Maximal slope of tensor product of Hermitian vector bundles

2007/06/05 by Huayi Chen, Chen, Huayi · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #math.AG

paper · pdf · doi:10.48550/arxiv.0706.0690

openalex publication_date 2007/06/05 · arxiv created 2008/01/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an upper bound for the maximal slope of the tensor product of several non-zero Hermitian vector bundles on the spectrum of an algebraic integer ring. By Minkowski's theorem, we need to estimate the Arakelov degree of an arbitrary Hermitian line subbundle M of the tensor product. In the case where the generic fiber of M is semistable in the sense of geometric invariant theory, the estimation is established by constructing, through the classical invariant theory, a special polynomial which does not vanish on the generic fibre of M. Otherwise we use an explicte version of a result of Ramanan and Ramanathan to reduce the general case to the former one.

Citations

• On the heat equation and the index theorem
• Harder-Narasimhan categories
• Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

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