Strictly nef divisors on singular threefolds

Abstract

Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. First, we show the ampleness of $K_X+t{L}_X$ with sufficiently large $t$ if either the Kodaira dimension $\kappa (X)\ne 0$ or the augmented irregularity $q^{\circ }(X)\ne 0$. Second, we show that, if $(X,\Delta )$ is a projective klt threefold pair with the anti-log canonical divisor $-(K_X+\Delta )$ being strictly nef, then $X$ is rationally connected.

arXiv comment28 pages; Theorem 1.7 has been strengthened and Section 6 has been greatly shortened and improved; comments are welcome!