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Uniformity of vector bundles of finite rank on complete intersections of finite codimension in a linear indGrassmannian
S. M. Yermakova^{} ^{} P. G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Abstract:
A linear projective indvariety $\mathbf X$ 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 indvariety $\mathbf X$ 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 indvariety ${\mathbf X}$ that is a complete intersection in the linear indGrassmannian $\mathbf{G}=\underrightarrow{\lim}G(k_m,n_m)$. By definition, ${\mathbf X}$
is an intersection of ${\mathbf{G}}$ with a finite number
of indhypersufaces $\mathbf{Y_i}=\underrightarrow{\lim}Y_{i,m}, {m\geq1}$, of fixed degrees $d_i$, $i=1,...,l$, in the space $\mathbf{P}^{\infty}$, in which the indGrassmannian $\mathbf{G}$ 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 indGrassmannians.
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512.7 Received: 25.11.2014
Citation:
S. M. Yermakova, “Uniformity of vector bundles of finite rank on complete intersections of finite codimension in a linear indGrassmannian”, 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 indGrassmannian
\jour Model. Anal. Inform. Sist.
\yr 2015
\vol 22
\issue 2
\pages 209218
\mathnet{http://mi.mathnet.ru/mais436}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3417822}
\elib{http://elibrary.ru/item.asp?id=23405829}
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This publication is cited in the following articles:

S. M. Ermakova, “FiniteRank Vector Bundles on Complete Intersections of Finite Codimension in the Linear IndGrassmannian”, Math. Notes, 98:5 (2015), 852–856

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