Quasi-elementary contractions of Fano manifolds

Quasi elementary contractions of Fano manifolds

Smooth complex Fano varieties of dimension n form a limited family, so they have only a finite number of possible topological types. However not much is known about their topological invariants in higher dimensions. In particular we consider here the second Betti number b2 of a smooth Fano variety X, which coincides with the Picard number ρX . Recall that a Del Pezzo surface S has ρS ≤ 9. Fano ...

Mirror Symmetry and Fano Manifolds

Quasi-projectivity of the Moduli Space of Smooth Kähler-einstein Fano Manifolds

In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space M of smoothable Kähler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on M and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kähler-Einstein Fano mani...

Quasi-sectorial contractions

We revise the notion of the quasi-sectorial contractions. Our main theorem establishes a relation between semigroups of quasi-sectorial contractions and a class of m−sectorial generators. We discuss a relevance of this kind of contractions to the theory of operator-norm approximations of strongly continuous semigroups.

Existence of Einstein metrics on Fano manifolds

This is largely an expository paper and dedicated to my friend J. Cheeger for his 65th birthday. The purpose of this paper is to discuss some of my works on the existence of Kähler-Einstein metrics on Fano manifolds and some related topics. I will describe a program I have been following for the last twenty years. It includes some of my results and speculations which were scattered in my previo...