A Base Point Free Theorem of Reid Type, II

Abstract

Let $X$ be a complete algebraic variety over {\bf C}. We consider a log variety $(X,\Delta)$ that is weakly Kawamata log terminal. We assume that $K_X+\Delta$ is a {\bf Q}-Cartier {\bf Q}-divisor and that every irreducible component of $\lfloor \Delta \rfloor$ is {\bf Q}-Cartier. A nef and big Cartier divisor $H$ on $X$ is called {\it nef and log big} on $(X,\Delta)$ if $H |_B$ is nef and big for every center $B$ of non-"Kawamata log terminal" singularities for $(X,\Delta)$. We prove that, if $L$ is a nef Cartier divisor such that $aL-(K_X+\Delta)$ is nef and log big on $(X,\Delta)$ for some $a \in$ {\bf N}, then the complete linear system $| mL |$ is base point free for $m \gg 0$.