Uniformity of Vector Bundles of Finite Rank on Complete Intersections of Finite Codimension in a Linear ind-Grassmannian
A linear projective ind-variety X is called 1-connected if any two points on it can be connected by a chain of lines $l_1, l_2, \cdot, l_k$ in X, such that $l_i$ intersects $l_{i+1}$. A linear projective ind-variety X is called 2-connected if any point of X lies on a projective line in X and for any two lines $l$ and $l'$ in X there is a chain of lines $l = l_1, l_2, \cdot, l_k = l'$, such that any pair $(l_i, l_{i+1})$ is contained in a projective plane $P^2$ in X. In this work we study an ind-variety X that is a complete intersection in the linear ind-Grassmannian $G = \varinjlim G(k_m,n_m)$. By definition, X is an intersection of G with a finite number of ind-hypersufaces $Y_i = \varinjlim Y_{i,m}, m \ge 1$, of fixed degrees $d_i, i = 1, \cdot, l$, in the space $P^{\infty}$, in which the ind-Grassmannian G is embedded by Plucker. One can deduce from work [17] that X is 1-connected. Generalising this result we prove that X is 2-connected. We deduce from this property that any vector bundle E of finite rank on X is uniform, i. e. the restriction of E to all projective lines in 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.