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Stable G-bundles and projective connections. (English) Zbl 0790.14019

This paper gives a self-contained, detailed account of the construction and compactification of the moduli space of Higgs bundle on (families of) curves. It is divided into five parts: I. The theorem of the cube; II. G-bundles; III. Abelianisation; IV. Projective connections; V. Infinitesimal parabolic structure.

Part I starts with a theorem on the determinant of the cohomology of coherent sheaves on a curve. It is shown that the theorem of the cube follows from this result. A second application is at the basis of the construction, via theta-functions, of global sections of determinant bundles on the moduli space of Higgs bundles. More precisely, let S be a noetherian base scheme for the family of curves π:CS, with π proper, all fibers of dimension 1, and such that π * (𝒪 C )=𝒪 S . A Higgs bundle on C is a vector bundle together with a section θ of Γ(C,nd()ω C ). The coefficients of the characteristic polynomial of θ define global sections f i Γ(C,ω i ), and the affine space classifying such sections is called the characteristic variety 𝒞har (it depends on rk()), and defines a point char() in 𝒞har. Such a Higgs bundle will often be denoted (,θ). One has the notion of (semi)-stability for Higgs bundles, and any semistable Higgs bundle admits a Jordan- Hölder (JH) filtration by subbundles with stable quotieents of constant ratio degree/rank. The isomorphism classes and multiplicities of these stable components are independent of the filtration. Two semi-stable Higgs bundles are called JH-equivalent if these coincide. A result on JH- equivalence is derived and used to show that the theta-functions separate points in the moduli space of Higgs bundles. The moduli-space of stable Higgs bundles of given rank and degree is constructed as an algebraic space θ 0 . Then θ 0 embeds as an open subscheme into the onrmalization θ of N ×𝒞har (for suitable N) in θ 0 .

In part II one considers a reductive connected algebraic group 𝒢 over a smooth projective connected curve C over a field k. A 𝒢-torsor P on C, together with an element

θΓ(C,Lie(𝒢 P )×ω C )

is called semistable if (Lie(𝒢 P ), ad(θ)) is a semistable Higgs bundle of degree zero. One also has the notion of stable P. The main result on semistable pairs (P,θ) is the following semistable reduction theorem: If V is a complete discrete valuation ring with fraction field K, CV a smooth projective curve, (P K ,θ K ) a semistable pair (associated with a connected reductive group 𝒢 over C) whose characteristic is integral over V, then there exists a finite extension V ' of V such that the base extension of (P K ,θ K ) extends to a semistable pair on C V ' . Furthermore, if the special fiber of this extension is stable, then any other semistable extension is isomorphic to it. For stable (P,θ) one is led to construct an algebraic moduli stack θ 0 (𝒢) and the coarse moduli space M θ 0 (𝒢) which is shown to be quasi-projective of explicitly calculated relative dimension over a suitable base. As before one defines a M θ (𝒢) as the normalisation of a N in M θ 0 (𝒢). Then M θ (𝒢) is projective over 𝒞har and contains M θ 0 (𝒢) as an open subscheme. Then, for example, if C has genus >2, the boundary M θ (𝒢)-M θ 0 (𝒢) has codimension 4. Many other results are derived.

In part III the theory is extended to exceptional groups. As a corollary of the theory one obtains, with the notations above, that the set of connected components of the moduli space 0 (𝒢) of stable (Higgs) 𝒢-bundles coincides with that of θ 0 (𝒢), M θ (𝒢) as well as that of a generic fiber of θ 0 (𝒢)𝒞har, under the natural mappings. Among many other results, one application of abelianisation is given by another corollary: On each connected component of θ 0 (𝒢), all global functions are obtained by pullback from 𝒞har.

In part IV the accent is on 0 (𝒢), where 𝒢 is the twisted form of some semi-simple G. The notion of Ω C - connections on 𝒢-torsors P is introduced. 0 (𝒢) denotes the moduli stack of such pairs (P,) with P stable. It is fibered over 0 (𝒢). Over , 0 (𝒢) classifies bundles with integrable connections, i.e. representations of π 1 (C). A locally faithful 𝒢-representation defines a line bundle =() on 0 (𝒢). Then the pullback of to 0 (𝒢) has a connection . Its curvature can be described explicitly.

The final part V discusses parabolic structures in the sense of C. Seshadri. The parabolic analogue of a Higgs bundle is introduced and a theory parallel to the one in the foregoing parts is sketched.


MSC:
14H10Families, algebraic moduli (curves)
14H60Vector bundles on curves and their moduli
14F05Sheaves, derived categories of sheaves, etc.
53C07Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills)