Abstract: A semi-Fano variety is a normal projective variety X admitting a klt boundary \Delta such that –(K_X+\Delta) is nef (defined by Shokurov-Prokhorov). It is expected that the Albanese map and the MRC fibration of X induce a decomposition of its universal cover into a product of C^q by a klt projective variety with trivial canonical divisor and a rationally connected variety. The interest of studying the strucutre of these varieties arises from the classical Beauville-Bogomolov decomposition theorem and from structure theorems on manifolds with nonnegative curvature such as works of Mori, Siu-Yau, Mok, Campana-Peternell /Demailly-Peternell-Schneider, etc.. In this talk, I will talk about our recent works (one by myself and the other by a joint work with Shin-ichi Matsumura) in this problem, which establish the strucutre theorem of the Albanese maps and the MRC fibrations of semi-Fano varieties.
