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