Stable vector bundles of rank 2 on P 3

If P 1 is the projective ind-space, i.e. P 1 is the inductive limit of linear embeddings of complex projective spaces, the Barth-Van de Ven-Tyurin (BVT) Theorem claims that every finite rank vector bundle on P 1 is isomorphic to a direct sum of line bundles. We extend this theorem to general sequences of morphisms between projective spaces by proving that, if there are infinitely many morphisms of degree higher than one, every vector bundle of finite rank on the inductive limit is trivial. We then establish a relative version of these results, and apply it to the study of vector bundles on inductive limits of grassmannians. In particular we show that the BVT Theorem extends to the ind-grassmannian of subspaces commensurable with a fixed infinite dimensional and infinite codimensional subspace in C 1 . We also show that, for a class of twisted ind-grassmannians, every finite rank vector bundle is trivial. 2000 AMS Subject Classification: Primary 32L05, 14J60, Secondary 14M15.

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Finite rank vector bundles on inductive limits of grassmannians

An ind-Grassmannian X = lim→ G(k i ; V n i ) is linear if almost all defining embeddings ϕ m : G(k m ; V n m )→G(k m + 1; V n m + 1) are of degree 1, and is twisted if infinitely many defining embeddings ϕ m are of degree higher than 1. In this paper we give a complete description of finite-rank vector bundles on any linear ind-Grassmannian, and prove that any vector bundle of rank 2 on a twisted ind-Grassmannian is trivial. Our work extends work by W. Barth, J. Donin, I. Penkov, E. Sato, A. Tyurin

Rank-2 Vector Bundles on ind-Grassmannians

Twisted ind-Grassmannians are ind-varieties obtained as direct limits of Grassmannians for under embeddings of degree greater than . It has been conjectured by Donin and Penkov (2003) that any vector bundle of finite rank on a twisted ind-Grassmannian is trivial. We prove this conjecture. Bibliography: 16 titles.

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Triviality of vector bundles on twisted ind-Grassmannians

Тривиальность векторных расслоений на скрученных инд-грассманианах@@@Triviality of vector bundles on twisted ind-Grassmannians

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Ei-ichi Sato

Papers

On infinitely extendable vector bundles on $G/P$