Coxeter transformations of the derived categories of coherent sheaves (Q397836)

The authors study the Coxeter transformation of the derived category \(\mathcal{D}^b(X)\) of a smooth, projective \(n\)-dimensional variety \(X\). This transformation is given by \(S_X[-1] = - \otimes \omega_X[n-1]\). In the article, a short survey is given about the representation-theoretic origins of these transformations: They were introduced by Coxeter to study compact Lie groups. Passing representation theory of finite-dimensional algebras, where they are known as Auslander-Reiten translations, the found their way to algebraic geometry by tilting theory.NEWLINENEWLINEHere, the corresponding linear map on the Grothendieck group \(K_0(X)_{\mathbb Q}\) is the main object of study, in the case that its rank \(m\) is finite. The main result is that its characteristic polynomial is \((x+(-1)^n)^m\). Therefore, one obtains a necessary condition that \(\mathcal{D}^b(X)\) is equivalent to the derived category of a finite-dimensional algebra.NEWLINENEWLINEAnother central topic is the Jordan canonical form of the Coxeter transformation. The authors can describe it expicitly in the case of toric varieties in terms of the combinatorics of the associated fans, whenever the (anti)canonical divisor is a Lefschetz element (e.g.\ if ample). The key ingredients to the proof are the hard Lefschetz theorem and that the cohomology \(H^\bullet(X,\mathbb Z)\) is concentrated in even degrees. As an example, the Jordan blocks for all toric del Pezzo surfaces and toric Fano three-folds are given.NEWLINENEWLINEThese ideas are also applied to rational surfaces \(X\), where \(K_X\) is a Lefschetz element if and only if the rank of the Picard group is not 10. This induces also a dichotomy of the structure of the Jordan blocks.NEWLINENEWLINEFinally, as an application, a formula is given for the Jordan normal form of tensor products of invertible matrices.