Geodesic
Additive Riemann–Hilbert Problem in Line Bundles Over CP1
Canadian mathematical bulletin, Tome 49 (2006) no. 1, pp. 72-81

Voir la notice de l'article provenant de la source Cambridge University Press

In this note we consider $\bar{\partial }$ -problem in line bundles over complex projective space $\mathbb{C}{{\mathbb{P}}^{1}}$ 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 $\mathbb{C}{{\mathbb{P}}^{2}}$ since by removing a point from it we get a line bundle over $\mathbb{C}{{\mathbb{P}}^{1}}$ .
DOI : 10.4153/CMB-2006-007-7
Mots-clés : 32F20, 14F05, 32C16, ∂̄-problem, cohomology groups, line bundles 
Dwilewicz, Roman J. Additive Riemann–Hilbert Problem in Line Bundles Over CP1. Canadian mathematical bulletin, Tome 49 (2006) no. 1, pp. 72-81. doi: 10.4153/CMB-2006-007-7
@article{10_4153_CMB_2006_007_7,
     author = {Dwilewicz, Roman J.},
     title = {Additive {Riemann{\textendash}Hilbert} {Problem} in {Line} {Bundles} {Over} {CP1}},
     journal = {Canadian mathematical bulletin},
     pages = {72--81},
     year = {2006},
     volume = {49},
     number = {1},
     doi = {10.4153/CMB-2006-007-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2006-007-7/}
}
TY  - JOUR
AU  - Dwilewicz, Roman J.
TI  - Additive Riemann–Hilbert Problem in Line Bundles Over CP1
JO  - Canadian mathematical bulletin
PY  - 2006
SP  - 72
EP  - 81
VL  - 49
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2006-007-7/
DO  - 10.4153/CMB-2006-007-7
ID  - 10_4153_CMB_2006_007_7
ER  - 
%0 Journal Article
%A Dwilewicz, Roman J.
%T Additive Riemann–Hilbert Problem in Line Bundles Over CP1
%J Canadian mathematical bulletin
%D 2006
%P 72-81
%V 49
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2006-007-7/
%R 10.4153/CMB-2006-007-7
%F 10_4153_CMB_2006_007_7

[AH] Andreotti, A. and Hill, C. D., E. E. Levi convexity and the Hans Lewy problem. Part I: Reduction to vanishing theorems. Ann. Scuola Norm. Sup. Pisa (3) 26(1972), 325–363. Google Scholar

[DM] Dwilewicz, R. and Merker, J., On the Hartogs–Bochner phenomenon for CR functions in P ℂ . Proc.Amer. Math. Soc. 130(2002), 1975–1980. Google Scholar

[GH] Griffiths, P. and Harris, J., Principles of Algebraic Geometry. John Wiley, New York, 1978. Google Scholar

[Gu] Gunning, R. C., Lectures on Riemann surfaces. Princeton University Press, Princeton, NJ, 1966. Google Scholar

[L] Laurent-Thiébaut, C., Phénomène de Hartogs-Bochner dans les variétés CR. In: Topics in Complex Analysis, Banach Center Publications 31, Warszawa, 1995, pp. 233–247. Google Scholar

[N] Narasimhan, R., Complex Analysis in One Variable. Birkhäuser Boston, Boston, MA, 1985. Google Scholar

[S1] Sarkis, F., CR meromorphic extension and the nonembeddability of the Andreotti-Rossi CR structure in the projective space. Internat. J. Math. 10(1999), 897–915. Google Scholar

[S2] Sarkis, F., Hartogs-Bochner type theorem in projective space. Ark. Mat. 41(2003), 151–163. Google Scholar

Cité par Sources :