Triviality of vector bundles on sufficiently twisted ind-Grassmannians

TL;DR

This paper proves that on sufficiently twisted ind-Grassmannians, all finite rank vector bundles are trivial, confirming a conjecture for a broad class of these infinite-dimensional varieties.

Contribution

It establishes the triviality of finite rank vector bundles on a class of ind-Grassmannians under a specific twisting condition, advancing understanding of vector bundles on ind-varieties.

Findings

All finite rank vector bundles are trivial on sufficiently twisted ind-Grassmannians.

The conjecture by PT and DP is confirmed under the condition that the ind-Grassmannian is sufficiently twisted.

The paper provides conditions under which vector bundles on ind-varieties are trivial.

Abstract

Twisted ind-Grassmannians are ind-varieties $\GG$ obtained as direct limits of Grassmannians $G(r_m,V^{r_m})$, for $m\in\ZZ_{>0}$, under embeddings $\phi_m:G(r_m,V^{r_m})\to G(r_{m+1}, V^{r_{m+1}})$ of degree greater than one. It has been conjectured in \cite{PT} and \cite{DP} that any vector bundle of finite rank on a twisted ind-Grassmannian is trivial. We prove this conjecture under the assumption that the ind-Grassmannian $\GG$ is sufficiently twisted, i.e. that $\lim_{m\to\infty}\frac{r_m}{\deg \phi_1...\deg\phi_m}=0$.