no. 1
A class of non-algebraic threefolds
Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 239-250.

Soit X une surface C-analytique, compacte, lisse, sans diviseurs, et E un fibré vectoriel holomorphe de rang 2 sur X. Le fibré projectif associé, P(E), n’aura pas de diviseurs si et seulement si E est “fortement” irréductible. On prouve l’existence de tels fibrés.

Let X be a compact complex nonsingular surface without curves, and E a holomorphic vector bundle of rank 2 on X. It turns out that the associated projective bundle PE has no divisors if and only if E is “strongly” irreducible. Using the results concerning irreducible bundles of [Banica-Le Potier, J. Crelle, 378 (1987), 1-31] and [Elencwajg- Forster, Annales Inst. Fourier, 32-4 (1982), 25-51] we give a proof of existence for bundles which are strongly irreducible.

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     author = {Toma, Matei},
     title = {A class of non-algebraic threefolds},
     journal = {Annales de l'Institut Fourier},
     pages = {239--250},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {1},
     year = {1989},
     doi = {10.5802/aif.1166},
     zbl = {0659.32024},
     mrnumber = {90k:32084},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1166/}
}
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Toma, Matei. A class of non-algebraic threefolds. Annales de l'Institut Fourier, Tome 39 (1989) no. 1, pp. 239-250. doi : 10.5802/aif.1166. https://aif.centre-mersenne.org/articles/10.5802/aif.1166/

[1] C. Bᾰnicᾰ & J. Le Potier, Sur l'existence des fibrés vectoriels holomorphes sur les surfaces non-algébriques, J. reine angew. Math., 378 (1987), 1-31. | MR | Zbl

[2] W. Barth, C. Peters & A. Van De Ven, Compact complex surfaces, Berlin-Heidelberg-New York, 1984. | MR | Zbl

[3] G. Elencwajg & O. Forster, Vector bundles on manifolds without divisors and a theorem on deformations, Ann. Inst. Fourier, 32-4 (1982), 25-51. | Numdam | MR | Zbl

[4] G. Fischer, Complex Analytic Geometry, LNM 538, Berlin-Heidelberg-New York, 1976. | MR | Zbl

[5] H. Grauert & R. Remmert, Coherent analytic sheaves, Berlin-Heidelberg-New-York, 1984. | MR | Zbl

[6] D. Mumford, Abelian varieties, Oxford Univ. Press, 1970. | MR | Zbl

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