Fujita-type conjectures and Seshadri constants

Let X be a smooth projective variety of dimension n and let L be an ample divisor on X. In 1988, Fujita conjectured that K+(n+1)L is globally generated and K+(n+2)L is very ample, where K is the canonical divisor on X. To tackle this conjecture, Demailly introduced Seshadri constants, which measure the local positivity of L at a point x in X. While examples of Miranda seemed to indicate that Seshadri constants could not be used to prove Fujita's conjecture, we present a new characteristic-free approach to Fujita's conjecture using Seshadri constants. Our technique recovers some known results toward Fujita's conjecture over the complex numbers, and proves new results for complex varieties with singularities. We also describe joint work with Yajnaseni Dutta and Mihai Fulger, in which we use Seshadri constants to prove generic positivity of direct images of pluricanonical bundles.