Hodge Bundles on Smooth Compactifications of Siegel Varieties and Applications

TL;DR

This paper constructs a canonical Hodge bundle on Siegel varieties, analyzes the Bergman metric's compatibility with the Hodge metric, and applies these results to study the varieties' general type and compactifications.

Contribution

It introduces a canonical Hodge bundle on Siegel varieties, embeds the tangent bundle into it, and explores metric compatibility and applications to compactifications and general type.

Findings

Bergman metric is compatible with the Hodge metric on Siegel varieties.

Explicit asymptotic estimates of the Bergman metric are obtained.

Applications to the general type classification of Siegel varieties.

Abstract

We study Hodge bundles on Siegel varieties and their various extensions to smooth toroidal compactifications. Precisely, we construct a canonical Hodge bundle on an arbitrary Siegel variety so that the holomorphic tangent bundle can be embedded into the Hodge bundle, and we observe that the Bergman metric on the Siegel variety is compatible with the induced Hodge metric. Therefore we obtain the asymptotic estimate of the Bergman metric explicitly. Depending on these properties and the uniformitarian of K\"ahler-Einstein manifold, we study extensions of the tangent bundle over any smooth toroidal compactification. We also apply this result, together with Siegel cusp modular forms, to study general type for Siegel varieties.