| ABSTRACT |
A Q-Cartier divisor D on a normal projective variety X is said to be strictly nef, if it has positive intersection with every integral curve. It has been a long history for people to measure how far a strictly nef divisor is from being ample. In my first talk, we will focus on the case when D is the anti-log canonical divisor -(K+B) of a projective klt pair (X,B); in this case, we show that X is rationally connected. In particular, when dim X=3, we prove that -(K+B) is ample, which generalizes Serrano’s theorem to the klt pair case and gives a positive answer to the singular version of Campana-Peternell’s conjecture in dimension three. In my second talk, we will study the singular version of Serrano’s conjecture, and show that, when X is a Q-factorial Gorenstein terminal projective threefold, every strictly nef divisor on X will be ample after a small turbulence of the canonical divisor K except that X is a weak Calabi-Yau threefold. My talks are based on some joint works with Jie Liu, Wenhao Ou, Juanyong Wang and Xiaokui Yang.
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