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 Model. Anal. Inform. Sist., 2015, Volume 22, Number 2, Pages 209–218 (Mi mais436)

Uniformity of vector bundles of finite rank on complete intersections of finite codimension in a linear ind-Grassmannian

S. M. Yermakova

P. G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia

Abstract: A linear projective ind-variety $\mathbf X$ is called $1$-connected if any two points on it can be connected by a chain of lines $l_1, l_2,...,l_k$ in $\mathbf X$, such that $l_i$ intersects $l_{i+1}$. A linear projective ind-variety $\mathbf X$ is called $2$-connected if any point of $\mathbf X$ lies on a projective line in $\mathbf X$ and for any two lines $l$ and $l'$ in $\mathbf X$ there is a chain of lines $l=l_1, l_2,...,l_k=l'$, such that any pair $(l_i,l_{i+1})$ is contained in a projective plane $\mathbb P^2$ in $\mathbf X$.
In this work we study an ind-variety ${\mathbf X}$ that is a complete intersection in the linear ind-Grassmannian $\mathbf{G}=\underrightarrow{\lim}G(k_m,n_m)$. By definition, ${\mathbf X}$ is an intersection of ${\mathbf{G}}$ with a finite number of ind-hypersufaces $\mathbf{Y_i}=\underrightarrow{\lim}Y_{i,m}, {m\geq1}$, of fixed degrees $d_i$, $i=1,...,l$, in the space $\mathbf{P}^{\infty}$, in which the ind-Grassmannian $\mathbf{G}$ is embedded by Plücker.
One can deduce from work [17] that $\mathbf X$ is $1$-connected. Generalising this result we prove that $\mathbf X$ is $2$-connected. We deduce from this property that any vector bundle $\mathbf{E}$ of finite rank on $\mathbf X$ is uniform, i. e. the restriction of $\mathbf{E}$ to all projective lines in $\mathbf X$ has the same splitting type.
The motiavtion of this work is to extend theorems of Barth–Van de Ven–Tjurin–Sato type to complete intersections of finite codimension in ind-Grassmannians.

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UDC: 512.7

Citation: S. M. Yermakova, “Uniformity of vector bundles of finite rank on complete intersections of finite codimension in a linear ind-Grassmannian”, Model. Anal. Inform. Sist., 22:2 (2015), 209–218

Citation in format AMSBIB
\Bibitem{Erm15} \by S.~M.~Yermakova \paper Uniformity of vector bundles of finite rank on complete intersections of~finite codimension in a linear ind-Grassmannian \jour Model. Anal. Inform. Sist. \yr 2015 \vol 22 \issue 2 \pages 209--218 \mathnet{http://mi.mathnet.ru/mais436} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3417822} \elib{http://elibrary.ru/item.asp?id=23405829} 

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This publication is cited in the following articles:
1. S. M. Ermakova, “Finite-Rank Vector Bundles on Complete Intersections of Finite Codimension in the Linear Ind-Grassmannian”, Math. Notes, 98:5 (2015), 852–856
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