Continuity of the complex Monge-Ampère operator

Source abstract

The complex Monge-Ampère operator ( d d c ) n (dd^c)^n is an important tool in complex analysis. It would be interesting to find the right notion of convergence u j → u u_j\to u such that ( d d c u j ) n → ( d d c u ) n (dd^cu_j)^n\to (dd^cu)^n in the weak topology. In this paper, using the C n − 1 C_{n-1} -capacity, we give a sufficient condition of the weak convergence ( d d c u j ) n → ( d d c u ) n (dd^cu_j)^n\to (dd^cu)^n . We also show that our condition is quite sharp in some case.