rank affine invariant submanifolds
• Realizability of tropical canonical divisors
• Cohomology classes of strata of differentials
• Infinitesimal structure of the pluricanonical double ramification locus
• Tautological relations via r-spin structures
• Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
• Affine geometry of strata of differentials
• Pixton's formula and Abel-Jacobi theory on the Picard stack
• Equations of linear subvarieties of strata of differentials
• Moduli spaces for Lamé functions and Abelian integrals of the second kind
• The boundary of linear subvarieties
• On the effective cone of ${\overline{\mathcal{M}}}_{g,n}$
• The Hurwitz space Picard rank conjecture for $d>g-1$
• The WYSIWYG compactification
• diffstrata -- a Sage package for calculations in the tautological ring of the moduli space of Abelian differentials
• Volumes and Siegel-Veech constants of $\mathcal{H}(2g-2)$ and Hodge integrals
• Measure bound for translation surfaces with short saddle connections
• Hyperelliptic Gorenstein curves and logarithmic differentials
• The image in the moduli space of curves of strata of meromorphic and quadratic differentials
• Moduli space of meromorphic differentials with marked horizontal separatrices
• Maximal gonality on strata of differentials and uniruledness of strata in low genus
• Moduli spaces of residueless meromorphic differentials and the KP hierarchy
• Reduction and lifting problem for differential forms on Berkovich curves
• Gorenstein curve singularities of genus three
• Effective Unique Ergodicity and Weak Mixing of Translation Flows
• Dynamical invariants and intersection theory on the flex and gothic loci
• Effective divisors in ${\overline{\mathcal{M}}}_{g,n}$ from abelian differentials
• Incidence varieties in the projectivized $k$-th Hodge bundle over curves with rational tails
• Tautological ring of strata of differentials
• Spin and hyperelliptic structures of log twisted differentials
• Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles
• Meromorphic quadratic differentials and measured foliations on a Riemann surface