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

Abstract

Let \(\overline{{\mathcal {H}}}_{g,n}\) denote the Hodge bundle over \(\overline{{\mathfrak {M}}}_{g,n}\), and \({\mathbb {P}}\overline{{\mathcal {H}}}_{g,n}\) its associated projective bundle. Let \({\mathcal {H}}_{g,n}\) and \({\mathbb {P}}{\mathcal {H}}_{g,n}\) be respectively the restriction of \(\overline{{\mathcal {H}}}_{g,n}\) and \({\mathbb {P}}\overline{{\mathcal {H}}}_{g,n}\) to the smooth part \({\mathfrak {M}}_{g,n}\) of \(\overline{{\mathfrak {M}}}_{g,n}\). The Hodge norm provides us with a Hermtian metric on \({\mathscr {O}}(-1)_{{\mathbb {P}}{\mathcal {H}}_{g,n}}\). Let \(\Theta \) denote the curvature form of this metric. In this paper, we show that if \(\overline{{\mathcal {N}}}\) is a subvariety of \({\mathbb {P}}\overline{{\mathcal {H}}}_{g,n}\) that intersects \({\mathcal {H}}_{g,n}\), then the integral of the top power of \(\Theta \) over the smooth part of \(\overline{{\mathcal {N}}}\cap {\mathbb {P}}{\mathcal {H}}_{g,n}\) equals the self-intersection number of the tautological divisor \(c_1({\mathscr {O}}(-1)_{{\mathbb {P}}\overline{{\mathcal {H}}}_{g,n}})\cap \overline{{\mathcal {N}}}\) in \(\overline{{\mathcal {N}}}\). This implies that the volume of a linear subvariety of \({\mathbb {P}}{\mathcal {H}}_{g,n}\) whose local coordinates do not involve relative periods can be computed by the intersection number of its closure in \({\mathbb {P}}\overline{{\mathcal {H}}}_{g,n}\) with some power of any divisor representing the tautological line bundle. We also genralize this statement to the bundles \({\mathbb {P}}\overline{{\mathcal {H}}}^{(k)}_{g,n}\), \(k \in {\mathbb {Z}}_{\ge 2}\), of k-differentials with poles of order at most \((k-1)\) over \(\overline{{\mathfrak {M}}}_{g,n}\). To obtain these results, we use the existence of an appropriate desingularization of \(\overline{{\mathcal {N}}}\) and a deep result of Kollár (Subadditivity of the Kodaira Dimension: Fiber of General Type, Algebraic Geometry, Sendai, 1985, Advanced studies in Pure Math. (1987)) on variation of Hodge structure.