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  1. arXiv:1707.01767 (Published 2017-07-05)

    Proof of Grothendieck--Serre conjecture on principal bundles over regular local rings containing a finite field

    Ivan Panin
    Comments: arXiv admin note: text overlap with arXiv:1406.0247
    Categories: math.AG

    Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. If the regular local ring R contains an infinite field this result is proved in [FP]. Thus the conjecture is true for regular local rings containing a field.

  2. arXiv:1406.0247 (Published 2014-06-02)

    Proof of Grothendieck--Serre conjecture on principal G-bundles over regular local rings containing a finite field

    Ivan Panin
    Comments: arXiv admin note: substantial text overlap with arXiv:1211.2678
    Categories: math.AG

    Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of pointed sets H^1_{et}(R,G) \to H^1_{et}(K,G), induced by the inclusion of R into K, has a trivial kernel. Certain arguments used in the present preprint do not work if the ring R contains a characteristic zero field. In that case and, more generally, in the case when the regular local ring R contains an infinite field this result is proved in joint work due to R.Fedorov and I.Panin (see [FP]). Thus the Grothendieck--Serre conjecture holds for regular local rings containing a field.

  3. arXiv:1308.3078 (Published 2013-08-14, updated 2016-11-13)

    Affine Grassmannians of group schemes and exotic principal bundles over A^1

    Roman Fedorov
    Comments: Introduction re-written. Other minor improvements. Final version
    Journal: Amer. J. Math., Vol. 138, No. 4, 2016, pp. 879-906
    Categories: math.AG

    Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.

  4. arXiv:1003.3823 (Published 2010-03-19)

    Principal bundles over finite fields

    Indranil Biswas, S. Subramanian
    Comments: Final version
    Categories: math.AG

    Let M be an irreducible smooth projective variety defined over \bar{{\mathbb F}_p}. Let \pi(M, x_0) be the fundamental group scheme of M with respect to a base point x_0. Let G be a connected semisimple linear algebraic group over \bar{{\mathbb F}_p}. Fix a parabolic subgroup P \subsetneq G, and also fix a strictly anti-dominant character \chi of P. Let E_G \to M be a principal G-bundle such that the associated line bundle E_G(\chi) \to E_G/P is numerically effective. We prove that E_G is given by a homomorphism \pi(M, x_0)\to G. As a consequence, there is no principal G-bundle E_G \to M such that degree(\phi^*E_G(\chi)) > 0 for every pair (Y ,\phi), where Y is an irreducible smooth projective curve, and \phi: Y\to E_G/P is a nonconstant morphism.

  5. arXiv:0907.1136 (Published 2009-07-07)

    On principal bundles over a projective variety defined over a finite field

    Indranil Biswas
    Comments: Final version of preprint (2007)
    Categories: math.AG
    Subjects: 14L15, 14F05

    Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x_0. Let \varpi(M,x_0) denote the corresponding fundamental group--scheme introduced by Nori. Let E_G be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization \xi on M. We prove that the following three statements are equivalent: The principal G-bundle E_G over M is given by a homomorphism \varpi(M,x_0) --> G. There are integers b > a > 0 such that the principal G-bundle (F^b_M)^*E_G is isomorphic to (F^a_M)^*E_G, where F_M is the absolute Frobenius morphism of M. The principal G-bundle E_G is strongly semistable, degree(c_2(ad(E_G))c_1(\xi)^{d-2}) = 0, where d = \dim M, and degree(c_1(E_G(\chi))c_1(\xi)^{d-1}) = 0 for every character \chi of G, where E_G(\chi) is the line bundle over $M$ associated to $E_G$ for \chi. The equivalence between the first statement and the third statement was proved by S. Subramanian under the extra assumption that dim(M) = 1 and $G$ is semisimple.

  6. arXiv:math/0112096 (Published 2001-12-11)

    Projective moduli space of semistable principal sheaves for a reductive group

    Tomas L. Gomez, Ignacio Sols
    Comments: 10 pages, LaTeX2e. Submitted to the conference proceedings of "Commutative Algebra and Algebraic Geometry", Catania, April 2001
    Categories: math.AG
    Subjects: 14D22, 14D20

    Let X be a smooth projective complex variety, and let G be an algebraic reductive complex group. We define the notion of principal G-sheaf, that generalises the notion of principal G-bundle. Then we define a notion of semistability, and construct the projective moduli space of semistable principal G-sheaves on X. This is a natural compactification of the moduli space of principal G-bundles. This is the announcement note presented by the second author in the conference held at Catania (11-13 April 2001), dedicated to the 60th birthday of Silvio Greco. Detailed proofs will appear elsewhere.