We show that Q-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable Q-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.
Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper
we consider cohomological rank functions of Q-twisted (complexes of) coherent sheaves on abelian
varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincar´e
transform, which has several consequences. In many concrete geometric contexts these functions
provide useful invariants. We illustrate this with two different applications, the first one to GVsubschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.
For each integer we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the A2 quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the A2 singularity.