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

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 C 2 \mathbb {C}^2 or C P 2 \mathbb {C}\mathbb {P}^2 . In particular, we show that the strong L 2 L^2 Dolbeault cohomology group on the Hartogs triangle in C P 2 \mathbb {C}\mathbb {P}^2 is infinitely dimensional.