Notices of the International Consortium of Chinese Mathematicians

Volume 7 (2019)

Number 2

Hodge bundles on smooth compactifications of Siegel varieties and applications

Pages: 1 – 18

DOI: https://dx.doi.org/10.4310/ICCM.2019.v7.n2.a1

Authors

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Yi Zhang (School of Mathematical Sciences, Fudan University, Shanghai, China)

Abstract

Siegel varieties are locally symmetric varieties. They are important and interesting in algebraic geometry and number theory. We construct a canonical Hodge bundle on a Siegel variety so that the holomorphic tangent bundle can be embedded into the Hodge bundle; we obtain that the canonical Bergman metric on a Siegel variety is same as the induced Hodge metric and we describe asymptotic behavior of this unique Kähler–Einstein metric explicitly; depending on these properties and the uniformitarian of Kähler–Einstein manifold, we study extensions of the tangent bundle over any smooth toroidal compactification We apply these results of Hodge bundles, to study dimension of Siegel cusp modular forms and general type for Siegel varieties.

Published 8 August 2019