The anticanonical complex - a combinatorial tool for Fano varieties

Toric Fano varieties are in one to one correspondence with certain lattice polytopes, the so called Fano polytopes. Moreover classification of toric Fano varieties with respect to their singularity type turns out to be purely combinatorial: the position of lattice points in the Fano polytope determines the singularity type. The anticanonical complex has been introduced as a natural generalisation of the toric Fano polytope and so far has been succesfully used for the study of varieties with a torus action of complexity one. We enlarge the area of application of the anticanonical complex to Fano varieties with torus action of arbitrary complexity, for example arrangement varieties. In particular, we show that the possibility to apply the anticanonical complex to these varieties is connected to certain properties of their quotients.