Twists of symmetric bundles

Following Serre's initial work, a number of authors have considered twists of quadratic forms on a scheme Y by torsors of a finite group G, together with formulas for the Hasse-Witt invariants of the twisted form. In this paper we take the base scheme Y to be affine and consider non-constant groups schemes G. There is a fundamental new feature in this case - in that the torsor may now be ramified over Y. The natural framework for handling the case of a non-constant group scheme over the affine base is provided by the quadratic theory of Hopf-algebras.

Pure and Applied Mathematics Quarterly, 2009

Journal of the London Mathematical Society

We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall-crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall-crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulas for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant). Contents 14 6. Motives of moduli spaces of (parabolic) Higgs bundles 29 Appendix A. A local-to-global trick 34 References 36

Journal of Topology and Analysis, 2017

We investigate how one can twist [Formula: see text]-invariants such as [Formula: see text]-Betti numbers and [Formula: see text]-torsion with finite-dimensional representations. As a special case we assign to the universal covering [Formula: see text] of a finite connected [Formula: see text]-complex [Formula: see text] together with an element [Formula: see text] a [Formula: see text]-twisted [Formula: see text]-torsion function [Formula: see text], provided that the fundamental group of [Formula: see text] is residually finite and [Formula: see text] is [Formula: see text]-acyclic.

Duke Mathematical Journal, 2009

Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold X which depend in addition on a vector bundle over X and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov-Witten invariants of the bundle, and to genus-zero one-point invariants of complete intersections in X . We develop tools for computing genus-zero twisted Gromov-Witten invariants of orbifolds and apply them to several examples. We prove a "quantum Lefschetz theorem" which expresses genus-zero one-point Gromov-Witten invariants of a complete intersection in terms of those of the ambient orbifold X . We determine the genus-zero Gromov-Witten potential of the type A surface singularityˆC 2 /Zn˜. We also compute some genus-zero invariants ofˆC 3 /Z 3˜, verifying predictions of Aganagic-Bouchard-Klemm. In a self-contained Appendix, we determine the relationship between the quantum cohomology of the An surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and Bryan-Graber in this case.

2009

Proceedings of the London Mathematical Society, 2014

Let Y be a scheme in which 2 is invertible and let V be a rank n vector bundle on Y endowed with a non-degenerate symmetric bilinear form q. The orthogonal group O(q) of the form q is a group scheme over Y whose cohomology ring H * (B O(q) , Z/2Z) ≃ AY [HW1(q), ..., HWn(q)] is a polynomial algebra over the étale cohomology ring AY := H * (Yet, Z/2Z) of the scheme Y. Here the HWi(q)'s are Jardine's universal Hasse-Witt invariants and B O(q) is the classifying topos of O(q) as defined by Grothendieck and Giraud. The cohomology ring H * (B O(q) , Z/2Z) contains canonical classes det[q] and [Cq] of degree 1 and 2 respectively, which are obtained from the determinant map and the Clifford group of q. The classical Hasse-Witt invariants wi(q) live in the ring AY. Our main theorem provides a computation of det[q] and [Cq] as polynomials in HW1(q) and HW2(q) with coefficients in AY written in terms of w1(q), w2(q) ∈ AY. This result is the source of numerous standard comparison formulas for classical Hasses-Witt invariants of quadratic forms. Our proof is based on computations with (abelian and non-abelian) Cech cocycles in the topos B O(q). This requires a general study of the cohomology of the classifying topos of a group scheme, which we carry out in the first part of this paper.

2017

In this note we extend the theory of twists of elliptic curves as presented in various standard texts for characteristic not equal to two or three to the remaining characteristics. For this, we make explicit use of the correspondence between the twists and the Galois cohomology set $H^1\big(\operatorname{G}_{\overline{K}/K}, \operatorname{Aut}_{\overline{K}}(E)\big)$. The results are illustrated by examples.

Transformation Groups, 2006

Consider the diagonal action of SO n (K) on the affine space X = V ⊕m where V = K n , K an algebraically closed field of characteristic = 2. We construct a "standard monomial" basis for the ring of invariants K[X] SOn(K) . As a consequence, we deduce that K[X] SOn(K) is Cohen-Macaulay. As the first application, we present the first and second fundamental theorems for SO n (K)-actions. As the second application, assuming that the characteristic of K is = 2, 3, we give a characteristicfree proof of the Cohen-Macaulayness of the moduli space M 2 of equivalence classes of semi-stable, rank 2, degree 0 vector bundles on a smooth projective curve of genus > 2. As the third application, we describe a K-basis for the ring of invariants for the adjoint action of SL 2 (K) on m copies of sl 2 (K) in terms of traces.

Algebraic & Geometric Topology, 2014

We use a spectral sequence to compute twisted equivariant K-Theory groups for the classifying space of proper actions of discrete groups. We study a form of Poincaré Duality for twisted equivariant K-theory studied by Echterhoff, Emerson and Kim in the context of the Baum-Connes Conjecture with coefficients and verify it for the Group Sl 3 Z. In this work, we examine computational aspects relevant to the computation of twisted equivariant K-theory and K-homology groups for proper actions of discrete groups. Twisted K-theory was introduced by Donovan and Karoubi [DK70] assigning to a torsion element α ∈ H 3 (X, Z) abelian groups α K * (X) defined on a space by using finite dimensional matrix bundles. After the growing interest by physicists in the 1990s and 2000s, Atiyah and Segal [AS04] introduced a notion of twisted equivariant K-theory for actions of compact Lie Groups. In another direction, orbifold versions of twisted K-theory were introduced by Adem and Ruan [AR03], and progress was made to develop computational tools for Twisted Equivariant K-Theory with the construction of a spectral sequence in [BEUV13]. The paper [BEJU12] introduces Twisted equivariant K-theory for proper actions, allowing a more general class of twists, classified by the third integral Borel cohomology group H 3 (X × G EG, Z). We concentrate in the case of twistings given by discrete torsion, which is given by cocycles α ∈ Z 2 (G, S 1) representing classes in the image of the projection map H 2 (G, S 1) ∼ = → H 3 (BG, Z) → H 3 (X× G EG, Z). Under this assumption on the twist, a version of Bredon cohomology with coefficients in twisted representations can be used to approximate twisted equivariant K-Theory, by means of a spectral sequence studied in [BEUV13] and [Dwy08]. The Bredon (co)-homology groups relevant to the computation of twisted equivariant K-theory, and its homological version, twisted equivariant K-homology satisfy a Universal Coefficient Theorem, 1.13. We state it more generally for a pair of coefficient systems satisfying conditions 1.12. Theorem (Universal Coefficient Theorem). Let X be a proper, finite G-CW complex. Let M ? and M ? be a pair of functors satisfying Conditions 1.12. Then, there exists a short exact sequence of abelian groups 0 → Ext Z (H G n−1 (X, M ?), Z) → H n G (X, M ?) → Hom Z (H G n (X, M ?), Z) → 0