Least negative intersections of positive closed currents on compact Kähler manifolds

2014/04/10 by Truong, Tuyen Trung
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1404.2875

Abstract

Let X be a compact Kähler manifold of dimension k. Let R be a positive closed (p,p) current on X, and T₁,… ,T_k-p be positive closed (1,1) currents on X. We define a so-called least negative intersection of the currents T₁,T₂,… ,T_k-p and R, as a sublinear bounded operator \bigwedge (T₁,… ,T_k-p,R):~C⁰(X)→ ℝ. This operator is \bf symmetric in T₁,… ,T_k-p. It is \bf independent of the choice of a quasi-potential u_i of T_i, of the choice of a smooth closed (1,1) form θ_i in the cohomology class of T_i, and of the choice of a Kähler form on X. Its total mass is the intersection in cohomology \T₁\\T₂\… \T_k-p\.\R\. It has a semi-continuous property concerning approximating T_i by appropriate smooth closed (1,1) forms, plus some other good properties. If p=0 and T₁=… =T_k=T, we have a least negative Monge-Ampere operator MA(T)=\bigwedge (T,… ,T). If the set where T has positive Lelong numbers does not contain any curve, then MA(T) is positive. Several examples are given.

Related

• Intersection of Positive Closed Currents
• Quasi-potentials and regularization of currents, and applications
• Density of positive closed currents, a theory of non-generic intersections
• Intersection of positive closed currents of higher bidegree
• Uniqueness of tangent currents for positive closed currents