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Pseudoconvex domains with smooth boundary in projective spaces (2004.14694v3)
Published 30 Apr 2020 in math.CV
Abstract: Given a pseudoconvex domain U with C1-boundary in Pn, n>2, we show that if H{2n-2}_\dR}(U)\not=0, then there is a strictly psh function in a neighborhood of boundary U. We also solve the \dbar-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.