论文标题
在项目活动的Abelian差异的模量空间中的线性亚变量的体积上
On the volumes of linear subvarieties in moduli spaces of projectivized Abelian differentials
论文作者
论文摘要
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带有$ n $标记点的固定的Riemman表面$ g $（所有电线杆都必须位于标记的点）。捆绑$ \ MATHCAL {h}^{（k）} _ {g，n} $及其关联的投影捆绑包$ \ Mathbb {p} \ Mathcal {h}^{h}^{（k）} _ {g，n} $ \ OVERLINE {\ MATHCAL {h}}}^{（k）} _ {g，n} $和$ \ Mathbb {p} \ overline {\ Mathcal {\ Mathcal {h}}}}^{（K） $ \ OVERLINE {\ MATHFRAK {M}} _ {G，N} $ of $ \ MATHFRAK {M} _ {G，N} $。我们证明了以下语句：让$ \ Mathcal {M} $是尺寸$ d $的子变量。用$ \ mathscr {o}（ - 1）_ {\ mathbb {p} \ overline {\ mathcal {\ mathcal {h}}}^{（k）} _ {g，n}} $ the {g，n}} $重武线束$ \ mathbb {p} \ overline {\ mathcal {h}}}^{（k）} _ {g，n} $。然后，$ d $ th hodge narm的$ d $ th的积分在$ \ mathscr {o}（ - 1）_ {\ mathbb {p} \ edromline {\ mathcal {\ mathcal {h}}}}}^（k）^{（k）} _ {g，n}} $ complote $ complote $ complote $ complote $ earts usect除数代表$ \ Mathscr {o}（ - 1）_ {\ Mathbb {p} \ edromline {\ Mathcal {h}}^{（k）} _ {g，n}} $的$ d $ -th功率$ \ mathbb {p} \ overline {\ mathcal {h}}}^{（k）} _ {g，n} $。结果，如果$ \ nathcal {m} $是项目活动的hodge捆绑$ \ mathbb {p} \ mathcal {h} _ {g，n}（= \ m马理bbb {p} {p} \ mathcal {时期，然后可以通过$ \ Mathcal {m} $ in in in Mathcal {m} $的$ \ MATHCAL {M} $的体积计算$ \ mathbb {p} \ overline {\ mathcal {h}} _ {g，n}（= \ m athbb {p} \ overline {\ mathcal {h}}}}}}^{（1）} _ {g，n}）$。
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 $\mathbb{P}\overline{\mathcal{H}}_{g,n}(=\mathbb{P}\overline{\mathcal{H}}^{(1)}_{g,n})$.
