Pseudoconvex domains with smooth boundary in projective spaces

Abstract

Given a pseudoconvex domain U with $$\mathscr {C}^1$$ -boundary in $${\mathbb {P}}^n,$$ $$n\geqslant 3,$$ we show that if $$H^{2n-2}_{\mathrm{dR}}(U)\,\,\ne \,0,$$ then there is a strictly psh function in a neighborhood of $$\partial U.$$ We also solve the $${{{\overline{\partial }}}}$$ -equation in $$X={\mathbb {P}}^n\setminus U,$$ for data in $$\mathscr {C}^\infty _{(0,1)}(X).$$ We discuss Levi-flat domains in surfaces. If Z is a real algebraic hypersurface in $${\mathbb {P}}^2,$$ (resp a real-analytic hypersurface with a point of strict pseudoconvexity), then there is a strictly psh function in a neighborhood of Z.