In this article we establish an analogue of the Barth-Van de Ven-Tyurin-Sato theorem.We prove that a finite rank vector bundle on a complete intersection of finite codimension in a linear ind-Grassmannian is isomorphic to a direct sum of line bundles.
@article{bwmeta1.element.doi-10_1515_coma-2015-0007,
author = {Svetlana Ermakova},
title = {Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians},
journal = {Complex Manifolds},
volume = {2},
year = {2015},
zbl = {06476709},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0007}
}
Svetlana Ermakova. Vector bundles of finite rank on complete intersections of finite codimension in ind-Grassmannians. Complex Manifolds, Tome 2 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0007/
[1] Barth W., Van de Ven A. On the geometry in codimension 2 in Grassmann manifolds, Lecture Notes in Math. 412. Springer- Verlag, 1974. P. 1-35. | Zbl 0299.14024
[2] Donin J., Penkov I. Finite rank vector bundles on inductive limits of Grassmannians, IMRN. 2003. No 34. P. 1871-1887. | Zbl 1074.14530
[3] Griffiths P. A., Harris J. Principles of Algebraic Geometry, New York: Wiley, 1978. | Zbl 0408.14001
[4] Sato E. On the decomposability of infinitely extendable vector bundles on projective spaces and Grassmann varieties, J. Math. Kyoto Univ. 1977. No 17. P. 127-150. | Zbl 0362.14005
[5] Penkov I., Tikhomirov A.S. Linear ind-Grassmannians, Pure and Applied Mathematics Quarterly. 2014. 10. N-2. P.289-323. [WoS]
[6] Penkov I., Tikhomirov A.S. On the Barth–Van de Ven–Tyurin–Sato theorem, arXiv:1405.3897 [math.AG]. [WoS]
[7] Penkov I., Tikhomirov A. S. Rank-2 vector bundles on ind-Grassmannians, Algebra, arithmetic,and geometry: in honor of Yu. I. Manin, V II, Progr. Math., V. 270. Birkhaeuser, Boston-Basel-Berlin, 2009. P. 555-572. | Zbl 1200.14102
[8] Tyurin A. N. Vector bundles of finite rank over infinite varieties, Math. USSR. Izvestija. 1976. No 10. P. 1187-1204. | Zbl 0379.14004
[9] Hartshorne R. Algebraic Geometry, New York: Springer-Verlag, 1977.
[10] Ермакова С.М. О пространстве путей на полных пересечениях в грассманианах, МАИС. 2014. Т. 21. No 4. C.35-46 (English translation: Yermakova S.M. On the variety of paths on complete intersections in Grassmannians, MAIS. 2014. V. 21. No 4. P. 35-46).
[11] Ермакова С.М. Равномерность векторных расслоений конечного ранга на полных пересечениях конечной коразмерности в линейных инд-грассманианах, МАИС. 2015. Т. 22. No 2. C. 209-2018 (English translation: Yermakova S.M. Uniformity of vector bundles of finite rank on complete intersections of finite codimension in a linear ind- Grassmannian, MAIS. 2015. V. 22. No 2. P. 209-2018).
[12] Пенков И.Б., Тихомиров А.С. Тривиальность векторных расслоений на скрученных инд-грассманианах, Математический сборник. 2011. 202. No 1. C. 65-104 (English translation: Penkov I.B., Tikhomirov A.S. Triviality of vector bundles on twisted ind-Grassmannians, Sbornik: Mathematics. 2011. 202, No 1. P. 61-99).
[13] Шафаревич И. Р. Основы алгебраической геометрии,МЦНМО, Москва, 2007. (English translation: Shafarevich I.R. Foundations of Algebraic Geometry, MCCME, Moscow. 2007).