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

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 $\mathbb{P}\overline{\mathcal{H}}{g,n}(=\mathbb{P}\overline{\mathcal{H}}^{(1)}{g,n})$.