Coxeter transformations of the derived categories of coherent sheaves

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.