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Working paper

On the Barth-Van de Ven-Tyurin-Sato Theorem

Tikhomirov A. S., Penkov I.
The Barth–Van de Ven–Tyurin–Sato Theorem claims that any finite rank vector bundle on the infinite complex projective space $\mathbf{P}^{\infty}$ is isomorphic to a direct sum of line bundles. We establish sufficient conditions on a locally complete linear ind-variety $\mathbf{X}$ which ensure that the same result holds on $\mathbf{X}$. We then exhibit natural classes of locally complete linear ind-varieties which satisfy these sufficient conditions.