Time: 2022-11-04, 3:30–4:30 pm；Venue: Tencent meeting 282-558-096

Time: 2022-11-07, 3:30–4:30 pm；Venue: Tencent meeting 149-219-706

Time: 2022-11-09, 3:30–4:30 pm；Venue: Tencent meeting 447-645-799

Abstract

Complex varieties with semipositive curvature have long been realized as having certain rigidity. From the differential-geometric viewpoint, this principle can be illustrated by the Bonnet-Myers theorem and the Cheeger-Colding theory. As for the algebro-geometric aspect, it is first revealed by Mori’s and Siu-Yau’s works on the Hartshorne-Frankel conjecture and people realized that the rigidity comes from the presence of rational curves. Inspired by the successive works to generalize this result (Mok-Zhong, Mok, Campana-Peternell, Demailly-Peternell-Schneider, etc.), and by the philosophy of the Minimal Model Program (MMP), mathematicians are interested in studying varieties of semi-Fano type. Here a normal projective variety X is called of semi-Fano type if there is an effective Q-divisor D on X such that (X,D) is klt and (KX + D) is nef. In addition to what is mentioned above, another motivation of studying these varieties is to find the log version of the Beauville-Bogomolov decomposition. In this talk, I’ll present the structure theorem of these varieties by revealing the structure of their rational curves; moreover I’ll give some applications of this structure theorem. These are outcomes of my thesis work and of a series of joint works with Jie Liu, Shin-ichi Matsumura, Wenhao Ou, Xiaokui Yang and Guolei Zhong.