Rank 2 vector bundles on ind-Grassmannians

TL;DR

This paper investigates the structure of rank 2 vector bundles on ind-Grassmannians, extending classical results about vector bundles on infinite projective spaces to more complex infinite flag varieties.

Contribution

It provides new insights into the classification and properties of rank 2 vector bundles on ind-Grassmannians, a class of infinite-dimensional homogeneous spaces.

Findings

Extension of BVT theorem to ind-Grassmannians

Classification results for rank 2 vector bundles

Analysis of vector bundle structures on infinite flag varieties

Abstract

The simplest example of an ind-Grassmannian is the infinite projective space $\mathbf P^\infty$. The Barth-Van de Ven-Tyurin (BVT) Theorem, proved more than 30 years ago \cite{BV}, \cite{T}, \cite{Sa} (see also a recent proof by A. Coand\u{a} and G. Trautmann, \cite{CT}), claims that any vector bundle of finite rank on $\mathbf P^\infty$ is isomorphic to a direct sum of line bundles. In the last decade natural examples of infinite flag varieties (or flag ind-varieties) have arisen as homogeneous spaces of locally linear ind-groups, \cite{DPW}, \cite{DiP}. In the present paper we concentrate our attention to the special case of ind-Grassmannians, i.e. to inductive limits of Grassmannians of growing dimension.