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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 $\pi :C\to S$, with $\pi$ proper, all fibers of dimension $\le 1$, and such that ${\pi }_{*}\left({𝒪}_{C}\right)={𝒪}_{S}$. A Higgs bundle on $C$ is a vector bundle $ℱ$ together with a section $\theta$ of ${\Gamma }\left(C,ℰnd\left(ℱ\right)\otimes {\omega }_{C}\right)$. The coefficients of the characteristic polynomial of $\theta$ define global sections ${f}_{i}\in {\Gamma }\left(C,{\omega }^{i}\right)$, and the affine space classifying such sections is called the characteristic variety $𝒞har$ (it depends on $\text{rk}\left(ℱ\right)\right)$, and $ℱ$ defines a point $\text{char}\left(ℱ\right)$ in $𝒞har$. Such a Higgs bundle will often be denoted $\left(ℱ,\theta \right)$. 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 ${ℳ}_{\theta }^{0}$. Then ${ℳ}_{\theta }^{0}$ embeds as an open subscheme into the onrmalization ${ℳ}_{\theta }$ of ${ℙ}^{N}×𝒞har$ (for suitable $N\right)$ in ${ℳ}_{\theta }^{0}$.

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

$\theta \in {\Gamma }\left(C,\text{Lie}\left({𝒢}_{P}\right)×{\omega }_{C}\right)$

is called semistable if $\left(\text{Lie}\left({𝒢}_{P}\right)$, ad$\left(\theta \right)\right)$ is a semistable Higgs bundle of degree zero. One also has the notion of stable $P$. The main result on semistable pairs $\left(P,\theta \right)$ 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, $\left({P}_{K},{\theta }_{K}\right)$ a semistable pair (associated with a connected reductive group $𝒢$ over $C\right)$ whose characteristic is integral over $V$, then there exists a finite extension ${V}^{\text{'}}$ of $V$ such that the base extension of $\left({P}_{K},{\theta }_{K}\right)$ extends to a semistable pair on ${C}_{{V}^{\text{'}}}$. Furthermore, if the special fiber of this extension is stable, then any other semistable extension is isomorphic to it. For stable $\left(P,\theta \right)$ one is led to construct an algebraic moduli stack ${ℳ}_{\theta }^{0}\left(𝒢\right)$ and the coarse moduli space ${M}_{\theta }^{0}\left(𝒢\right)$ which is shown to be quasi-projective of explicitly calculated relative dimension over a suitable base. As before one defines a ${M}_{\theta }\left(𝒢\right)$ as the normalisation of a ${ℙ}^{N}$ in ${M}_{\theta }^{0}\left(𝒢\right)$. Then ${M}_{\theta }\left(𝒢\right)$ is projective over $𝒞har$ and contains ${M}_{\theta }^{0}\left(𝒢\right)$ as an open subscheme. Then, for example, if $C$ has genus $>2$, the boundary ${M}_{\theta }\left(𝒢\right)-{M}_{\theta }^{0}\left(𝒢\right)$ has codimension $\ge 4$. 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 ${ℳ}^{0}\left(𝒢\right)$ of stable (Higgs) $𝒢$-bundles coincides with that of ${ℳ}_{\theta }^{0}\left(𝒢\right)$, ${M}_{\theta }\left(𝒢\right)$ as well as that of a generic fiber of ${ℳ}_{\theta }^{0}\left(𝒢\right)\to 𝒞har$, under the natural mappings. Among many other results, one application of abelianisation is given by another corollary: On each connected component of ${ℳ}_{\theta }^{0}\left(𝒢\right)$, all global functions are obtained by pullback from $𝒞har$.

In part IV the accent is on ${ℳ}^{0}\left(𝒢\right)$, where $𝒢$ is the twisted form of some semi-simple $G$. The notion of ${{\Omega }}_{C}$- connections $\nabla$ on $𝒢$-torsors $P$ is introduced. ${ℳ}_{\nabla }^{0}\left(𝒢\right)$ denotes the moduli stack of such pairs $\left(P,\nabla \right)$ with $P$ stable. It is fibered over ${ℳ}^{0}\left(𝒢\right)$. Over $ℂ$, ${ℳ}_{\nabla }^{0}\left(𝒢\right)$ classifies bundles with integrable connections, i.e. representations of ${\pi }_{1}\left(C\right)$. A locally faithful $𝒢$-representation $ℱ$ defines a line bundle $ℒ=ℒ\left(ℱ\right)$ on ${ℳ}^{0}\left(𝒢\right)$. Then the pullback of $ℒ$ to ${ℳ}_{\nabla }^{0}\left(𝒢\right)$ 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.

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