A Hirzebruch proportionality principle in Arakelov geometry

Generalized automorphic sheaves and the proportionality principle of Hirzebruch-Mumford

We axiomatize the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalized automorphic sheaf is proposed which includes sheaves of (meromorphic) sections of automorphic vector bundles with prescribed vanishing and pole orders along strata in the compactification, and their quotients. These include, for instance, sheaves of Jacobi forms and weakly holomorphic modular forms. Using this machinery we give a short and purely algebraic proof of the proportionality theorem of Hirzebruch and Mumford.

Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type

For many classical moduli spaces of orthogonal type there are results about the Kodaira dimension. But nothing is known in the case of dimension greater than 19. In this paper we obtain the first results in this direction. In particular the modular variety defined by the orthogonal group of the even unimodular lattice of signature (2, 8m+2) is of general type if m is at least 5.

数域上 Hirzebruch-Kleinschmidt 簇的有理点计数

Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields

本文研究了数域上 Picard 秩为 2 的光滑射影分裂环面簇在 Arakelov 高度下的有理点渐近增长，证明其可分解为有限个不相交子簇并给出显式渐近公式，同时提供了包含 Hirzebruch 曲面的实例。

The Hirzebruch-Mumford volume for the orthogonal group and applications

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice L of rank at least 3. If Γis an arithmetic subgroup of the group O(L) of isometries of L and L has signature (2,n), then an application of Hirzebruch-Mumford proportionality allows us to determine the leading term of the growth of the dimension of the spaces S_k(Γ) of cusp forms of weight k, as k goes to infinity. We compute this in a number of examples, which are important for geometric applications.