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

TL;DR

This paper extends the Barth-Van de Ven-Tyurin-Sato theorem, showing that under certain conditions, vector bundles on specific ind-varieties decompose into sums of line bundles, generalizing known results to broader contexts.

Contribution

It provides sufficient conditions on locally complete linear ind-varieties for the theorem to hold, and identifies classes of ind-varieties satisfying these conditions.

Findings

The theorem extends to certain ind-varieties under specified conditions.

Natural classes of ind-varieties satisfying these conditions are identified.

The paper broadens understanding of vector bundle decompositions in infinite-dimensional settings.

Abstract

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. Keywords: ind-variety, vector bundle