Stable \(G\)-bundles and projective connections.

*(English)*Zbl 0790.14019This 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- HĂ¶lder (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.

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- HĂ¶lder (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.

Reviewer: W.W.J.Hulsbergen (Breda)

##### MSC:

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) |