Very Ampleness Part of Fujita Conjecture and Multiplier Ideal Sheaves of Higher Order

Abstract

There are two parts in the Fujita conjecture of 1987. One is the freeness part which since 1995 has been proved with various weaker bounds. The other is the very ampleness part, which remains open even with weaker bounds. Here we will present a proof of the very ampleness part of the Fujita conjecture with a weaker bound by the method of multiplier ideal sheaves of higher order. Moreover, we prove the following more general main result. For an ample line bundle L on a compact complex manifold X of complex dimension n, any prescribed (q_j − 1)-jets at prescribed points P_j of X for 1 ≤ j ≤ k can be achieved by a global holomorphic section of mL + K_X for m ≥ m₀ with m₀ explicitly computable from n,q₁,…,q_k. This article is a write-up of my talk given on April 21, 2024 in the Conference on Several Complex Variables and Complex Geometry at Beijing Normal University. This write-up keeps the conversational style of my talk.