Let G be a split connected semisimple group over a field. We give a conjectural formula for the motive of the stack of G-bundles over a curve C, in terms of special values of the motivic zeta function of C. The formula is true if C = P 1 or G = SL n . If k = C, upon applying the Poincaré or Serre characteristic, the formula reduces to results of Teleman and Atiyah-Bott on the gauge group. If k = F q , upon applying the counting measure, it reduces to the fact that the Tamagawa number of G over the function field of C is |π 1 (G)|.

Abstract. Let X be a smooth projective geometrically connected curve over a finite field with function field K. Let G be a connected semisimple group scheme over X. Under certain hypotheses we prove the equality of two numbers associated with G. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of Gtorsors on X. Our results are most useful for studying connected components as much is known about Tamagawa numbers.

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curve C, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Thélène, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne-Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curve C is elliptic.2000 Mathematics Subject Classification. 14D23, 14D20. Key words and phrases. Essential dimension, moduli stack, endomorphism algebra, curve. I. B. is supported by the J. C. Bose Fellowship. A. D. is partially supported by NSERC. N. H. was partially supported by SFB 647: Space -Time -Matter in Berlin. He thanks TIFR Bombay for hospitality, and Bernd Kreussler for a useful discussion on bundles over elliptic curves. Projective Modules over Right-Artinian RingsLet R be a ring. Our rings are always associative, and they always have a unit, but they are not necessarily commutative. By an R-module, we mean a right R-module, unless stated otherwise. Let n ⊂ R be a nilpotent two-sided ideal.Lemma 2.1. Every element q ∈ R/n with q 2 = q admits a lift p ∈ R with p 2 = p.Proof. By assumption, there is an integer n ≥ 1 such that n n = 0. Using induction over n, we may assume n 2 = 0 without loss of generality.

Abstract. We compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SL n is one.

Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of O ⊕r X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on C r . The group PGL(r, C) acts on Q via the action of GL(r, C) on O ⊕r X . We prove that this subgroup PGL(r, C) is the connected component, containing the identity element, of the holomorphic automorphism group Aut(Q). As an application of it, we prove that the isomorphism class of the complex manifold Q uniquely determines the isomorphism class of the Riemann surface X.

Using the description of the category of quasi-coherent sheaves on a root stack, we compute the G-theory of root stacks via localisation methods. We apply our results to the study of equivariant K-theory of algebraic varieties under certain conditions. Proof. This follows by combining Corollaries 3.10 and 3.15. So we have:G(X L, r ) ∼ = K(EP(X, L, r)), and we reduced the problem to describing the K-theory of the (abelian) category of coherent extendable pairs EP(X, L, r).We are going to use several splittings of the category of coherent extendable pairs to simplify the latter K-theory. The first step is this Lemma 3.29. If X is a locally noetherian scheme, then in notation of section 3.3 one has:

We compute the motive of the classifying stack of an orthogonal group in the Grothendieck ring of stacks over a field of characteristic different from two.

We find upper bounds for the essential dimension of various moduli stacks of SLn-bundles over a curve. When n is a prime power, our calculation computes the essential dimension of the stack of stable bundles exactly and the essential dimension is not equal to the dimension in this case.