An analogue of the Narasimhan-Seshadri theorem in higher dimensions and some applications

Balaji, V.; Parameswaran, A. J.

doi:10.1112/jtopol/jtq036

Cited by 12 publications

(16 citation statements)

References 41 publications

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“…It is quite possible that this follows from the results in [10] but we give an easy proof of it here using the restriction theorem.…”

Section: Claim the Sequence Of Integers D M K Is A Decreasing Functimentioning

confidence: 70%

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Restriction theorems for principal bundles in arbitrary characteristic

Gurjar

2015

Journal of Algebra

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The aim of this paper is to give a proof of the restriction theorems for principal bundles with a reductive algebraic group as structure group in arbitrary characteristic. Let G be a reductive algebraic group over any field k =k, let X be a smooth projective variety over k, let H be a very ample line bundle on X and let E be a semistable (resp. stable) principal G-bundle on X w.r.t. H. The main result of this paper is that the restriction of E to a general smooth curve which is a complete intersection of ample hypersurfaces of sufficiently high degrees is again semistable (resp. stable).

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“…It is quite possible that this follows from the results in [10] but we give an easy proof of it here using the restriction theorem.…”

Section: Claim the Sequence Of Integers D M K Is A Decreasing Functimentioning

confidence: 70%

“…A version of the restriction theorem for strongly semistable bundles exists in [10]. In another direction A. Langer proved an effective restriction theorem for torsion-free sheaves in positive characteristic (see [9]).…”

Section: Introductionmentioning

confidence: 97%

Restriction theorems for principal bundles in arbitrary characteristic

Gurjar

2015

Journal of Algebra

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“…The category

\overset{C}{true}

could be one of the categories in the following examples. Examples Take

\overset{C}{true}

to be the subcategory of

V e c t_{Y_{1}}

consisting of trivial vector bundles. Take

\overset{C}{true}

to be the subcategory of

V e c t_{Y_{1}}

consisting of essentially finite vector bundles. When Y 1 is smooth, take

\overset{C}{true}

to be the subcategory of

V e c t_{Y_{1}}

consisting of lf‐graded vector bundles of degree 0 (defined in ). Take

\overset{C}{true}

to be the subcategory of

V e c t_{Y_{1}}

consisting of numerically flat vector bundles . In this case, the category

scriptC

is the category of numerically flat vector bundles on Y .Let

G_{Y_{1}}^{0}

denote the dual group scheme for the category of numerically flat vector bundles on Y 1 , similarly on Y .…”

Section: Some Tannakian Group Schemes In Higher Dimensionsmentioning

confidence: 99%

“…(3) When Y 1 is smooth, takeC to be the subcategory of V ect Y 1 consisting of lf-graded vector bundles of degree 0 (defined in [1]). (4) TakeC to be the subcategory of V ect Y 1 consisting of numerically flat vector bundles [15].…”

Section: Examples 43mentioning

confidence: 99%

Holonomy group scheme of an integral curve

Bhosle

Parameswaran

2014

Mathematische Nachrichten

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Let Y be a projective variety over a field k (of arbitrary characteristic). Assume that the normalization X of Y is such that Xk¯ is normal, truek¯ 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 dimY=1, we show that strongly semistable vector bundles on Y form a neutral Tannakian category. We define the holonomy group scheme GY of Y to be the Tannakian group scheme for this category. For a strongly semistable principal G‐bundle EG, 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.

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“…As a consequence we have a homomorphism of O Y0 -algebras g 0 * (O X0 ) W 0 (cf. also [1,Section 6]). …”

Section: Lemma 23mentioning

confidence: 99%