Pseudoconvex domains with smooth boundary in projective spaces
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Mathematische Zeitschrift
Pseudoconvex domains with smooth boundary in projective spaces Nessim Sibony1,2 Received: 26 November 2019 / Accepted: 28 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Given a pseudoconvex domain U with C 1 -boundary in Pn , n ≥ 3, we show that if 2n−2 HdR (U ) = | 0, then there is a strictly psh function in a neighborhood of ∂U . We also solve ∞ (X ). We discuss Levi-flat domains in surfaces. the ∂-equation in X = Pn \U , for data in C(0,1) 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 . Keywords Levi-flat · ∂-Equation · Pseudo-concave sets · Strictly plurisubharmonic functions Mathematics Subject Classification Primary 32Q28 · 32U10 · 32U40 · 32W05 ; Secondary 37F75
1 Introduction In this paper we discuss the pluri-potential theory on a smooth hypersurface Z in Pn . This includes the question of existence of positive closed (resp. dd c -closed) currents supported on Z and also the question of the existence of a strictly plurisubharmonic function (psh) in a neighborhood of Z . We will sometimes need a pseudo-convexity hypothesis on a component of the complement. We also give some results in the case where Z is a closed set satisfying some geometric assumptions. Recall that a complex manifold of dimension n is strongly q-complete if it admits a smooth exhaustion function ρ whose Levi form at each point has at least (n − q + 1) strictly positive eigenvalues. The main result in that theory is the following Theorem. Theorem (Andreotti–Grauert [1]) Let U be a strongly q-complete manifold. Then for every coherent analytic sheaf S over U , H k (U , S ) = 0, for k ≥ q.
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Nessim Sibony [email protected]
1
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405 Orsay, France
2
Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
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N. Sibony
In particular, if k ≥ q, H n,k (U , C) = 0 Indeed, for a holomorphic bundle E, H n,k (U , E) = H k (U , n,0 U ⊗ E). Our main result will use the above theorem. Theorem 1.1 Let U be a domain in Pn , n ≥ 3, with C 1 boundary. Assume U is strongly (n − 2)-complete (i.e. it admits a smooth exhaustion function whose Levi form has at least 3 2n−2 strictly positive eigenvalues at each point). Assume also that HdR (U , C) = 0. Then there is a strictly psh function near the boundary of U . As a consequence of Theorem 1.1 and of Proposition 2.1 below, we get the following result. Corollary 1.2 In Pn , n ≥ 3, there is no C 1 hypersurface Z such that the two components U ± 2 (U − ) = 0. of Pn \ Z are both strongly (n − 2)-complete and one them, say U − , satisfies HdR 2 (U − ) = 0 for a domain with C 1 boundary implies Observe that in Pn the hypothesis HdR 2n−2 + that HdR (U , C) = 0. See the proof of Proposition 2.1 below. Y. T. Siu has proved the following result, [19].
Theorem (Siu) In Pn , n ≥ 3, there
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