The General Definition of the Complex Monge-Amp`ere Operator on Compact Kahler Manifolds.pdf

a r X i v : 0 7 0 5 .2 0 9 9 v 1 [ m a t h .C V ] 1 5 M a y 2 0 0 7 The General Definition of the Complex Monge-Ampe?re Operator on Compact Ka?hler Manifolds Yang Xing Abstract. We introduce a wide subclass F(X,ω) of quasi-plurisubharmonic func- tions in a compact Ka?hler manifold, on which the complex Monge-Ampe?re operator is well-defined and the convergence theorem is valid. We also prove that F(X,ω) is a convex cone and includes all quasi-plurisubharmonic functions which are in the Cegrell class. 1. Introduction Let X be a compact connected Ka?hler manifold of dimension n, equipped with the fundamental form ω given in local coordinates by ω = i 2 ∑ α,β gαβ?dz α ∧ dz?β , where (gαβ?) is a positive definite Hermitian matrix and dω = 0. The smooth volume form associated to this Ka?hler metric is the nth wedge product ωn. Denote by PSH(X,ω) the set of upper semi-continuous functions u : X → R ∪ {?∞} such that u is integrable in X with respect to the volume form ωn and ωu := ω + dd cu ≥ 0 on X , where d = ? + ?? and dc = i (?? ? ?). These functions are called quasi-plurisubharmonic functions (quasi-psh for short) and play an important role in the study of positive closed currents in X , see Demailly’s paper [D1]. A quasi-psh function is locally the difference of a plurisubhar- monic function and a smooth function. Therefore, many properties of plurisubharmonic functions hold also for quasi-psh functions. Following Bedford and Taylor [BT2], the com- plex Monge-Ampe?re operator (ω + ddc)n is locally and hence globally well defined for all bounded quasi-psh functions in X . Some important results of the complex Monge-Ampe?re operator for bounded quasi-psh functions have been obtained by Kolodziej [KO1-2] and Blocki [BL1]. It is also known that the complex Monge-Ampe?re operator does not work well for all unbounded quasi-psh functions. Otherwise, we shall lose some of the essential properties that the complex Monge-Ampe?re operator should have, see Kiselman’s pap