Moduli Spaces for Principal Bundles in Arbitrary Characteristic (original) (raw)

Moduli space of principal sheaves over projective varieties

Annals of Mathematics, 2005

Let G be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal G-bundle on a Riemann surface and constructed a projective moduli space of such objects. We generalize Ramanathan's notion and construction to higher dimension, allowing also objects which we call semistable principal G-sheaves, in order to obtain a projective moduli space: a principal G-sheaf on a projective variety X is a triple (P, E, ψ), where E is a torsion free sheaf on X, P is a principal G-bundle on the open set U where E is locally free and ψ is an isomorphism between E|U and the vector bundle associated to P by the adjoint representation.

Moduli spaces of principal F-bundles

Selecta Mathematica, 2004

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) F-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected curve over a finite field, a split reductive group G, and an irreducible algebraic representation ω of (G) n /Z(G). Our spaces generalize moduli spaces of F-sheaves, studied by Drinfeld and Lafforgue, which correspond to the case G = GL r and ω is the tensor product of the standard representation and its dual. The importance of the moduli spaces of F-bundles is due to the belief that Langlands correspondence is realized in their cohomology.

A Torelli theorem for moduli spaces of principal bundles over a curve

Annales de l’institut Fourier, 2012

Let X and X ′ be compact Riemann surfaces of genus ≥ 3, and let G and G ′ be nonabelian reductive complex groups. If one component M d G (X) of the coarse moduli space for semistable principal G-bundles over X is isomorphic to another component M d ′ G ′ (X ′), then X is isomorphic to X ′ .

The line bundles on moduli stacks of principal bundles on a curve

Documenta Mathematica, 2010

As long as moduli spaces of bundles on a smooth projective algebraic curve C have been studied, their Picard groups have attracted some interest. The first case was the coarse moduli scheme of semistable vector bundles with fixed determinant over a curve C of genus gC ...

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

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2009

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumptio...

Projective moduli space of semistable principal sheaves for a reductive group

2001

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.

On Semistable Principal Bundles over a Complex Projective Manifold

International Mathematics Research Notices, 2010

Let G be a simple linear algebraic group defined over the field of complex numbers. Fix a proper parabolic subgroup P of G, and also fix a nontrivial antidominant character χ of P. We prove that a holomorphic principal G-bundle E G over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class c 2 (ad(E G)) ∈ H 4 (M, Q) vanishes if and only if the line bundle over E G /P defined by χ is numerically effective. Also, a principal G-bundle E G over M is semistable with c 2 (ad(E G)) = 0 if and only if for every pair of the form (Y , ψ), where ψ is a holomorphic map to M from a compact connected Riemann surface Y , and for every holomorphic reduction of structure group E P ⊂ ψ * E G to the subgroup P , the line bundle over Y associated to the principal P-bundle E P for χ is of nonnegative degree. Therefore, E G is semistable with c 2 (ad(E G)) = 0 if and only if for each pair (Y , ψ) of the above type the G-bundle ψ * E G over Y is semistable. Similar results remain valid for principal bundles over M with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka, [Mi], where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.

On semistable principal bundles over a complex projective manifold, II

Geometriae Dedicata, 2010

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and E G −→ X a holomorphic principal G-bundle, where G is a connected reductive linear algebraic group defined over C. Let Z(G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P ⊂ G and a holomorphic reduction of structure group E P

Moduli spaces of semistable sheaves on singular genus one curves

2008

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle EN of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves O(−1) supported on one irreducible component. Finally, we prove that the reduced subscheme of the connected component of the moduli space that contains vector bundles of rank r is isomorphic to the r-th symmetric product of the rati...