Pseudoconvex domains with smooth boundary in projective spaces
Abstract
Given a pseudoconvex domain U with C1-boundary in Pn, n>2, we show that if H2n-2(U)=0, then there is a strictly psh function in a neighborhood of boundary U. We also solve the -equation in X=Pn\ U, for data smooth (0,1) forms on X. We also discuss Levi-flat domains in surfaces. If Z is a real algebraic hypersurface in P2, (resp a real-analytic hypersurface with a point of strict pseudoconvexity), then there is a strictly psh function in a neighborhood of Z.
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