Plurisubharmonic function

Abstract: — LetX be a compact complex manifold and let T be a closed positive current of bidegree (1, 1) on X . It is shown that T is the weak limit of a sequence (Tk) of smooth closed real (1, 1)-currents with small negative part. The negative part of the Tk ’s can be bounded in terms of the Lelong numbers of T , once a lower bound for the curvature of the tangent bundle TX is known. Moreover, Kiselman’s procedure for killing Lelong numbers of a plurisubharmonic function is extended to manifolds by an alternative method based on Hormander’s L estimates for ∂. These results are then applied to derive various results concerning divisors or intersection theory in the context of analytic geometry. Especially, we obtain a relation between effective and numerically effective divisors on arbitrary compact manifolds, and we show that every manifold X in the Fujiki class C with nef tangent bundle is Kahler. If D is an effective divisor in a Kahler manifold, we also obtain a general self-intersection inequality giving a bound of the degrees of the constant multiplicity strata of D, in terms of a polynomial in the cohomology class {D} ∈ H(X, IR).

Abstract: This chapter is a survey article on the theory of Lelong numbers, viewed as a tool for studying intersection theory by complex differential geometry. We have not attempted to make an exhaustive compilation of the existing literature on the subject, nor to present a complete account of the state-of-the-art. Instead, we have tried to present a coherent unifying frame for the most basic results of the theory, based in part on our earlier works [7–10] and on Siu’s fundamental work [30]. To a large extent, the asserted results are given with complete proofs, many of them substantially shorter and simpler than their original counterparts. We only assume that the reader has some familiarity with differential calculus on complex manifolds and with the elementary facts concerning analytic sets and plurisubharmonic functions. The reader can consult Lelong’s books [25, 26] for an introduction to the subject. Most of our results still work on arbitrary complex analytic spaces, provided that suitable definitions are given for currents, plurisubharmonic functions, etc., in this more general situation. We have refrained ourselves from doing so for simplicity of exposition; we refer the reader to Ref. 9 for the technical definitions required in the context of analytic spaces.

Abstract: In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly. The multiplier ideal sheaf associated to a plurisubharmonic function, which is an invariant of the singularities of the psh function, plays an important role in several complex variables and complex geometry. Various properties about the multiplier ideal sheaves associated to plurisubharmonic functions have been discussed (e.g., see [25], [7], [30], [31]). Demailly’s strong openness conjecture means that the strong openness property about the multiplier ideal sheaf holds. In the present article, we establish such a useful strong openness property on the multiplier ideal sheaf associated to any plurisubharmonic function; i.e., Demailly’s strong openness conjecture is true. 1.1. Organization of the paper. The paper is organized as follows. In the present section, we recall the statement of the strong openness conjecture posed by Demailly and present the main result of the present paper: a solution of the strong openness conjecture. In Section 2, we recall or give some preliminary lemmas used in the proof of the main result. In Section 3, we give the proof of the strong openness conjecture and present some consequences by combining the conjecture with some well-known results.

Abstract: We establish the convexity of Mabuchi’s K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold, thus confirming a conjecture of Chen, and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampere equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.