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# Additive Riemann--Hilbert Problem in Line Bundles Over $\mathbb{CP}^1$

Published:2006-03-01
Printed: Mar 2006
• Roman J. Dwilewicz
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## Abstract

In this note we consider $\overline\partial$-problem in line bundles over complex projective space $\mathbb{CP}^1$ and prove that the equation can be solved for $(0,1)$ forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{CP}^2$ since by removing a point from it we get a line bundle over $\mathbb{CP}^1$.
 Keywords: $\overline\partial$-problem, cohomology groups, line bundles
 MSC Classifications: 32F20 - unknown classification 32F2014F05 - Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 32C16 - unknown classification 32C16