Qi Zhang

A variety $X$ is said to be uniruled (resp : rationally connected) if for any point $x$ (resp : a pair of points $x$ and $y$) in $X$, there exists a rational curve which contains $x$ (resp : $x$ and $y$).

Q-Fano varieties are those which are similar to del Pezzo surfaces (but higher dimensional with certain singularities). Q-Fano varieties play an increasingly important role in Mori's program. They are known to be uniruled (by the work of Miyaoka-Mori). A famous conjecture of Kollar-Miyaoka-Mori predicts that they should be rationally connected.

In this talk I shall explore the history and some ramifications of the conjecture. I shall also explain my recent work in which I was able to give an affirmative answer to the conjecture (in any dimension).