Moduli of representations of the fundamental group of a smooth projective variety. I. (English) Zbl 0891.14005
This is a step by step construction with full proofs of moduli spaces and associated objects of representations of the fundamental group of a smooth projective variety. It leads to various incarnations:
(1) The Betti moduli space \(\mathbf{M}_B\) as the moduli space of representations of a finitely generated group,
(2) the Dolbeault moduli space \(\mathbf{M}_{\text{Dol}}\) and
(3) the de Rham moduli space \(\mathbf{M}_{\text{DR}}\) if the group is the fundamental group of a smooth projective variety.
\(\mathbf{M}_{\text{Dol}}\), parametrizes so-called Higgs bundles on the variety \(X\), i.e. holomorphic vector bundles \(E\) together with a holomorphic map \(\theta:E\rightarrow E\otimes\Omega_X^1\) such that \(\theta\wedge\theta=0\). \(\mathbf{M}_B\) is the moduli space for vector bundles on \(X\) with integrable connection. There are comparison morphisms between the various moduli spaces. The actual moduli spaces are constructed in part II of the paper (see the following review). The necessary general formalism realized on the basis of Mumford’s geometric invariant theory of the Hilbert scheme is described in this part I.
In the sequel all schemes are supposed to be separated and of finite type over \(\text{Spec}({\mathbb C})\). For \(X\rightarrow S=\text{Spec}({\mathbb C})\) a projective scheme with very ample invertible sheaf \({\mathcal O}_X(1)\) and a coherent sheaf \({\mathcal E}\) on \(X\), one defines the Hilbert polynomial of \({\mathcal E}\) by \(p({\mathcal E},n)=\dim H^0(X,{\mathcal E}(n))\) for \(n\gg 0\). One writes \(d=d({\mathcal E})\) for the dimension of the support of \({\mathcal E}\), \(r=\text{rank}({\mathcal E})\), and then, \(p({\mathcal E},n)=rn^d/n!+an^{d-1}/(d-1)!+\cdots\), and \(\mu:=a/r\) is called the slope of \({\mathcal E}\). \({\mathcal E}\) is called of pure dimension \(d\) if for any non-zero subsheaf \({\mathcal F}\subset{\mathcal E}\) one has \(d({\mathcal F})=d({\mathcal E})\). Following Mumford, Gieseker and other authors one has the notions of \(p\)-(semi)stability, \(\mu\)-(semi)stability for \({\mathcal E}\) on \(X\), and when there is a reductive group acting on \(X\) etc. one has the notion of (semi)stable points. Any sheaf \({\mathcal E}\) of pure dimension \(d\) admits a unique filtration \(0={\mathcal E}_0\subset{\mathcal E}_1\ldots\subset{\mathcal E}_k={\mathcal E}\) with \(p\)-semistable quotients of pure dimension \(d\), the so-called Harder-Narasimhan filtration, with associated \(\text{gr}({\mathcal E})\).
An important categorical notion is the following: Let \(Y^{\natural}\) be a functor from the category of \(S\)-schemes to the category of sets. Let \(Y\) be a scheme over \(S\) and \(\phi:Y^{\natural}(S')\rightarrow Y(S')\) a natural transformation of functors. Then \(Y\) is said to corepresent \(Y^{\natural}\) if for every \(S\)-scheme \(W\) and natural transformation of functors \(\psi:Y^{\natural}\rightarrow W\) there is a unique morphism of schemes \(f:Y\rightarrow W\) giving a factorization \(\psi=f\circ\phi\). Define the fiber product of functors \(V\times_YY^{\natural}(S'):=V(S') \times_{Y(S')}Y^{\natural}(S')\). Then \(Y\) is said to universally corepresent \(Y^{\natural}\) if for every morphism of schemes \(V\rightarrow Y\), \(V\) corepresents the functor \(V\times_YY^{\natural}\).
A useful result for the whole paper is that the set of \(\mu\)-semistable sheaves with given Hilbert polynomial is bounded. The theory as sketched above may be generalized to projective schemes \(X\rightarrow S\), where \(S\) is any scheme of finite type over \(\text{Spec}({\mathbb C})\) by imposing flatness and then considering the fibers \(X_s\) separately. For a coherent sheaf \({\mathcal W}\) on a projective scheme \(X\rightarrow S\), Grothendieck introduced the Hilbert scheme \(\mathbf{Hilb}({\mathcal W},P)\) parametrizing quotients \({\mathcal W}\rightarrow{\mathcal F}\rightarrow 0\) with Hilbert polynomial \(P\). \(\mathbf{Hilb}({\mathcal W},P)\) represents a functor such that for any connected scheme \(\sigma:S'\rightarrow S\), the \(S'\)-valued points of \(\mathbf{Hilb}({\mathcal W},P)\) are isomorphism classes of quotients on \(X\times_SS'\), \(\sigma^*{\mathcal W}\rightarrow{\mathcal F}\rightarrow 0\), with \({\mathcal F}\) flat over \(S'\), and \(p({\mathcal F},n)=P(n)\). For \(m\gg 0\) one has a projective embedding \(\psi_m:\mathbf{Hilb}({\mathcal W},P)\rightarrow\text\textbf{Grass}(H^0(X/S,{\mathcal W}(m)),P(m))\).
For a finite dimensional vector space \(V\) one may consider \(\mathbf{Hilb}(V\otimes{\mathcal W},P)\) with \(Sl(V)\)-action. One defines open sets \(Q_2\subset Q_1\subset\mathbf{Hilb}(V\otimes{\mathcal W},P)\) as follows: \(Q_1\) consists of coherent sheaves \({\mathcal E}\) on \(X'=X\times_SS'\), flat over \(S'\) with Hilbert polynomial \(P\) and morphism \(\alpha:V\otimes{\mathcal O}_S'\rightarrow H^0(X'/S',{\mathcal E}(N))\) such that the sections in the image of \(\alpha\) generate \({\mathcal E}(N)\) for suitable large \(N\), and such that \({\mathcal W}={\mathcal O}_X(-N)\) and \(V={\mathbb C}^{P(N)}\), and such that \({\mathcal E}\) has pure dimension \(d\) and is \(p\)-semistable. \(Q_2\subset Q_1\) is the open subset where \(\alpha\) is an isomorphism. Let \(\mathbf{M}^{\natural}({\mathcal O}_X,P)\) be the functor which associates to any \(S\)-scheme \(S'\) the set of semistable sheaves \({\mathcal E}\) on \(X'/S'\), of pure dimension \(d\) with Hilbert polynomial \(P\). The next result summarizes its main properties: Let \(\mathbf{M}({\mathcal O}_X,P):=Q_2/Sl(V)\) be the good quotient (in Mumford’s language) applied to the group action on \(\mathbf{Hilb}(V\otimes{\mathcal W},P,d)\) (= the closure of the set of quotients \({\mathcal E}\) in \(\mathbf{Hilb}(V\otimes{\mathcal W},P)\) of pure dimension \(d\)). Then:
(1) There exists a natural transformation \(\phi:\mathbf{M}^{\natural}({\mathcal O}_X,P)\rightarrow\text\textbf{M}({\mathcal O}_X,P)\) such that
\((\mathbf{M}({\mathcal O}_X,P),\phi)\) universially corepresents \(\mathbf{M}({\mathcal O}_X,P)\).
(2) \(\mathbf{M}({\mathcal O}_X,P)\) is a projective scheme.
(3) The points of \(\mathbf{M}({\mathcal O}_X,P)\) represent the equivalence classes of semistable sheaves under the relation \({\mathcal E}_1\sim{\mathcal E}_2\) if \(\text{gr}({\mathcal E}_1)=\text{gr}({\mathcal E}_2)\) (for the Harder-Narasimhan filtration).
(4) There is an open subset \(\mathbf{M}^s({\mathcal O}_X,P)\subset \text\textbf{M}({\mathcal O}_X,P)\) with inverse image \(Q_2^s\) whose points represent isomorphism classes of \(p\)-stable sheaves.
(5) If \(x\in\mathbf{M}^s({\mathcal O}_X,P)\) is a point such that \(Q_2^s\) is smooth at the inverse image of \(x\), then \(\mathbf{M}^s({\mathcal O}_X,P)\) is smooth at \(x\).
To treat the most important cases of Higgs bundles and bundles with integrable connection, one needs sheaves of rings of differential operators \(\Lambda\) on \(X\rightarrow S\). This is extensively discussed and a privileged kind of such rings consisting of so-called split almost polynomial sheaves of differential operators \((\Lambda,\zeta)\), play a major role for the cases at hand. One would like to have a result as above for semistable \(\Lambda\)-modules. Here the notions of (semi)stability, pure dimension etc. carry over to \(\Lambda\)-modules on \(X\). One can define a unique filtration on \(\Lambda\)-modules and two \(p\)-semistable \(\Lambda\)-modules are called Jordan equivalent if they have isomorphic associated \(\text{gr}\)’s. Again there is a boundedness result and one may generalize to the relative case. For \(N\gg 0\) consider the functor which associates to each \(S\)-scheme \(S'\) the set of isomorphism classes of pairs \(({\mathcal E},\alpha)\) with \({\mathcal E}\) a \(p\)-semistable \(\Lambda\)-module with Hilbert polynomial \(P\) on \(X'=X\times_SS'\), such that \(\alpha:({\mathcal O}_{S'})^{p(N)}\widetilde{\longrightarrow} H^0(X'/S',{\mathcal E}(N))\). Then this functor is representable by a quasi-projective scheme \(Q\) over \(S\). Let \(\mathbf{M}^{\natural}(\Lambda,P)\) denote the functor which associates to \(S'\rightarrow S\) the set of isomorphism classes of \(p\)-semistable \(\lambda'\)-modules on \(X'\rightarrow S'\) with Hilbert polynomial \(P\). Then one has the main result:
Let \(\mathbf{M}(\Lambda,P)\) be the good quotient \(Q/Sl(V)\). One has a morphism of functors \(\phi:\mathbf{M}{\natural}(\Lambda,P)\rightarrow\text\textbf{M} (\lambda,P)\) such that \((\mathbf{M}(\Lambda,P),\phi)\) universally corepresents the functor \(\mathbf{M}^{\natural}(\Lambda,P)\). Furthermore:
(1) \(\mathbf{M}(\Lambda,P)\) is a quasi-projective variety.
(2) The geometric points of \(\mathbf{M}(\Lambda,P)\) represent the equivalence classes of \(p\)-semistable \(\Lambda\)-modules with Hilbert polynomial \(P\) on the fibers \(X_s\) under Jordan equivalence.
(3) The closed orbits in \(Q\) are exactly those corresponding to semisimple objects \({\mathcal F}\) with \(\text{ gr}({\mathcal F})\simeq{\mathcal F}\).
(4) There is an open subset \(\mathbf{M}^s(\Lambda,P)\subset\text\textbf{M}(\Lambda,P)\) whose points represent isomorphism classes of \(p\)-stable \(\Lambda\)-modules.
Rigidifying the moduli functors one may construct fine moduli spaces. Assuming that the fibers \(X_s\) are irreducible this leads to representation spaces \(\mathbf{R}(\Lambda,\xi,P)\), where \(\xi:S\rightarrow X\) is a section such that the Jordan-Hölder factors of the \(\Lambda\)-modules are locally free near \(\xi\) (condition \(LF(\xi)\)). These spaces classify pairs \(({\mathcal E},\beta)\) with \({\mathcal E}\) a \(p\)-semistable \(\Lambda\)-module with Hilbert polynomial \(P\), satisfying \(LF(\xi)\), and such that \(\beta\) is a frame for \(\xi^*({\mathcal E})\).
The paper closes with a detailed section on analytic aspects of the theory to be used in the following part II.
(1) The Betti moduli space \(\mathbf{M}_B\) as the moduli space of representations of a finitely generated group,
(2) the Dolbeault moduli space \(\mathbf{M}_{\text{Dol}}\) and
(3) the de Rham moduli space \(\mathbf{M}_{\text{DR}}\) if the group is the fundamental group of a smooth projective variety.
\(\mathbf{M}_{\text{Dol}}\), parametrizes so-called Higgs bundles on the variety \(X\), i.e. holomorphic vector bundles \(E\) together with a holomorphic map \(\theta:E\rightarrow E\otimes\Omega_X^1\) such that \(\theta\wedge\theta=0\). \(\mathbf{M}_B\) is the moduli space for vector bundles on \(X\) with integrable connection. There are comparison morphisms between the various moduli spaces. The actual moduli spaces are constructed in part II of the paper (see the following review). The necessary general formalism realized on the basis of Mumford’s geometric invariant theory of the Hilbert scheme is described in this part I.
In the sequel all schemes are supposed to be separated and of finite type over \(\text{Spec}({\mathbb C})\). For \(X\rightarrow S=\text{Spec}({\mathbb C})\) a projective scheme with very ample invertible sheaf \({\mathcal O}_X(1)\) and a coherent sheaf \({\mathcal E}\) on \(X\), one defines the Hilbert polynomial of \({\mathcal E}\) by \(p({\mathcal E},n)=\dim H^0(X,{\mathcal E}(n))\) for \(n\gg 0\). One writes \(d=d({\mathcal E})\) for the dimension of the support of \({\mathcal E}\), \(r=\text{rank}({\mathcal E})\), and then, \(p({\mathcal E},n)=rn^d/n!+an^{d-1}/(d-1)!+\cdots\), and \(\mu:=a/r\) is called the slope of \({\mathcal E}\). \({\mathcal E}\) is called of pure dimension \(d\) if for any non-zero subsheaf \({\mathcal F}\subset{\mathcal E}\) one has \(d({\mathcal F})=d({\mathcal E})\). Following Mumford, Gieseker and other authors one has the notions of \(p\)-(semi)stability, \(\mu\)-(semi)stability for \({\mathcal E}\) on \(X\), and when there is a reductive group acting on \(X\) etc. one has the notion of (semi)stable points. Any sheaf \({\mathcal E}\) of pure dimension \(d\) admits a unique filtration \(0={\mathcal E}_0\subset{\mathcal E}_1\ldots\subset{\mathcal E}_k={\mathcal E}\) with \(p\)-semistable quotients of pure dimension \(d\), the so-called Harder-Narasimhan filtration, with associated \(\text{gr}({\mathcal E})\).
An important categorical notion is the following: Let \(Y^{\natural}\) be a functor from the category of \(S\)-schemes to the category of sets. Let \(Y\) be a scheme over \(S\) and \(\phi:Y^{\natural}(S')\rightarrow Y(S')\) a natural transformation of functors. Then \(Y\) is said to corepresent \(Y^{\natural}\) if for every \(S\)-scheme \(W\) and natural transformation of functors \(\psi:Y^{\natural}\rightarrow W\) there is a unique morphism of schemes \(f:Y\rightarrow W\) giving a factorization \(\psi=f\circ\phi\). Define the fiber product of functors \(V\times_YY^{\natural}(S'):=V(S') \times_{Y(S')}Y^{\natural}(S')\). Then \(Y\) is said to universally corepresent \(Y^{\natural}\) if for every morphism of schemes \(V\rightarrow Y\), \(V\) corepresents the functor \(V\times_YY^{\natural}\).
A useful result for the whole paper is that the set of \(\mu\)-semistable sheaves with given Hilbert polynomial is bounded. The theory as sketched above may be generalized to projective schemes \(X\rightarrow S\), where \(S\) is any scheme of finite type over \(\text{Spec}({\mathbb C})\) by imposing flatness and then considering the fibers \(X_s\) separately. For a coherent sheaf \({\mathcal W}\) on a projective scheme \(X\rightarrow S\), Grothendieck introduced the Hilbert scheme \(\mathbf{Hilb}({\mathcal W},P)\) parametrizing quotients \({\mathcal W}\rightarrow{\mathcal F}\rightarrow 0\) with Hilbert polynomial \(P\). \(\mathbf{Hilb}({\mathcal W},P)\) represents a functor such that for any connected scheme \(\sigma:S'\rightarrow S\), the \(S'\)-valued points of \(\mathbf{Hilb}({\mathcal W},P)\) are isomorphism classes of quotients on \(X\times_SS'\), \(\sigma^*{\mathcal W}\rightarrow{\mathcal F}\rightarrow 0\), with \({\mathcal F}\) flat over \(S'\), and \(p({\mathcal F},n)=P(n)\). For \(m\gg 0\) one has a projective embedding \(\psi_m:\mathbf{Hilb}({\mathcal W},P)\rightarrow\text\textbf{Grass}(H^0(X/S,{\mathcal W}(m)),P(m))\).
For a finite dimensional vector space \(V\) one may consider \(\mathbf{Hilb}(V\otimes{\mathcal W},P)\) with \(Sl(V)\)-action. One defines open sets \(Q_2\subset Q_1\subset\mathbf{Hilb}(V\otimes{\mathcal W},P)\) as follows: \(Q_1\) consists of coherent sheaves \({\mathcal E}\) on \(X'=X\times_SS'\), flat over \(S'\) with Hilbert polynomial \(P\) and morphism \(\alpha:V\otimes{\mathcal O}_S'\rightarrow H^0(X'/S',{\mathcal E}(N))\) such that the sections in the image of \(\alpha\) generate \({\mathcal E}(N)\) for suitable large \(N\), and such that \({\mathcal W}={\mathcal O}_X(-N)\) and \(V={\mathbb C}^{P(N)}\), and such that \({\mathcal E}\) has pure dimension \(d\) and is \(p\)-semistable. \(Q_2\subset Q_1\) is the open subset where \(\alpha\) is an isomorphism. Let \(\mathbf{M}^{\natural}({\mathcal O}_X,P)\) be the functor which associates to any \(S\)-scheme \(S'\) the set of semistable sheaves \({\mathcal E}\) on \(X'/S'\), of pure dimension \(d\) with Hilbert polynomial \(P\). The next result summarizes its main properties: Let \(\mathbf{M}({\mathcal O}_X,P):=Q_2/Sl(V)\) be the good quotient (in Mumford’s language) applied to the group action on \(\mathbf{Hilb}(V\otimes{\mathcal W},P,d)\) (= the closure of the set of quotients \({\mathcal E}\) in \(\mathbf{Hilb}(V\otimes{\mathcal W},P)\) of pure dimension \(d\)). Then:
(1) There exists a natural transformation \(\phi:\mathbf{M}^{\natural}({\mathcal O}_X,P)\rightarrow\text\textbf{M}({\mathcal O}_X,P)\) such that
\((\mathbf{M}({\mathcal O}_X,P),\phi)\) universially corepresents \(\mathbf{M}({\mathcal O}_X,P)\).
(2) \(\mathbf{M}({\mathcal O}_X,P)\) is a projective scheme.
(3) The points of \(\mathbf{M}({\mathcal O}_X,P)\) represent the equivalence classes of semistable sheaves under the relation \({\mathcal E}_1\sim{\mathcal E}_2\) if \(\text{gr}({\mathcal E}_1)=\text{gr}({\mathcal E}_2)\) (for the Harder-Narasimhan filtration).
(4) There is an open subset \(\mathbf{M}^s({\mathcal O}_X,P)\subset \text\textbf{M}({\mathcal O}_X,P)\) with inverse image \(Q_2^s\) whose points represent isomorphism classes of \(p\)-stable sheaves.
(5) If \(x\in\mathbf{M}^s({\mathcal O}_X,P)\) is a point such that \(Q_2^s\) is smooth at the inverse image of \(x\), then \(\mathbf{M}^s({\mathcal O}_X,P)\) is smooth at \(x\).
To treat the most important cases of Higgs bundles and bundles with integrable connection, one needs sheaves of rings of differential operators \(\Lambda\) on \(X\rightarrow S\). This is extensively discussed and a privileged kind of such rings consisting of so-called split almost polynomial sheaves of differential operators \((\Lambda,\zeta)\), play a major role for the cases at hand. One would like to have a result as above for semistable \(\Lambda\)-modules. Here the notions of (semi)stability, pure dimension etc. carry over to \(\Lambda\)-modules on \(X\). One can define a unique filtration on \(\Lambda\)-modules and two \(p\)-semistable \(\Lambda\)-modules are called Jordan equivalent if they have isomorphic associated \(\text{gr}\)’s. Again there is a boundedness result and one may generalize to the relative case. For \(N\gg 0\) consider the functor which associates to each \(S\)-scheme \(S'\) the set of isomorphism classes of pairs \(({\mathcal E},\alpha)\) with \({\mathcal E}\) a \(p\)-semistable \(\Lambda\)-module with Hilbert polynomial \(P\) on \(X'=X\times_SS'\), such that \(\alpha:({\mathcal O}_{S'})^{p(N)}\widetilde{\longrightarrow} H^0(X'/S',{\mathcal E}(N))\). Then this functor is representable by a quasi-projective scheme \(Q\) over \(S\). Let \(\mathbf{M}^{\natural}(\Lambda,P)\) denote the functor which associates to \(S'\rightarrow S\) the set of isomorphism classes of \(p\)-semistable \(\lambda'\)-modules on \(X'\rightarrow S'\) with Hilbert polynomial \(P\). Then one has the main result:
Let \(\mathbf{M}(\Lambda,P)\) be the good quotient \(Q/Sl(V)\). One has a morphism of functors \(\phi:\mathbf{M}{\natural}(\Lambda,P)\rightarrow\text\textbf{M} (\lambda,P)\) such that \((\mathbf{M}(\Lambda,P),\phi)\) universally corepresents the functor \(\mathbf{M}^{\natural}(\Lambda,P)\). Furthermore:
(1) \(\mathbf{M}(\Lambda,P)\) is a quasi-projective variety.
(2) The geometric points of \(\mathbf{M}(\Lambda,P)\) represent the equivalence classes of \(p\)-semistable \(\Lambda\)-modules with Hilbert polynomial \(P\) on the fibers \(X_s\) under Jordan equivalence.
(3) The closed orbits in \(Q\) are exactly those corresponding to semisimple objects \({\mathcal F}\) with \(\text{ gr}({\mathcal F})\simeq{\mathcal F}\).
(4) There is an open subset \(\mathbf{M}^s(\Lambda,P)\subset\text\textbf{M}(\Lambda,P)\) whose points represent isomorphism classes of \(p\)-stable \(\Lambda\)-modules.
Rigidifying the moduli functors one may construct fine moduli spaces. Assuming that the fibers \(X_s\) are irreducible this leads to representation spaces \(\mathbf{R}(\Lambda,\xi,P)\), where \(\xi:S\rightarrow X\) is a section such that the Jordan-Hölder factors of the \(\Lambda\)-modules are locally free near \(\xi\) (condition \(LF(\xi)\)). These spaces classify pairs \(({\mathcal E},\beta)\) with \({\mathcal E}\) a \(p\)-semistable \(\Lambda\)-module with Hilbert polynomial \(P\), satisfying \(LF(\xi)\), and such that \(\beta\) is a frame for \(\xi^*({\mathcal E})\).
The paper closes with a detailed section on analytic aspects of the theory to be used in the following part II.
Reviewer: W.W.J.Hulsbergen (Haarlem)
MSC:
| 14F35 | Homotopy theory and fundamental groups in algebraic geometry |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14L24 | Geometric invariant theory |
Keywords:
representation space; moduli space; corepresentation; geometric invariant theory; Hilbert scheme; good quotient; fine moduli spacesCitations:
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