https://eprints.ucm.es/20702/1/0506511v4.pdf

Moduli spaces for principal bundles in arbitrary characteristic

This survey is devoted to recent developments in constructing moduli spaces of geometric objects such as vector bundles or principal G-bundles on a fixed variety (or on a fixed family of varieties). The choice of topics was dictated by the author’s taste and it ignores much of the recent progress. In particular, we avoid analytic techniques and present only an algebraic part of the theory, with special emphasis on positive characteristic. Also we do not say anything about geometry of obtained moduli spaces. One of the reasons is that in many cases they were just constructed and they were not studied yet. We will adopt the framework of the Geometric Invariant Theory (GIT) which forces us to restrict to a specially nice class of objects that can be fit to form a nice projective scheme. The well known idea behind is that in general there are either far too many objects to fit into a projective scheme (we need a parameter space) or they degenerate so badly that the moduli space would not be separated (quotients do not exist in general). People willing to sacrifice projective varieties using algebraic spaces or stacks still run into difficulties. They either talk about the moduli space of simple sheaves (this still restricts the class of sheaves and produces huge non-separated algebraic spaces) or they talk about the moduli stack of sheaves (which even after restricting to stable sheaves is not a Deligne–Mumford stack). In this last case it is sometimes possible to get useful information, usually looking at stratifications of the moduli stack coming from GIT (this presents the stack as a direct limit of moduli stacks) and using the full machinery of semistable sheaves developed to study projective moduli spaces. For example, the moduli stack is not separated although it satisfies the other half of the valuative properness criterion. Similarly, the moduli stack of semistable sheaves satisfies this part of the properness criterion (this follows from the construction of the moduli space although it was known before as Langton’s theorem). Only after restricting to stable sheaves we get an Artin stack and in this case GIT says that it represents the group quotient. Obviously in this last case we have a very nice quasiprojective moduli scheme which universally corepresents the moduli functor. The above explanation obviously does not mean that moduli stacks are useless in the context of vector

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Moduli spaces of sheaves and principal G-bundles

We prove that the cohomology of the moduli stack of G-bundles on a smooth projective curve is freely generated by the Atiyah-Bott classes in arbitrary characteristic. The main techni- cal tool needed is the construction of coarse moduli spaces for bun- dles with parabolic structure in arbitrary characteristic. Using these spaces we show that the cohomology of the moduli stack is pure and satisfies base-change for curves defined over a discrete valuation ring. Thereby we get an algebraic proof of the theorem of Atiyah and Bott and conversely this can be used to give a geometric proof of the fact that the Tamagawa number of a Chevalley group is the number of connected components of the moduli stack of principal bundles. 2010 Mathematics Subject Classification: 14D23, 14F20, 14D20

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The cohomology rings of moduli stacks of principal bundles over curves

Let G be a reductive group over an algebraically closed field k. Consider the moduli space of stable principal G-bundles on a smooth projective curve C over k. We give necessary and sufficient conditions for the existence of Poincare bundles over open subsets of this moduli space, and compute the orders of the corresponding obstruction classes. This generalizes the previous results of Newstead, Ramanan and Balaji–Biswas–Nagaraj–Newstead to all reductive groups, to all topological types of bundles, and also to all characteristics.

Poincaré families of G-bundles on a curve

Let k be an algebraically closed field of characteristic zero. Let f:X→S be a flat, projective morphism of k-schemes of finite type with integral geometric fibers. We prove the existence of a projective relative moduli space for semistable singular principal bundles on the fibers of f. This generalizes the result of A. Schmitt, who studied the case when X is a nodal curve.

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Moduli spaces of principal bundles on singular varieties

This paper addresses the problem of constructing and compactifying moduli spaces of stable principal G -bundles over smooth projective schemes for an arbitrary reductive group G in both zero and nonzero characteristic. This involves refining the methods of the third author [Transform. Groups 9 (2004), no. 2, 167–209; Int. Math. Res. Not. 2004, no. 62, 3327–3366;] and creating a unified formulation of the results of these papers and a paper of the first and fourth authors [Ann. of Math. (2) 161 (2005), no. 2, 1037–1092;]. The paper also complements a previous article by all four authors [Adv. Math. 219 (2008), no. 4, 1177–1245]. Particular features of the paper are the inclusion of non-semisimple groups and non-faithful representations of G , a new approach to obtaining the semistable reduction theorem in characteristic zero and in large positive characteristic and a construction of the moduli space of decorated sheaves over projective varieties in arbitrary characteristic. The first main theorem states that, in characteristic zero and in large positive characteristic, coarse moduli spaces for (semi)stable singular principal G -bundles can be defined using faithful representations (under some conditions) and exist as projective schemes. This covers many known cases, for example when G is semisimple, for G=GL(V) , also for G one of the classical groups O r (k) , SO r (k) and Sp r (k) provided the characteristic is not 2 and for groups of adjoint type. In order to use a non-faithful representation ρ , singular principal G -bundles must be replaced by principal ρ -sheaves. The second main theorem states that the corresponding coarse moduli spaces exist as quasi-projective schemes provided the base scheme is a curve or the characteristic is zero. Both types of moduli space can be considered as compactifications of the moduli space of slope-stable rational principal G -bundles.

L. Gómez Tomás

Papers

Moduli spaces for principal bundles in large characteristic