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

For $k\in {\mathbb{Z}}_{>0}$, let ${\mathcal{H}}_{g,n}^{(k)}$ 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 ${\mathcal{H}}_{g,n}^{(k)}$ and its associated projective bundle $\mathbb{P}{\mathcal{H}}_{g,n}^{(k)}$ admit natural extensions, denoted by ${\bar{\mathcal{H}}}_{g,n}^{(k)}$ and $\mathbb{P}{\bar{\mathcal{H}}}_{g,n}^{(k)}$ respectively, to the Deligne-Mumford compactification ${\bar{\mathfrak{M}}}_{g,n}$ of ${\mathfrak{M}}_{g,n}$. We prove the following statement: let $\mathcal{M}$ be a subvariety of dimension $d$ of the projective bundle $\mathbb{P}{\mathcal{H}}_{g,n}^{(k)}$. Denote by $\mathcal{O}(-1{)}_{\mathbb{P}{\bar{\mathcal{H}}}_{g,n}^{(k)}}$ the tautological line bundle over $\mathbb{P}{\bar{\mathcal{H}}}_{g,n}^{(k)}$. Then the integral of the $d$-th power of the curvature form of the Hodge norm on $\mathcal{O}(-1{)}_{\mathbb{P}{\bar{\mathcal{H}}}_{g,n}^{(k)}}$ over the smooth part of $\mathcal{M}$ is equal to the intersection number of the $d$-th power of the divisor representing $\mathcal{O}(-1{)}_{\mathbb{P}{\bar{\mathcal{H}}}_{g,n}^{(k)}}$ and the closure of $\mathcal{M}$ in $\mathbb{P}{\bar{\mathcal{H}}}_{g,n}^{(k)}$. As a consequence, if $\mathcal{M}$ is a linear subvariety of the projectivized Hodge bundle $\mathbb{P}{\mathcal{H}}_{g,n}(=\mathbb{P}{\mathcal{H}}_{g,n}^{(1)})$ 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 $\mathbb{P}{\bar{\mathcal{H}}}_{g,n}(=\mathbb{P}{\bar{\mathcal{H}}}_{g,n}^{(1)})$.