Non-annulation effective et positivit\'e locale des fibr\'es en droites amples adjoints

Abstract

We prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. The main tools are (some known cases of) the Kawamata's effective non-vanishing conjecture and the adjunction theory. We prove the non-vanishing conjecture in dimension 3 in the case of line bundles of "high" volume using Kawamata's subadjunction formula.