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 \(\pi:C\to S\), with \(\pi\) proper, all fibers of dimension \(\leq 1\), and such that \(\pi_ *({\mathcal O}_ C)={\mathcal O}_ S\). A Higgs bundle on \(C\) is a vector bundle \({\mathcal F}\) together with a section \(\theta\) of \(\Gamma(C,{\mathcal E}nd({\mathcal F})\otimes\omega_ C)\). The coefficients of the characteristic polynomial of \(\theta\) define global sections \(f_ i\in\Gamma(C,\omega^ i)\), and the affine space classifying such sections is called the characteristic variety \({\mathcal C}har\) (it depends on \(\text{rk}({\mathcal F}))\), and \({\mathcal F}\) defines a point \(\text{char}({\mathcal F})\) in \({\mathcal C}har\). Such a Higgs bundle will often be denoted \(({\mathcal F},\theta)\). 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 \({\mathcal M}^ 0_ \theta\). Then \({\mathcal M}_ \theta^ 0\) embeds as an open subscheme into the onrmalization \({\mathcal M}_ \theta\) of \(\mathbb{P}^ N\times{\mathcal C}har\) (for suitable \(N)\) in \({\mathcal M}^ 0_ \theta\).
In part II one considers a reductive connected algebraic group \({\mathcal G}\) over a smooth projective connected curve \(C\) over a field \(k\). A \({\mathcal G}\)-torsor \(P\) on \(C\), together with an element \[ \theta\in\Gamma(C,\text{Lie}({\mathcal G}_ P)\times\omega_ C) \] is called semistable if \((\text{Lie}({\mathcal G}_ P)\), ad\((\theta))\) is a semistable Higgs bundle of degree zero. One also has the notion of stable \(P\). The main result on semistable pairs \((P,\theta)\) is the following semistable reduction theorem: If \(V\) is a complete discrete valuation ring with fraction field \(K\), \(C\to V\) a smooth projective curve, \((P_ K,\theta_ K)\) a semistable pair (associated with a connected reductive group \({\mathcal G}\) over \(C)\) whose characteristic is integral over \(V\), then there exists a finite extension \(V'\) of \(V\) such that the base extension of \((P_ K,\theta_ K)\) 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 \((P,\theta)\) one is led to construct an algebraic moduli stack \({\mathcal M}^ 0_ \theta({\mathcal G})\) and the coarse moduli space \(M^ 0_ \theta({\mathcal G})\) which is shown to be quasi-projective of explicitly calculated relative dimension over a suitable base. As before one defines a \(M_ \theta({\mathcal G})\) as the normalisation of a \(\mathbb{P}^ N\) in \(M^ 0_ \theta({\mathcal G})\). Then \(M_ \theta({\mathcal G})\) is projective over \({\mathcal C}har\) and contains \(M^ 0_ \theta({\mathcal G})\) as an open subscheme. Then, for example, if \(C\) has genus \(>2\), the boundary \(M_ \theta({\mathcal G})-M^ 0_ \theta({\mathcal G})\) has codimension \(\geq 4\). 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 \({\mathcal M}^ 0({\mathcal G})\) of stable (Higgs) \({\mathcal G}\)-bundles coincides with that of \({\mathcal M}^ 0_ \theta({\mathcal G})\), \(M_ \theta({\mathcal G})\) as well as that of a generic fiber of \({\mathcal M}^ 0_ \theta({\mathcal G})\to{\mathcal C}har\), under the natural mappings. Among many other results, one application of abelianisation is given by another corollary: On each connected component of \({\mathcal M}^ 0_ \theta({\mathcal G})\), all global functions are obtained by pullback from \({\mathcal C}har\).
In part IV the accent is on \({\mathcal M}^ 0({\mathcal G})\), where \({\mathcal G}\) is the twisted form of some semi-simple \(G\). The notion of \(\Omega_ C\)- connections \(\nabla\) on \({\mathcal G}\)-torsors \(P\) is introduced. \({\mathcal M}^ 0_ \nabla({\mathcal G})\) denotes the moduli stack of such pairs \((P,\nabla)\) with \(P\) stable. It is fibered over \({\mathcal M}^ 0({\mathcal G})\). Over \(\mathbb{C}\), \({\mathcal M}^ 0_ \nabla({\mathcal G})\) classifies bundles with integrable connections, i.e. representations of \(\pi_ 1(C)\). A locally faithful \({\mathcal G}\)-representation \({\mathcal F}\) defines a line bundle \({\mathcal L}={\mathcal L}({\mathcal F})\) on \({\mathcal M}^ 0({\mathcal G})\). Then the pullback of \({\mathcal L}\) to \({\mathcal M}^ 0_ \nabla({\mathcal G})\) has a connection \(\nabla\). 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.


14H10 Families, moduli of curves (algebraic)
14H60 Vector bundles on curves and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)