The ∂-equation for (p,q)-forms on a non-reduced analytic space

2020/02/05 by Mats Andersson, Andersson, Mats, Richard Lärkäng +5
Mathematics · #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2002.01797

openalex publication_date 2020/02/05 · openalex created_date 2020/02/14 · openalex updated_date 2026/07/28

Abstract

On any pure n-dimensional, possibly non-reduced, analytic space X we introduce the sheaves \mathscrE_X^p,q of smooth (p,q)-forms and certain extensions \mathscrA_X^p,q of them such that the corresponding Dolbeault complex is exact, i.e., the ∂-equation is locally solvable in \mathscrA_X. The sheaves \mathscrA_X^p,q are modules over the smooth forms, in particular, they are fine sheaves. We also introduce certain sheaves \mathscrB_X^n-p,n-q of currents on X that are dual to \mathscrA_X^p,q in the sense of Serre duality. More precisely, we show that the compactly supported Dolbeault cohomology of \mathscrB^n-p,n-q(X) in a natural way is the dual of the Dolbeault cohomology of \mathscrA^p,q(X).

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