Positivity of the Chow-Mumford line bundle for families of K-stable Fano varieties

Giulio Codogni
(Universita' Roma Tre)

Abstract: The Chow-Mumford (CM) line bundle is a functorial line bundle defined on the base of any families of Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-polystable Fano varieties, whose existence has been recently proven by C. Xu and co-authors, building on the work of C. Birkar. According to the Yau-Tian-Donaldson conjecture, K-polystable Fano varieties are exactly the Fano varieties admitting a Kaehler-Einstein metric. In this talk, after giving an overview about K-stability, I will present a result that I have recently obtained with Zs. Patakfalvi. We have shown that the CM line bundle is nef on the moduli space of K-polystable Fano varieties, and big on the components which intersect non-trivially the open locus of uniformly K-stable Fano varieties. This boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-polystable/uniformly K-stable fibers.