A Dolbeault lemma for temperate currents

2020/03/25 by Skoda, Henri
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.11437

Abstract

We consider a bounded open Stein subset Ω of a complex Stein manifold X of dimension n. We prove that if f is a current on X of bidegree (p,q+1), ∂-closed on Ω, we can find a current u on X of bidegree (p,q) which is a solution of the equation ∂ u=f in Ω. In other words, we prove that the Dolbeault complex of temperate currents on Ω (i.e. currents on Ω which extend to currents on X) is concentrated in degree 0. Moreover if f is a current on X= Cⁿ of order k, then we can find a solution u which is a current on Cⁿ of order k+2n+1.

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