Generalizations of the Nakano Vanishing Theorem

Abstract

In Chapter II, we saw that the Kodaira Vanishing Theorem holds for semi-positive or semi-negative line bundles (Corollary (2.32)). However, the Nakano Vanishing Theorem does not hold for these bundles, as shown below in Example (3.23). This chapter presents some generalizations of the Nakano Vanishing Theorem without strict positivity or negativity. First, we give a generalization to k-positive and k-negative line bundles. We then introduce a powerful “slicing technique” in order to further extend the Nakano Theorem. We also introduce the concept of k-ampleness and show the relationship between the First Lefschetz Theorem and the Nakano Theorem.