On Logarithmic Canonical Divisors on Threefolds
Hironobu Maeda · Tokyo Journal of Mathematics · 1985
We prove the following THEOREM.Let (V, $D$ ) be a non-singular pair of dimension 3. Then (i) under the condition that $\kappa(K+D, V)\geqq 0,$ $K+D$ is ample if and only if $K+D$ is numerically positive; i.e. $(K+D)\cdot C>0$ for all curves $C$ on $V$ , (ii) under the condition that $\kappa(-K-D, V)\geqq 0,$ $-K-D$ is ample if and only if $-K-D$ is numerically positive.COROLLARY (cf.[4]).Let (V, $D$ ) be as in the Theorem.Then (V, $D$ ) is a logarithmic Fano threefold if and only if the following two conditions are satisfied. (a) The linear system $|-K-D|$ is non-empty. (b) $-K-D$ is numerically positive.PROOF.The if part follows from the Theorem.Let (V, $D$ ) be a logarithmic Fano threefold.Applying Norimatsu Vanishing ([5, Theorem 1]) we deduce $H^{\ell}(V, P_{V}(-K-D))=0$ for $i>0$