Coxeter transformations of the derived categories of coherent sheaves

TL;DR

This paper investigates the Coxeter transformation in derived categories of coherent sheaves on smooth varieties, revealing its characteristic polynomial and Jordan form relations to Betti numbers, with applications to matrix tensor products.

Contribution

It provides explicit formulas for Coxeter transformations' characteristic polynomials and Jordan forms, linking algebraic and topological invariants of varieties.

Findings

Characteristic polynomial is $(\,\lambda+(-1)^n\)^m for finite rank Grothendieck groups.

Jordan forms of Coxeter transformations relate to Betti numbers of varieties.

Computed Jordan forms for tensor products of matrices.

Abstract

In this paper, we study the Coxeter transformation of the derived categories of coherent sheaves on smooth complete varieties. We first obtain that if the rank of the Grothendieck group is finite, say $m$, then its characteristic polynomials is $(\lambda+(-1)^{n})^m$, where $n$ is dimension of the variety. We then study the Jordan canonical forms of the Coxeter transformations for rational surfaces, smooth complete toric varieties with ample canonical or anticanonical bundles, and prove that their Jordan canonical forms can determine and be determined by the Betti numbers of these varieties. As an application, we compute the Jordan canonical forms of tensor products of matrices.