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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 $\overline\partial$-problem, cohomology groups, line bundles
MSC Classifications: 32F20, 14F05, 32C16 show english descriptions unknown classification 32F20
Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
unknown classification 32C16
32F20 - unknown classification 32F20
14F05 - Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
32C16 - unknown classification 32C16
 

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