# On the structure of varieties of semi-Fano type

# On the structure of varieties of semi-Fano type

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**Zoom link:**** https://princeton.zoom.us/j/91248028438**

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 –(K_X+D) is nef. The motivation of studying these varieties is two-folded : on one hand, to extend the singular Beauville-Bogomolov decomposition to the log case, on the other hand, to comprehend the structure of varieties with semi-positive curvature. If X is smooth, the structure of these varieties is obtained by the successive works of Demailly-Peternell-Schneider, Păun, Debarre, Zhang, Cao, Cao-Höring, Campana-Cao-Matsumura, etc.. In fact, the Albanese map and the MRC fibration of these varieties are shown to be locally trivial fibrations, and induce a decomposition of their universal covers. By the philosophy of MMP, it is then natural to extend these results to the singular case, that is, the case where X is of semi-Fano type. In a recent work with Shin-ichi Matsumura, we obtain a structure theorem for these varieties by proving that the MRC fibration is a locally trivial fibration ; moreover, by combining this with the results of Ambro (2005), we obtain a splitting theorem for varieties of log Calabi-yau type. In this talk, I will explain these results and present some applications to related problems.