On the volumes of linear subvarieties in moduli spaces of projectivized Abelian differentials

TL;DR

This paper establishes a geometric method to compute volumes of linear subvarieties in moduli spaces of projectivized Abelian differentials using intersection theory and curvature integrals, extending to compactifications.

Contribution

It proves a precise equality between curvature integrals and intersection numbers for subvarieties in the Hodge bundle, enabling volume calculations via algebraic geometry.

Findings

Curvature integrals equal intersection numbers for subvarieties.

Volumes of linear subvarieties can be computed through intersection theory.

Method applies to subvarieties with local coordinates excluding relative periods.

Abstract

For $k \in \mathbb{Z}_{>0}$, let $\mathcal{H}^{(k)}_{g,n}$ denote the vector bundle over $\mathfrak{M}_{g,n}$ whose every fiber consists of meromorphic $k$-differentials with poles of order at most $k-1$ on a fixed Riemman surface of genus $g$ with $n$ marked points (all the poles must be located at the marked points). The bundle $\mathcal{H}^{(k)}_{g,n}$ and its associated projective bundle $\mathbb{P}\mathcal{H}^{(k)}_{g,n}$ admit natural extensions, denoted by $\overline{\mathcal{H}}^{(k)}_{g,n}$ and $\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}$ respectively, to the Deligne-Mumford compactification $\overline{\mathfrak{M}}_{g,n}$ of $\mathfrak{M}_{g,n}$. We prove the following statement: let $\mathcal{M}$ be a subvariety of dimension $d$ of the projective bundle $\mathbb{P}\mathcal{H}^{(k)}_{g,n}$. Denote by $\mathscr{O}(-1)_{\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}}$ the tautological line bundle over $\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}$. Then the integral of the $d$-th power of the curvature form of the Hodge norm on $\mathscr{O}(-1)_{\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}}$ over the smooth part of $\mathcal{M}$ is equal to the intersection number of the $d$-th power of the divisor representing $\mathscr{O}(-1)_{\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}}$ and the closure of $\mathcal{M}$ in $\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}$. As a consequence, if $\mathcal{M}$ is a linear subvariety of the projectivized Hodge bundle $\mathbb{P}\mathcal{H}_{g,n}(=\mathbb{P}\mathcal{H}^{(1)}_{g,n})$ whose local coordinates do not involve relative periods, then the volume of $\mathcal{M}$ can be computed by the self-intersection number of the tautological line bundle on the closure of $\mathcal{M}$ in $\mathbb{P}\overline{\mathcal{H}}_{g,n}(=\mathbb{P}\overline{\mathcal{H}}^{(1)}_{g,n})$.