Strictly nef divisors on singular threefolds

Abstract

Let $X$ be a normal projective variety with only klt singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. In this paper, we study the singular version of Serrano's conjecture, i.e., the ampleness of $K_X+t{L}_X$ for sufficiently large $t\gg 1$. We show that, if $X$ is assumed to be a $\mathbb{Q}$-factorial Gorenstein terminal threefold, then $K_X+t{L}_X$ is ample for $t\gg 1$ unless $X$ is a weak Calabi-Yau variety (i.e., the canonical divisor $K_X{\sim }_{\mathbb{Q}}0$ and the augmented irregularity $q^{\circ }(X)=0$) with $L_X\cdot c_2(X)=0$.

arXiv comment23 pages; this article is a revision of the previous version arXiv: 2105.07681; new results on singular weak Calabi-Yau threefolds are added in Section 5; comments are welcome! arXiv admin note: substantial text overlap with arXiv:2105.07681

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