Uniformity of Vector Bundles of Finite Rank on Complete Intersections of Finite Codimension in a Linear ind-Grassmannian

• Main Author: S. M. Yermakova
• Format: Article
• Language: English
• Published: Yaroslavl State University 2015-04-01
• Series: Modelirovanie i Analiz Informacionnyh Sistem
• Subjects: ind-grassmannian; vector bundle; uniform bundle; fano variety of lines
• Online Access: https://www.mais-journal.ru/jour/article/view/241

A linear projective ind-variety X is called 1-connected if any two points on it can be connected by a chain of lines l1, l2, ..., lk in X, such that li intersects li+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 0 in X there is a chain of lines l = l1, l2, ..., lk = l 0 , such that any pair (li , li+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 = lim−→G(km, nm). By definition, X is an intersection of G with a finite number of ind-hypersufaces Yi = lim−→Yi,m, m ≥ 1, of fixed degrees di , i = 1, ..., l, in the space P∞, in which the ind-Grassmannian G is embedded by Pl¨ucker. 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.