Differential Forms and Dolbeault Theory

  • Hans Grauert
  • Reinhold Remmert
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 236)


In this chapter Dolbeault cohomology theory is presented. One of the basic tools is the \(\bar \partial \)-integration lemma for closed (p, q)-forms (Theorem 4.1). The proof of this lemma is based on the existence of bounded solutions of the inhomogeneous Cauchy-Riemann differential equation \(\partial g/\partial \bar z = f.\)This solution is constructed in Paragraph 3 by means of the classical integral operator
$$Tf(z,u) = \frac{1}{{2\pi i}}\iint\limits_B {\frac{{f(\zeta ,u)}}{{\zeta - z}}}d\zeta \wedge d\bar \zeta .$$


Tangent Vector Differential Form Complex Manifold Differentiable Manifold Stein Manifold 
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Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • Hans Grauert
    • 1
  • Reinhold Remmert
    • 2
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenFederal Republic of Germany
  2. 2.Mathematisches InstitutWestfälischen Wilhelms-UniversitätMünsterFederal Republic of Germany

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