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Principal bundles whose restrictions to a curve are isomorphic. (English) Zbl 1273.14086

Given an algebraic variety \(X\) it has long been known (and is widely used, e.g. in birational geometry), that one can learn about the geometry of \(X\) by studying curves \(C \subseteq X\). In the article under review, an instance of this principle is studied generalizing earlier work of T. Graber et al., [Ann. Sci. Cole Norm. Sup. (4) 38, No. 5, 671–692 (2005; Zbl 1092.14062)] and I. Biswas and Y. Holla [Math. Z. 251(3), 607–614 (2005; Zbl 1106.14028)].
By first considering the special case of vector bundles (which can be dealt with using the lemma of Enriques-Severi-Zariski) and reducing the general case to this, the author proves the following result:
Let \(K\) be an algebraically closed field and let \(X\) be a normal projective variety over \(K\). Let \(G\) be an affine algebraic \(K\)-group and let \(E_G\) and \(F_G\) be two principal \(G\)-bundles over \(X\). Then there exists an integer \(n>>0\) such that for a general curve \(C \in |\mathcal{O}_X(n)|\) the restrictions of \(E_G\) and \(F_G\) to \(C\) are isomorphic if and only if \(E_G\) and \(F_G\) are isomorphic.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
20G07 Structure theory for linear algebraic groups
14C20 Divisors, linear systems, invertible sheaves
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References:

[1] Balaji V, Principal bundles on projective varieties and the Donaldson–Uhlenbeck compactification, J. Differ. Geom. 76(3) (2007) 351–398 · Zbl 1121.14037
[2] Balaji V, Lectures on principal bundles. Moduli spaces and vector bundles, London Math. Soc. Lecture Note Ser. 359 (Cambridge: Cambridge Univ. Press) (2009) pp. 2–28 · Zbl 1187.14020
[3] Biswas I and Holla Y I, Principal bundles whose restriction to curves are trivial, Math. Z. 251(3) (2005) 607–614 · Zbl 1106.14028 · doi:10.1007/s00209-005-0825-6
[4] Biswas I and Subramanian S, Flat holomorphic connections on principal bundles over a projective manifold, Trans. Am. Math. Soc. 356(10) (2004) 3995–4018 · Zbl 1058.53020 · doi:10.1090/S0002-9947-04-03567-6
[5] Graber T, Harris J, Mazur B and Starr J, Rational connectivity and sections of families over curves, Ann. Sci. Cole Norm. Sup. (4) 38(5) (2005) 671–692 · Zbl 1092.14062
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