Triviality of vector bundles on sufficiently twisted ind-Grassmannians

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 ${\varphi }_m:G(r_m,V^{r_m})\to G(r_{m+1},V^{r_{m+1}})$ 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 ${\lim }_{m\to \infty }\frac{r_m}{°{\varphi }_1...\mathrm{deg}{\varphi }_m}=0$.

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