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Holonomy group scheme of an integral curve
Author(s) -
Bhosle Usha N.,
Parameswaran A. J.
Publication year - 2014
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300117
Subject(s) - holonomy , mathematics , vector bundle , pure mathematics , group (periodic table) , scheme (mathematics) , tensor (intrinsic definition) , principal bundle , algebra over a field , topology (electrical circuits) , combinatorics , mathematical analysis , chemistry , organic chemistry
Let Y be a projective variety over a field k (of arbitrary characteristic). Assume that the normalization X of Y is such that X k ¯is normal, k ¯ being the algebraic closure of k . We define a notion of strong semistability for vector bundles on Y . We show that a vector bundle on Y is strongly semistable if and only if its pull back to X is strongly semistable and hence it is a tensor category. In case dim Y = 1 , we show that strongly semistable vector bundles on Y form a neutral Tannakian category. We define the holonomy group scheme G Y of Y to be the Tannakian group scheme for this category. For a strongly semistable principal G ‐bundle E G , we construct a holonomy group scheme. We show that if Y is an integral complex nodal curve, then the holonomy group of a strongly semistable vector bundle on Y is the Zariski closure of the (topological) fundamental group of Y .