Solving \overline{∂} with prescribed support on Hartogs triangles in ℂ² and ℂℙ²

Laurent-Thiébaut, Christine; Shaw, Mei-Chi

doi:10.1090/tran/7545

Solving ${\overline {\partial }}$ with prescribed support on Hartogs triangles in ${\mathbb {C}}^2$ and ${\mathbb {C}}\mathbb {P}^2$
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Trans. Amer. Math. Soc. 371 (2019), 6531-6546 Request permission

Abstract:

In this paper, we consider the problem of solving the Cauchy–Riemann equation with prescribed support in a domain of a complex manifold for forms or currents. We are especially interested in the case when the domain is a Hartogs triangle in $\mathbb {C}^2$ or $\mathbb {C}\mathbb {P}^2$. In particular, we show that the strong $L^2$ Dolbeault cohomology group on the Hartogs triangle in $\mathbb {C}\mathbb {P}^2$ is infinitely dimensional.

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