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

S. M. Yermakova

Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 2, pp. 209-218

Résumé

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.