## ERC StG 307119 - mini-workshop at Pavia
## 26 November 2015, Aula Beltrami, Dipartimento di Matematica "F. Casorati", Pavia We will review some results and open problems in the study of the geometry of (smooth, complex) Fano 4-folds, in particular in the case of large second Betti number. An important result of Chen-Donaldson-Sun and Tian relates the existence of Kaehler-Einstein metrics on Fano varieties to an algebro-geometric notion called K-stability. K-stability is however understood in very few cases. We show that certain finite covers of K-stable Fano varieties are K-stable. This talk is about the following converse of Riemann's theorem: let A be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum Theta=C+Y of a curve C and a codimension two subvariety Y. Then C is smooth and A is isomorphic to the Jacobian of C. As an easy consequence, we deduce that an irreducible theta divisor admits a dominant rational map from a product of curves if and only if the corresponding principally polarized abelian variety is the Jacobian of a curve. |