Solutions of the ∂ -equation on Stein and on Kähler manifold with compact support

2019/02/07 by Éric Amar, Eric Amar, Amar, Eric
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.CV

paper · pdf · doi:10.48550/arxiv.1902.02724

A Section on Stein manifold was added to clarify the links with a previous work done by C. Laurent-Tiebaut. So the title was also changed

arxiv created 2020/01/23 · arxiv updated 2020/01/24

Abstract

We study the ∂ -equation first in Stein manifold then in complete Kähler manifolds. The aim is to get L^r and Sobolev estimates on solutions with compact support. In the Stein case we get that for any (p,q)-form ω in L^r with compact support and ∂ -closed there is a (p,q-1)-form u in W^1,r with compact support and such that ∂ u=ω. In the case of Kähler manifold, we prove and use estimates on solutions on Poisson equation with compact support and the link with ∂ equation is done by a classical theorem stating that the Hodge laplacian is twice the ∂ (or Kohn) Laplacian in a Kähler manifold. This uses and improves, in special cases, our result on Andreotti-Grauert type theorem.

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