Surfae group representations, Higgs bundles, and holomorphi triples Steven B. Bradlow 1,2 Department of Mathematis, University of Illinois, Urbana, IL 61801, USA E-mail: bradlow@math.uiu.edu Osar Garía Prada 1,3,5,6 Departamento de Matemátias, Universidad Autónoma de Madrid, 28049 Madrid, Spain E-mail: osar.garia-prada@uam.es Peter B. Gothen 1,4,5 Departamento de Matemátia Pura, Fauldade de Ciênias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal E-mail: pbgothen@f.up.pt June 3, 2002 1 Members of VBAC (Vetor Bundles on Algebrai Curves), whih is partially supported by EAGER (EC FP5 Contrat no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contrat no. HPRN-CT-2000-00101). 2 Partially supported by the National Siene Foundation under grant DMS-0072073 3 Partially supported by the Ministerio de Cienia y Tenología (Spain) under grant BFM2000-0024 4 Partially supported by the Fundação para a Ciênia e a Tenologia (Portugal) through the Centro de Matemátia da Universidade do Porto and through grant no. SFRH/BPD/1606/2000. 5 Partially supported by the Portugal/Spain bilateral Programme Aiones Integradas, grant nos. HP2000-0015 and AI-01/24 6 Partially supported by a British EPSRC grant (Otober-Deember 2001)
Abstrat. Using the L 2 norm of the Higgs field as a Morse funtion, we study the moduli spaes of U(p, q)-higgs bundles over a Riemann surfae. We require that the genus of the surfae be at least two, but plae no onstraints on (p, q). A key step is the identifiation of the funtion s loal minima as moduli spaes of holomorphi triples. We prove that these moduli spaes of triples are irreduible and non-empty. Beause of the relation between flat bundles and fundamental group representations, we an interpret our onlusions as results about the number of onneted omponents in the moduli spae of semisimple PU(p, q)- representations. The topologial invariants of the flat bundles bundle are used to label omponents. These invariants are bounded by a Milnor Wood type inequality. For eah allowed value of the invariants satisfying a ertain oprimality ondition, we prove that the orresponding omponent is nonempty and onneted. If the oprimality ondition does not hold, our results apply to the irreduible representations.
Contents 1 Introdution 1 2 Representations of surfae groups 10 2.1 Definitions............................. 10 2.2 Invariants on R Γ (U(p, q)) and relation to R(PU(p, q))..... 13 3 Higgs bundles and flat onnetions 14 3.1 GL(n)-Higgs bundles....................... 14 3.2 U(p, q)-higgs bundles....................... 16 3.3 Deformation theory of Higgs bundles.............. 20 3.4 Bounds on the topologial invariants.............. 21 3.5 Moduli spae for p = q and τ = τ M.............. 24 3.6 Rigidity for extreme values of the Toledo invariant....... 24 4 Morse theory 26 4.1 The Morse funtion........................ 26 4.2 Critial points of the Morse funtion.............. 27 4.3 Loal minima and the adjoint bundle.............. 30 4.4 Stable Higgs bundles....................... 32 4.5 Reduible Higgs bundles..................... 34 4.6 Loal minima and onnetedness................ 38 5 Stable triples 39 5.1 Definitions and basi fats.................... 39 5.2 Minima as triples......................... 43 5.3 Extensions and deformations of triples............. 47 6 Crossing ritial values 52 6.1 Flip Loi.............................. 53 6.2 Codimension Estimates...................... 58 6.3 Holomorphi hains........................ 60 6.4 Estimate of χ(n, d, n, d ) and omparison of moduli spaes. 66
7 Speial values of α 68 7.1 The kernel of φ and the parameter α.............. 68 7.2 The okernel of φ and the parameter α............. 71 8 Moduli spae of triples with n 1 n 2 72 8.1 Moduli spae for α = α M..................... 72 8.2 Moduli spae for large α..................... 73 8.3 Moduli spae for 2g 2 α < α M............... 79 9 Moduli spae of triples with n 1 = n 2 80 9.1 Moduli spae for d 1 = d 2..................... 80 9.2 Bounds on E 1 and E 2 for α > α 0................ 81 9.3 Stabilization of moduli...................... 82 9.4 Moduli for large α and α 2g 2................ 84 10 Existene and onnetedness for U(p, q) and PU(p, q) moduli spaes 88 10.1 Toledo invariant and the number of omponents........ 89 10.2 Generiity of 2g-2 and oprimality onditions.......... 91 10.3 Moduli spaes of U(p, q)-higgs bundles............. 94 10.4 Moduli spaes of U(p, q) and PU(p, q) representations..... 96 Referenes 98
1 Introdution The ore of this paper is a Morse theoreti study of the the moduli spae of U(p, q)-higgs bundles over a Riemann surfae X of genus g 2. Our interest in this spae omes from two soures. The first is its relevane to questions onerning the representation variety for representations of π 1 X in the real Lie group PU(p, q). The seond has to do with the intrinsi geometry revealed by the Morse funtion and the methods we are able to use to arry out our analysis. Our main goals are to fully understand the minimal submanifolds of the Morse funtion and, thereby, to ount the number of onneted omponents in the representation varieties. A Higgs bundle onsists of a holomorphi bundle together with a Higgs field, i.e. a setion of a ertain assoiated vetor bundle. A U(p, q)-higgs bundle is a speial ase of the G-Higgs bundles defined by Hithin in [23], where G is a real form of a omplex redutive Lie group. Suh objets provide a natural generalization of holomorphi vetor bundles, whih orrespond to the ase G = U(n) and zero Higgs field. In partiular, they permit an extension to other groups of the Narasimhan and Seshadri theorem ([28]) on the relation between unitary representations of π 1 X and stable vetor bundles. By embedding U(p, q) in GL(p + q) we an give a onrete desription of a U(p, q)-higgs bundle as a pair (V W, Φ = ( 0 β γ 0 ) ) where V and W are holomorphi vetor bundles of rank p and q respetively, β is a setion in H 0 (Hom(W, V ) K), and γ H 0 (Hom(V, W ) K), so that Φ H 0 (End(V W ) K). Foremost among the key features of suh objets is (by the work of Hithin, [23, 24] Donaldson [12], Corlette [10] and Simpson [31, 32, 33, 34]) the existene of moduli spaes of polystable objets whih an be identified with moduli spaes of solutions to natural gauge theoreti equations. Moreover, sine the gauge theory equations amount to a projetive flatness ondition, these moduli spaes orrespond with a moduli spaes of flat strutures. In the ase of U(p, q)-higgs bundles, the flat strutures orrespond to semisimple representations of π 1 X into the group PU(p, q). The Higgs bundle moduli spaes an thus be used, in a way whih we make preise in Setions 2 and 3, as a tool to study the representation variety R(PU(p, q)) = Hom + (π 1 X, PU(p, q))/pu(p, q), where Hom + (π 1 X, PU(p, q)) denotes the semisimple representations into PU(p, q) and the the quotient is by the adjoint ation. 1
This relation between Higgs bundles and surfae group representations has been suessfully exploited by others, going bak originally to the work of Hithin and Simpson on omplex redutive groups. The use of Higgs bundle methods to study representation variety R(G) for real G was pioneered by Hithin in [24], and further developed in [18, 19] and by Xia and Xia- Markman in [37, 38, 39, 26]. Where we differ from these works is that in none of them is the general ase of PU(p, q) onsidered. What we have in ommon is that in all ases insight into the topology of the spae R(G) omes from a natural Morse funtion on the orresponding moduli spae of Higgs bundles. The natural Morse funtion measures the L 2 -norm of the Higgs field. This turns out to provide a suitably non-degenerate Morse funtion whih is, moreover, a proper map. In some ases (f. [23, 18]) all the ritial submanifolds are suffiiently well understood so as to permit the extration of detailed topologial information suh as the Poinaré polynomial. In our ase our understanding is onfined mostly to the submanifolds orresponding to the loal minima of the Morse funtion. Fortunately, this is suffiient for our purposes, namely to understand the number of omponents of the Higgs moduli spaes, and thus of the representation varieties. The Morse funtion is non-negative but annot always attain its zero lower bound. For GL(n)-Higgs bundles, this lower bound is attained, with the minimizing points in the moduli spae onsisting of semistable vetor bundles with zero Higgs field. However in the ase of U(p, q)-higgs bundles, the speial form of the underlying holomorphi bundle prevents a polystable Higgs bundle from having a vanishing Higgs field. The minimizers of the Morse funtion thus have a more ompliated struture than simply that of a stable bundle. Generalizing the results in [18, 19], we show that any minimizer onsists of a pair of bundles together with a morphism between them. That is, the minimizers orrespond preisely to a speial ase of the holomorphi triples introdued in [6]. The holomorphi triples admit moduli spaes of stable objets in their own right. In order to exploit the relation between these spaes and the minimal submanifolds in the moduli spaes of U(p, q)-higgs bundles, we need a suffiiently good understanding of the triples moduli spaes. A substantial part of this paper is devoted to aquiring just suh an understanding. The way we aquire the needed information is similar in spirit to tehniques used by Thaddeus in [35]. The key idea (desribed fully in setions 5 and 6) is that the moduli spaes of triples ome in disrete families, with the members of the families ordered by intervals in the range of a ontinuously varying real parameter. As the parameter moves to the large extreme of its range, 2
the struture of the orresponding moduli spaes simplify and we an obtain a detailed desription. Moreover, as the parameter dereases, we an trak, albeit somewhat rudely, how the moduli spaes hange. Combining these piees of data, we get just enough information about the moduli spae of relevane to our Higgs bundle problem. We now give a brief summary of the ontents and main results of this paper. In Setions 2 and 3 we give some bakground and desribe the basi objets of our study. In Setions 2 we desribe the natural invariants assoiated with representations of π 1 X into PU(p, q). We also disuss the invariants assoiated with representations of Γ, the universal entral extensions of π 1, into U(p, q). In both ases, these involve a pair of integers (a, b) whih an be interpreted respetively as degrees of rank p and rank q vetor bundles over X. In the ase of the PU(p, q) representations, the pair is well defined only as a lass in a quotient Z Z/(p, q)z. This leads us to define subspaes R[a, b] R(PU(p, q)) and R Γ (a, b) R Γ (U(p, q)). For fixed (a, b), the spae R Γ (a, b) fibers over R[a, b] with onneted fibers. In setion 3 we give preise definitions of the U(p, q)-higgs bundles and their moduli spaes and establish their essential properties. Thinking of a U(p, q)-higgs bundle as a pair (V W, Φ), the parameters (a, b) appear here as the degrees of the bundles V and W. We denote the moduli spae of polystable U(p, q)-higgs bundles with deg(v ) = a and deg W = b by M(a, b), and identify M(a, b) with the omponent R Γ (a, b) of R Γ (U(p, q)). This, together with the fibration over R Γ (U(p, q)) are the ruial links between the Higgs moduli and the surfae group representation varieties. Exept for the last setion, where we translate bak to the language of representation varieties, the rest of the paper is onerned with the spaes M(a, b). Fixing p, q, a and b, we begin the Morse theoreti analysis of M(a, b) in Setion 4. Using the L 2 -norm of the Higgs field Φ = ( ) 0 β γ 0 as the Morse funtion, the basi result we need (f. Proposition 4.2) is that this funtion has a minimum on eah onneted omponent of M(a, b), and if the subspae of loal minima is onneted then so is M(a, b). The next step is to identify the loal minima, the loi of whih we denote by N (a, b). We prove (f. Propositions 4.10 and 4.15) that these orrespond preisely to the polystable Higgs bundles in whih β = 0 or γ = 0. The data defining a Higgs bundle with β = 0 an thus be written as the triple (W K, V, γ). Similarly, the γ = 0 minima orrespond to triples (V K, W, β). This brings us to the theory of suh holomorphi triples. In setions 5-9 we develop the theory we need onerning holomorphi 3
triples and their moduli spaes. While only triples of a speifi speial kind orrespond to the minima on the U(p, q)-higgs moduli, we develop the theory for the general ase in whih a holomorphi triple is speified by the set T = (E 1, E 2, φ), where E 1 and E 2 are holomorphi bundles on X and φ: E 2 E 1 is holomorphi (see [15] and [6]). There is a notion of stability for triples whih depends on a real parameter α and there are moduli spaes of α- polystable triples, whih are shown in [6] (see also [15]) to be projetive varieties. In order for N α to be non-empty, one must have α α m with α m = d 1 /n 1 d 2 /n 2 0. In the ase n 1 n 2 there is also a finite upper bound α M. When the parameter α varies, the nature of the α-stability ondition only hanges for a disrete number of so-alled ritial values of α (see setion 5.1 for the preise statements). We denote by N α = N α (n 1, n 2, d 1, d 2 ) the moduli spae of α-polystable triples with rk(e i ) = n i and deg(e i ) = d i for i = 1, 2. The subspae of α-stable triples inside N α, denoted by N s α, is a quasi-projetive variety. In Theorem 5.21 we show that Theorem. N s α is smooth for all values of α greater than or equal to 2g 2. We show furthermore that the triples whih appear in N (a, b) are α-polystable with α = 2g 2. We must thus understand a moduli spae of α-stable triples, with α on the boundary of the range in whih the moduli spaes are smooth. We do this indiretly, by obtaining a desription of N s α when α is large and then examining how the moduli spae hanges as α dereases. In setion 6 we examine how the moduli spaes differ for values of α on opposite sides of a ritial value. If N α ± denote the moduli spaes for values of α above and below a ritial value α, we denote the loi along whih they differ by S α ± respetively. Our main results are Theorem (Theorem 6.19). Let α (α m, α M ) be a ritial value for triples of type (n 1, n 2, d 1, d 2 ). If α > 2g 2 then the loi S α ± N s are ontained α ± in subvarieties of odimension at least g 1. In partiular, they are ontained in subvarieties of stritly positive odimension if g 2. If α = 2g 2 then the same is true for S α +. Theorem (Corollary 6.20). Let α 1 and α 2 be any two values in (α m, α M ) suh that α m < α 1 < α 2 < α M and 2g 2 α 1. Then The moduli spaes N s α 1 and N s α 2 have the same number of onneted omponents, and 4
The moduli spae N s α 1 is irreduible if and only if N s α 2 is. Where by α M we denote the upper bound for α if n 1 n 2, or if n 1 = n 2. In setions 7-9 we look more losely at how the parameter α affets the nature of α-stable triples. There are three ases to onsider, namely n 1 < n 2, n 1 > n 2 and n 1 = n 2. However, using a duality result, it is enough to onsider n 1 > n 2 and n 1 = n 2. In the first ase, as mentioned above, there is a bounded interval [α m, α M ] outside of whih N α is empty. Within this interval we identify a number of speial values beyond whih the struture of α-stable triples simplify; by Corollary 7.3 the map φ : E 2 E 1 is injetive if α > α 0, by Proposition 7.5 the okernel is torsion free. Finally, for the largest values of α, i.e. for values greater than a bound whih we denote by α L, we show (f. Proposition 8.3) that α-stable triples have the form 0 E 2 φ E 1 F 0, with F loally free, and E 2 and F semistable. This leads to a desription of N α for any α in the range α L < α < α M. Denoting this moduli spae by N L, we get Theorem (Theorem 8.7). Let n 1 > n 2 and d 1 /n 1 > d 2 /n 2. The moduli spae N s L (n 1, n 2, d 1, d 2 ) is smooth, and is birationally equivalent to a P N -fibration over M s (n 1 n 2, d 1 d 2 ) M s (n 2, d 2 ), where M s (n, d) denotes the moduli spae of stable bundles of degree n and rank d, and the fiber dimension is N = n 2 d 1 n 1 d 2 + n 1 (n 1 n 2 )(g 1) 1. In partiular, N s L (n 1, n 2, d 1, d 2 ) is non-empty and irreduible. If GCD(n 1 n 2, d 1 d 2 ) = 1 and GCD(n 2, d 2 ) = 1, the birational equivalene is an isomorphism. Moreover, N L (n 1, n 2, d 1, d 2 ) is irreduible and hene birationally equivalent to N s L (n 1, n 2, d 1, d 2 ). Theorem (Theorem 8.9, Corollary 8.10). Let α be any value in the range 2g 2 α < α M. Then N s α is birationally equivalent to N s L. In partiular it is non-empty and irreduible. Let (n 1, n 2, d 1, d 2 ) be suh that GCD(n 2, n 1 + n 2, d 1 + d 2 ) = 1. If α is generi then N α is birationally equivalent to N L, and in partiular it is irreduible. The ase n 1 = n 2 differs from the n 1 > n 2 ase in two ways. The range for α is unbounded above, and in general there is no way to avoid torsion in the okernel of the map φ. The range for α presents no diffiulties sine 5
(f. Theorem 9.5) beyond a finite bound there are no hanges in the moduli spaes. It thus still makes sense to identify a large α moduli spae, N L. We prove the following. Theorem (Theorem 9.13). The moduli spae N L (n, n, d 1, d 2 ) is non-empty and irreduible. Moreover, it is birationally equivalent to a P N -fibration over M s (n, d 2 ) Div d (X), where the fiber dimension is N = n(d 1 d 2 ) 1. Theorem (Theorem 9.14). If α 2g 2 then the moduli spae N s α(n, n, d 1, d 2 ) is birationally equivalent to N L (n, n, d 1, d 2 ) and hene non-empty and irreduible. Moreover, N α (n, n, d 1, d 2 ) is birationally equivalent to N L (n, n, d 1, d 2 ), and hene irreduible, if also GCD(n, 2n, d 1 + d 2 ) = 1 and α 2g 2 is generi, or d 1 d 2 < α, In setion 10 we apply our results to the moduli spaes M(a, b), and hene to the omponents R Γ (a, b) and R[a, b] of the representation varieties R Γ (U(p, q)) and R(PU(p, q)), respetively. Some of the results depend only on the ombination aq bp τ = τ(a, b) = 2, p + q known as the Toledo invariant. Indeed, (a, b) is onstrained by the bounds 0 τ τ M, where τ M = 2 min{p, q}(g 1). Originally proved by Domi and Toledo in [11], these bounds emerge naturally from our point of view (f. Corollary 3.21 and Remark 5.13). After a disussion (in Setion 10.1) of the relation between (a, b) and τ, and (in setion 10.2) of the signifiane of the oprime ondition GCD(p + q, a + b) = 1, we assemble (in setion 10.3) our results for the Higgs Moduli spaes. Summarizing the results of setion 10.3 into one Theorem, we get Theorem. Let (a, b) be suh that τ(a, b) τ M. Unless further restritions are imposed, let (p, q) be any pair of positive integers. (1) If either of the following sets of onditions apply, then the moduli spae M s (a, b) is a non-empty, smooth manifold of the expeted dimension, with onneted losure M s (a, b): (i) 0 < τ(a, b) < τ M, (ii) τ(a, b) = τ M and p = q 6
(2) If any one of the following sets of onditions apply, then the moduli spae M(a, b) is non-empty and onneted: (i) τ(a, b) = 0, (ii) τ(a, b) = τ M and p q., (iii) (p 1)(2g 2) < τ τ M = p(2g 2) and p = q, (iv) GCD(p + q, a + b) = 1 (3) If τ(a, b) = τ M and p q then any element in M(a, b) is stritly semistable (i.e. M s (a, b) is empty). If p < q, then any suh representation deomposes as a diret sum of a U(p, p)-higgs bundle with maximal Toledo invariant and a polystable vetor bundle of rank q p. Thus, if τ = p(2g 2) then there is an isomorphism M(p, q, a, b) = M(p, p, a, a p(2g 2)) M(q p, b a + p(2g 2)), where the notation M(p, q, a, b) indiates the moduli spae of U(p, q)- Higgs bundles with invariants (a, b), and M(n, d) is the moduli spae of semistable vetor bundles of rank n and degree d. (A similar result holds if p > q, as well as if τ = p(2g 2)). (4) If GCD(p + q, a + b) = 1 then M(a, b) is a smooth manifold of the expeted dimension. Translating this into the language of representations of Γ and the fundamental group, we get the following. Theorem (Theorem 10.18). Let (a, b) be suh that τ(a, b) τ M. Unless further restritions are imposed, let (p, q) be any pair of positive integers. (1) If either of the following sets of onditions apply, then the moduli spae R Γ (a, b) of irreduible semi-simple representations, is a nonempty, smooth manifold of the expeted dimension, with onneted losure R Γ (a, b): (i) 0 < τ(a, b) < τ M, (ii) τ(a, b) = τ M and p = q (2) If any one of the following sets of onditions apply, then the moduli spae R Γ (a, b) of all semi-simple representations is non-empty and onneted: 7
(i) τ(a, b) = 0, (ii) τ(a, b) = τ M and p q., (iii) (p 1)(2g 2) < τ τ M = p(2g 2) and p = q, (iv) GCD(p + q, a + b) = 1 (3) If τ(a, b) = τ M and p q then any representation in R Γ (a, b) is reduible (i.e. R Γ (a, b) is empty). If p < q, then any suh representation deomposes as a diret sum of a semisimple representation of Γ in U(p, p) with maximal Toledo invariant and a semisimple representation in U(q p). Thus, if τ = p(2g 2) then there is an isomorphism R Γ (p, q, a, b) = R Γ (p, p, a, a p(2g 2)) R Γ (q p, b a + p(2g 2)), where the notation R Γ (p, q, a, b) indiates the moduli spae of representations of Γ in U(p, q) with invariants (a, b), and R Γ (n, d) denotes the moduli spae of degree d representations of Γ in U(n). (A similar result holds if p > q, as well as if τ = p(2g 2)). (4) If GCD(p + q, a + b) = 1 then R Γ (a, b) is a smooth manifold of the expeted dimension. Theorem (Theorem 10.19). Let (a, b) be suh that τ(a, b) τ M. Unless further restritions are imposed, let (p, q) be any pair of positive integers. (1) If either of the following sets of onditions apply, then the moduli spae R [a, b] of irreduible semi-simple representations, is non-empty, with onneted losure R [a, b]: (i) 0 < τ(a, b) < τ M, (ii) τ(a, b) = τ M and p = q (2) If any one of the following sets of onditions apply, then the moduli spae R[a, b] of all semi-simple representations is non-empty and onneted: (i) τ(a, b) = 0, (ii) τ(a, b) = τ M and p q., (iii) (p 1)(2g 2) < τ τ M = p(2g 2) and p = q, (iv) GCD(p + q, a + b) = 1 8
(3) If τ(a, b) = τ M and p q then any representation in R[a, b] is reduible (i.e. R [a, b] is empty). If p < q, then any suh representation redues to a semisimple representation of π 1 X in P(U(p, p) U(q p)), suh that the representation in PU(p, p) indued via projetion on the first fator has maximal Toledo invariant. (A similar result holds if p > q, as well as if τ = p(2g 2)). Statement (3) in the previous theorem is a generalization to arbitrary (p, q) of a result of D. Toledo [36] when p = 1 and L. Hernández [22] when p = 2. This rigidity phenomenon for the moduli spae of representations for the largest value of the Toledo invariant turns out to be of signifiane in relation to Hithin s Teihmüller omponents for the real split form of a omplex group [24] (this will be disussed somewhere else [16]). We note, finally, that our methods learly have wider appliability than to the U(p, q)-higgs bundles and representations into PU(p, q). A areful srutiny of the Lie algebra properties used in our proofs suggests a generalization to any real group G for whih G/K is hermitian symmetri, where K G is a maximal ompat subgroup. This will be addressed in a future publiation. The main results proved in this paper were announed in the note [7]. In that note we laim that the onnetedness results hold for the moduli spaes R(a, b) and R[a, b], whether or not the oprimality ondition GCD(p + q, a + b) = 1 is satisfied (and similarly for the orresponding moduli of triples and U(p, q)-higgs bundles). While we expet this to be true, we have not so far been able to prove it. We hope to ome bak to this question in a future publiation. Aknowledgements. We thank the mathematis departments of the University of Illinois at Urbana-Champaign, the Universidad Autónoma de Madrid and the University of Aarhus, the Department of Pure Mathematis of the University of Porto, the Mathematial Sienes Researh Institute of Berkeley and the Mathematial Institute of the University of Oxford for their hospitality during various stages of this researh. We thank Ron Donagi, Bill Goldman, Tomás Gómez, Rafael Hernández, Nigel Hithin, Alastair King, Eyal Markman, Viente Muñoz, Peter Newstead, S. Ramanan, Domingo Toledo, and Eugene Xia, for many insights and patient explanations. 9
2 Representations of surfae groups 2.1 Definitions Let X be a losed oriented surfae of genus g 2 and let G be either U(p, q) or PU(p, q) where p and q are any positive integers. We think of U(p, q) as the subgroup of GL(n) (with n = p + q) whih leaves invariant a hermitian form of signature (p, q). It is a non-ompat real form of GL(n) with enter S 1 and maximal ompat subgroup U(p) U(q). The quotient U(p, q)/u(p) U(q) is a hermitian symmetri spae. The adjoint form PU(p, q) is given by the exats sequene of groups 1 U(1) U(p, q) PU(p, q) 1. By a representation of π 1 X in G we mean a homomorphism ρ: π 1 X G. Fixing PU(p, q) PGL(n), we say a representation of π 1 X in PU(p, q) is semi-simple if it defines a semi-simple PGL(n) representation. The group PU(p, q) ats on the set of representations via onjugation. Restriting to the semi-simple representations, we get the harater variety, R(PU(p, q)) = Hom + (π 1 X, PU(p, q))/pu(p, q). (2.1) This an be desribed as follows: from the standard presentation π 1 X = A 1, B 1,..., A g, B g g [A i, B i ] = 1 i=1 we see that Hom + (π 1 X, PU(p, q)) an be embedded in PU(p, q) 2g via Hom + (π 1 X, PU(p, q)) PU(p, q) 2g ρ (ρ(a 1 ),... ρ(b g )). We give Hom + (π 1 X, PU(p, q)) the subspae topology and R(PU(p, q)) the quotient topology; this is Hausdorff beause we have restrited attention to semi-simple representations. We an similarly define where Γ is the entral extension R Γ (U(p, q)) = Hom + (Γ, U(p, q))/u(p, q), (2.2) 0 Z Γ π 1 X 1 (2.3) defined (as in [3]) by the generators A 1, B 1,..., A g, B g and a entral element J subjet to the relation g i=1 [A i, B i ] = J. Regarding U(p, q) as a subset 10
of GL(n), the representations in Hom + (Γ, U(p, q)) are diret sums of irreduible representations on C n. The first step in the study of the topologial properties of R(G) is to identify the appropriate topologial invariants of a representation ρ: π 1 X G. To do that, one uses the orrespondene between representations of π 1 X in G and flat prinipal G-bundles on X. We start with G = PU(p, q). Let ρ: π 1 X PU(p, q) be a representation. The orresponding flat prinipal PU(p, q)-bundle is P ρ = X ρ PU(p, q), where X is the universal over of X. Sine X has real dimension two, any PU(p, q)-bundle lifts to a U(p, q)-bundle. Moreover, a PU(p, q)-bundle with a flat onnetion an be lifted to a U(p, q)-bundle with a projetively flat onnetion, i.e. with a onnetion with onstant entral urvature. Now, for any smooth (not neessarily flat) U(p, q)-bundle there is a redution of struture group to the maximal ompat subgroup U(p) U(q). Taking the standard representation on C p C q, we get an assoiated vetor bundle of the form V W, where V and W are rank p and q omplex vetor bundles respetively. Suh bundles over a Riemann surfae are topologially lassified by a pair of integers (a, b) = (deg(v ), deg(w )). The lift to a U(p, q)-bundle, and therefore the pair (a, b), is however not uniquely determined. If we twist the assoiated vetor bundle (plus projetively flat onnetion) by a line bundle L with a onnetion with onstant urvature, then after projetivizing we obtain the same flat PU(p, q)-bundle. If the degree of L is l then the invariant assoiated to the twisted bundle is (a + pl, b + ql). In order to obtain a well defined invariant for the representation ρ we must thus take the quotient of (Z Z) by the Z-ation l (a, b) = (a + pl, b + ql), i.e. we must pass to the quotient (Z Z)/(p, q)z in the exat sequene 0 Z Z Z (Z Z)/(p, q)z 0. Sine the PU(p, q)-orbits in Hom(π 1 X, PU(p, q)) under the onjugation ation orrespond to isomorphism lasses of flat PU(p, q)-bundles, the above onstrution defines a map : R(PU(p, q)) (Z Z)/(p, q). (2.4) The map is ontinuous and is thus onstant on onneted omponents of R(PU(p, q)). 11
Remark 2.1. This map an be seen from a different point of view, from whih it seen that the target spae is π 1 PU(p, q). We begin with the observation that the flat bundle P ρ is desribed by loally onstant transition funtions. Thus the isomorphism lass of this bundle is represented by a lass in the (non-abelian) ohomology set H 1 (X, PU(p, q)), where, by abuse of notation, we denote the sheaf of loally onstant maps into PU(p, q) on X by the same symbol PU(p, q). Let PU(p, q) be the universal over of PU(p, q). The short exat sequene of groups π 1 PU(p, q) PU(p, q) PU(p, q) indues a sequene of ohomology sets and, sine π 1 PU(p, q) is Abelian, the oboundary map δ : H 1 (X, PU(p, q)) H 2 (X, π 1 PU(p, q)) an be defined. The obstrution to lifting the flat PU(p, q)-bundle P ρ to a flat PU(p, q)-bundle is exatly the image of the ohomology lass of P ρ under δ. We denote this lass by (ρ) H 2 (X, π 1 PU(p, q)) = π 1 PU(p, q). Next we reall the alulation of π 1 PU(p, q). The maximal ompat subgroup of U(p, q) is U(p) U(q) and the inlusion U(p) U(q) U(p, q) is a homotopy equivalene. The determinant gives an isomorphism of fundamental groups π 1 U(p) = π 1 U(1) = Z. Hene the map U(p, q) U(1) U(1) defined by U(p, q) U(1) U(1) ( ) x y (det(x), det(w)) z w (2.5) gives an isomorphism π 1 U(p, q) = Z Z. Furthermore, the omposition of the standard inlusion U(1) U(p, q) and the map given in (2.5) is the map λ (λ p, λ q ) from U(1) U(1) U(1). The indued map on fundamental groups is n (pn, qn). The short exat sequene U(1) U(p, q) PU(p, q) (2.6) is a fibration, so we see that π 1 U(p, q) π 1 PU(p, q) is surjetive. It follows that we have a ommutative diagram π 1 U(1) π 1 U(p, q) π 1 PU(p, q) = = = Z (p,q ) Z Z (Z Z)/(p, q)z and hene (ρ) defines a lass [a, b] (Z Z)/(p, q)z. This is the same lass as that defined by the map (2.4) though, sine we will not make use of this, we omit the proof. 12
2.2 Invariants on R Γ (U(p, q)) and relation to R(PU(p, q)) Putting together (2.6) and (2.3) we get the ommutative diagram U(1) U(p, q) PU(p, q) ρ ρ. Z Γ π 1 X From this we get a surjetion π : R Γ (U(p, q)) R(PU(p, q)). We an understand the fibers of this map as follows. By the same argument as in [3] 1, R Γ (U(p, q)) an be identified as the moduli spae of U(p, q)-bundles on X with projetively flat strutures. Taking the redution to the maximal ompat U(p) U(q), we thus assoiate to eah lass ρ R Γ (U(p, q)) a vetor bundle of the form V W, where V and W are rank p and q respetively, and thus a pair of integers (a, b) = (deg(v ), deg(w )). The map : ρ (a, b) fits in a ommutative diagram R Γ (U(p, q)) Z Z We an now define the subspaes R Γ (a, b) : = 1 (a, b) π R(PU(p, q)) (Z Z)/(p, q)z = { ρ R Γ (U(p, q)) ( ρ) = (a, b) Z Z}, R[a, b] : = 1 [a, b] Clearly we have surjetive maps Moreover, the pre-image = {ρ R(PU(p, q)) (ρ) = [a, b] Z Z/(p, q)z}. R Γ (a, b) R[a, b]. (2.7) π 1 (R[a, b]) = R Γ (a, b) (2.8) where the union is over all (a, b) in the lass [a, b] Z Z/(p, q)z. As mentioned above, tensoring by line bundles with onstant urvature onnetions of degree l gives an isomorphism (a,b) R Γ (a, b) = R Γ (a + pl, b + ql). 1 While [3] gives the argument for U(n) and PU(n), there are no essential hanges to be made in order to adapt for the ase of U(p, q) and PU(p, q). 13.
Notie that if the invariant (ρ) of a representation ρ R(PU(p, q)) an be represented by the pair (a, a), then the assoiated U(p, q)-bundle has degree zero and the projetively flat onnetion is atually flat. Thus ρ defines to a representation of π 1 X in U(p, q). Under the orrespondene between R(PU(p, q)) and R Γ (U(p, q)), ρ orresponds to a Γ representation in whih the entral element J ats trivially. Furthermore the subspaes R(a) = {ρ R(U(p, q)) (ρ) = (a, a) with a Z} an be identified with the subspaes R Γ (a, a) R Γ (U(p, q)). Finally, we observe that Ja(X), the moduli spae of flat degree zero line bundles, ats by tensor produt of bundles on R Γ (a, b). Sine Ja(X) is isomorphi to the torus U(1) 2g, we get the following relation between onneted omponents. Proposition 2.2. The map R Γ (a, b) R[a, b] given in (2.7) defines a U(1) 2g -fibration whih, if the total spae and base are smooth manifolds, is a smooth prinipal bundle. Thus the subspae R[a, b] R(PU(p, q)) is onneted if R Γ (a, b) is onneted. We will study R Γ (a, b) by hoosing a omplex struture on X and identifying this spae with a ertain moduli spae of Higgs bundles. This is arried out in the next setion. In the rest of the paper, the subspaes of irreduible representations are denoted by R. 3 Higgs bundles and flat onnetions 3.1 GL(n)-Higgs bundles Give X the struture of a Riemann surfae. We reall (from [12, 10, 23, 31, 33, 34]) the following basi fats about GL(n)-Higgs bundles. Definition 3.1. 1. A GL(n)-Higgs bundle on X is a pair (E, Φ), where E is a rank n holomorphi vetor bundle over X and Φ H 0 (End(E) K) is a holomorphi endomorphism of E twisted by the anonial bundle K of X. 2. The GL(n)-Higgs bundle (E, Φ) is stable if the slope stability ondition µ(e ) < µ(e) (3.1) holds for all proper Φ-invariant subbundles E of E. Here the slope is defined by µ(e) = deg(e)/ rk(e) and Φ-invariane means that Φ(E ) E K. 14
3. Semistability is defined by replaing the above strit inequality with a weak inequality. A Higgs bundle is alled polystable if is the diret sum of stable Higgs bundles with the same slope. 4. Given a hermitian metri H on E, let A denote the unique onnetion ompatible with the holomorphi struture and unitary with respet to H. Hithin s equations on (E, Φ) are F A + [Φ, Φ ] = 1µId E ω, A Φ = 0, (3.2) where ω is the Kähler form on X, Id E is the identity on E, µ = µ(e) and A is the antiholomorphi part of the ovariant derivative d A. Proposition 3.2. [12, 23, 31, 33, 34] 1. Let (E, Φ) be a GL(n)-Higgs bundle. Then (E, Φ) is polystable if and only if it admits a hermitian metri suh that Hithin s equation (3.2) is satisfied 2. There is a moduli spae of rank n degree d polystable Higgs bundles whih is a quasi-projetive variety of omplex dimension 2(d+n 2 (g 1)). 3. If we define a Higgs onnetion (as in [33]) by D = d A + θ (3.3) where θ = Φ + Φ, then Hithin s equations are equivalent to the onditions F D = 1µId E ω, d A θ = 0, (3.4) d Aθ = 0. 4. In partiular, sine X is a Riemann surfae, if A satisfies (3.2) then D is a projetively flat onnetion. If deg(e) = 0 then D is atually flat. It follows that in this ase the pair (E, D) defines a representation of π 1 X in GL(n). If deg(e) 0, then the pair (E, D) defines a representation of π 1 X in PGL(n), or equivalently, a representation of Γ in GL(n). By the theorem of Corlette ([10]), every semisimple representation of Γ (and therefore all semisimple representation of π 1 X) arise in this way. 15
5. There is thus a bijetive orrespondene between the moduli spae of polystable Higgs bundles of rank n and the moduli spae of (onjugay lasses of) semisimple representations of Γ in GL(n). If the degree of the Higgs bundle is zero, then the first moduli spae orresponds atually to the representation variety for representations of π 1 X in GL(n). 3.2 U(p, q)-higgs bundles If we fix integers p, q suh that n = p + q, then we an isolate a speial lass of GL(n)-Higgs bundles by the requirements that E = V W Φ = ( ) (3.5) 0 β γ 0 where V and W are holomorphi vetor bundles on X with rk(v ) = p, rk(w ) = q, deg(v ) = a, deg(w ) = b, β H 0 (Hom(W, V ) K), and γ H 0 (Hom(V, W ) K). We an desribe suh Higgs bundles more intrinsially as follows. Let P GL(p) and P GL(q) be the prinipal frame bundles for V and W respetively. Let P = P GL(p) P GL(q) be the fiber produt, and let Ad P = P Ad gl(n) be the adjoint bundle, where GL(p) GL(q) GL(n) ats by the Ad-ation on the the Lie algebra of GL(n). Let (gl(p) gl(q)) gl(n) be the orthogonal omplement with respet to the usual inner produt. This defines a subbundle P p,q := P Ad (gl(p) gl(q)) Ad P. (3.6) We an then make the following definition. Definition 3.3. A U(p, q)-higgs bundle 2 on X is a pair (P, Φ) where P is a holomorphi prinipal GL(p) GL(q) bundle, and Φ is a holomorphi setion of the vetor bundle P p,q K (where P p,q is the bundle defined in (3.6). Remark 3.4. We an always write P = P GL(p) P GL(q). If we let V and W be the standard vetor bundles assoiated to P GL(p) and P GL(q) respetively, then any Φ H 0 (P p,q K) an be written as in (3.5). We will usually adopt the vetor bundle desription of U(p, q)-higgs bundles. Remark 3.5. Definition 3.3 is ompatible with the definitions in [24] and in [18]. There they define a G-Higgs bundle for any real form of a omplex redutive Lie group. The bundle in their definition is a prinipal H C -bundle, 2 The reason for the name is explained by the following remarks and by Lemma 3.6 16
where H G is a maximal ompat subgroup and H C is its omplexifiation. Thus in the ase that G = U(p, q), we get that H C = GL(p) GL(q). The Higgs field is preisely a setion of the bundles whih appears in Definition 3.3. From a different perspetive, Definition 3.3 defines an example of a prinipal pair in the sense of [4] and [27]. Stritly speaking, sine the anonial bundle K plays the role of a fixed twisting bundle, what we get is a prinipal pair in the sense of [8]. The defining data for the pair are then (i) the prinipal GL(p) GL(q) GL(1)-bundle P GL(p) P GL(q) P K, where P K is the frame bundle for K and (ii) the assoiated vetor bundle P p,q K. Lemma 3.6. Let (V W, Φ) be a U(p, q)-higgs bundle with a Hermitian metri H = H V H W, i.e. suh that V W is a unitary deomposition. Let A be a unitary onnetion with respet to H, and let D = d A + θ be the orresponding Higgs onnetion, where θ = Φ + Φ. Then D is a U(p, q)- onnetion, i.e. in any unitary loal frame the onnetion 1-form takes its values in the Lie algebra of U(p, q). Proof. Fix a loal unitary frame (with respet to H = H V H W ). Then D = d + A + θ, where A takes its values in u(p) u(q) u(p, q) u(n), while θ takes its values in (u(p) u(q)) u(p, q). Definition 3.7. Let (E, Φ) be a U(p, q)-higgs bundle with E = V W and Φ = ( ) 0 β γ 0. We say (E, Φ) is a stable U(p, q)-higgs bundle if the slope stability ondition (3.1), i.e. µ(e ) < µ(e), is satisfied for all Φ-invariant subbundles of the form E = V W, i.e. for all subbundles V V and W W suh that β : W V K (3.7) γ : V W K. (3.8) Semistability for U(p, q)-higgs bundles is defined by replaing the above strit inequality with a weak inequality, and polystability means a diret sum of stable U(p, q)-higgs bundles all with the same slope. In partiular a polystable U(p, q)-higgs bundle is the diret sum of (lower rank) U(p, q )-Higgs bundles. We shall say that a polystable U(p, q)-higgs bundle whih is not stable is reduible. Two U(p, q)-higgs bundles (V W, Φ) and (V W, Φ ) are isomorphi if there are isomorphisms g V : V V and g W : W W whih intertwine Φ and Φ, i.e. suh that (g V g W ) I K Φ = Φ (g V g W ) where I K is the identity on K. Remark 3.8. The stability ondition for a U(p, q)-higgs bundle is a priori weaker than the stability ondition given in Definition 3.1 for GL(n)-Higgs 17
bundles. Namely, the slope ondition has to be satisfied only for all proper non-zero Φ-invariant subbundles whih respet the deomposition E = V W, that is, subbundles of the form E = V W with V V and W W. However, it is shown in [19, Setion 2.3] that the weaker ondition is in fat equivalent to the ordinary stability of (E, Φ). Proposition 3.9. Let (E, Φ) be a U(p, q)-higgs bundle with E = V W and Φ = ( ) 0 β γ 0. Then (E, Φ) is U(p, q)-polystable if and only if it admits a ompatible hermitian metri H = H V H W suh that Hithin s equation (3.2) is satisfied Proof. Though not expliitly proved there, this is a speial ase of the orrespondene invoked in [24] for G-Higgs bundles where G is a real form of a redutive Lie group. By Remark 3.5 it an also be seen as a speial ase of the Hithin Kobayashi orrespondene for prinipal pairs (f. [4] and [27] and [8]). We note finally that in one diretion the result follows immediately from Remark 3.8 : if (V W, Φ) supports a ompatible metri suh that (3.2) is satisfied, then it is polystable as a GL(n)-Higgs bundle, and hene it is U(p, q)-polystable. Remark 3.10. This orrespondene allows us, via the next theorem, to use U(p, q)-higgs bundles to study representations of the surfae groups π 1 X and Γ into U(p, q) and PU(p, q) Definition 3.11. Fix integers a and b. Let M(a, b) denote the moduli spae of isomorphism lasses of polystable U(p, q)-higgs bundles with deg(v ) = a and deg W = b. Proposition 3.12. The moduli spae M(a, b) an be identified with the moduli spae of U(p, q)-higgs bundles whih admit solutions to Hithin s equations. It is a quasi-projetive variety whih is smooth away from the points representing reduible U(p, q)-higgs bundles. There is an homeomorphism between R Γ (a, b) and M(a, b). This restrits to give a homeomorphism between the subspae R Γ (a, b) of irreduible elements in R Γ(a, b) and the subspae M s (a, b) of stable Higgs bundles in M(a, b). Proof. The first statement is a diret onsequene of Proposition 3.9. The onstrution of M(a, b) is essentially the same as in setion 9 of [33]. There the moduli spae of G-Higgs bundles is onstruted for any redutive group G. We take G = GL(p) GL(q). The differene between a U(p, q)-higgs bundle and a GL(p) GL(q)-Higgs bundle is entirely in the nature of the Higgs fields. Taking the standard embedding of GL(p) GL(q) in GL(p + q) 18
we see that in a GL(p) GL(q)-Higgs bundle the Higgs field Φ takes its values in the subspae (gl(p) gl(q)) gl(p+q), while in a U(p, q)-higgs bundle the Higgs field Φ takes its values in the omplementary subspae (gl(p) gl(q)). Sine both subspaes are invariant under the adjoint ation of GL(p) GL(q), the same method of onstrution works for the moduli spaes of both types of Higgs bundle. Suppose that (E = V W, Φ) represents a point in M(a, b), i.e. suppose that it is a U(p, q)-polystable Higgs bundle, and suppose that with metri H = H V H W Hithin s equation (3.2) is satisfied. Rewriting the equations in terms of the Higgs onnetion D = d A + θ, where A is the metri onnetion determined by H and θ = Φ + Φ, we see that D is projetively flat. By Lemma 3.6 it is a projetively flat U(p, q)-onnetion, and thus defines a point in R Γ (a, b). Conversely by Corlette s theorem [10], every representation in Hom + (π 1 X, PU(p, q)), or equivalently every representation in Hom + (Γ, U(p, q)), arises in this way. Remark 3.13. If GCD(p + q, a + b) = 1 then for purely numerial reasons there are no stritly semistable U(p, q)-higgs bundles in M(a, b). In this ase M s (a, b) = M(a, b). Proposition 3.14. With n = p + q and d = a + b, let M(d) denote the moduli spae of polystable GL(n)-Higgs bundles of degree d. If p q or a b then M(a, b) embeds as a losed subvariety in M(d). If p = q and a = b, then there is a finite morphism from M(a, a) to M(d). Proof. Let [V W, Φ] p,q denote the point in M(a, b) represented by the U(p, q)-higgs bundle (V W, Φ). Then (E = V W, Φ) is a polystable GL(n)-Higgs bundle and the map M(a, b) M(d) is defined by [V W, Φ] p,q [E, Φ] n, where [, ] n denotes the isomorphism lass in M(d). The only question is whether this map is injetive. Suppose that (E = V W, Φ) and (E = V W, Φ ) are isomorphi as GL(n)-Higgs bundles. Let the isomorphism be given by omplex gauge transformation g : E E. Sine we an regard the smooth splitting of E as fixed, we see that unless V = W and W = V, the gauge transformation must already be of the form ( g V 0 0 g W ), i.e. [V W, Φ] p,q = [V W, Φ ] p,q. But in order to have V = W and W = V we require p = q and a = b. In that ase, if V and W are non-isomorphi, then [V W, ( 0 β γ 0 )] n = [W V, ( 0 γ β 0 )] n but the Higgs bundles are not isomorphi as U(p, q)-higgs bundles. 19
3.3 Deformation theory of Higgs bundles A main tool in the study of the topology of moduli spaes of Higgs bundles is given by the Morse theoreti tehniques introdued by Hithin [24]. In order to use these methods, we first need to reall the deformation theory of Higgs bundles. We refer to the paper by Biswas and Ramanan [5] for details. Let (E, Φ) be a stable U(p, q)-higgs bundle in M s (a, b). Some times it will be onvenient to use the following notation: U = End(E) U + = End(V ) End(W ), U = Hom(W, V ) Hom(V, W ). (3.9) Clearly, U = U + U. Note that Φ H 0 (U K) and that ad(φ) interhanges U + and U. As it is shown in [5], the Zariski tangent spae to M s (a, b) at the point defined by (E, Φ) an be identified with the first hyperohomology of the omplex of sheaves C : U + One has the long exat sequene ad(φ) U K. (3.10) 0 H 0 (C ) H 0 (U + ) H 0 (U K) H 1 (C ) H 1 (U + ) H 1 (U K) H 2 (C ) 0, (3.11) from whih one obtains the following. Proposition 3.15. The moduli spae of stable U(p, q)-higgs bundles is a smooth omplex variety of dimension 1 + (p + q) 2 (g 1). Proof. Let (E, Φ) be a stable U(p, q)-higgs bundle. Then (E, Φ) is simple, that is, its only automorphisms are the non-zero salars. Thus, if (E, Φ) is stable, ker ( ad(φ): H 0 (U) H 0 (U K) ) = C. Sine U = U + U and ad(φ) interhanges these two summands it follows that ker ( ad(φ): H 0 (U + ) H 0 (U K) ) = C (3.12) ker ( ad(φ): H 0 (U ) H 0 (U + K) ) = 0 (3.13) Hene, if (E, Φ) is stable, (3.12) shows that H 0 (C ) = C. To show that the moduli spae is smooth at a neighbourhood of (E, Φ) we need to show that 20
H 2 (C ) = 0. But we have natural ad-invariant isomorphisms U + = (U + ) and U = (U ). Thus ad(φ): H 1 (U + ) H 1 (U K) is Serre dual to ad(φ): H 0 (U ) H 0 (U + K). Thus (3.13) shows that H 2 (C ) = 0. The dimension of the moduli spae is hene dim H 1 (C ) = 1 χ(u + ) + χ(u K) = 1 + (p 2 + q 2 )(g 1) + 2pq(g 1) = 1 + (p + q) 2 (g 1). Remark 3.16. Notie that the dimension of the moduli spae of stable U(p, q)- Higgs bundles is half the dimension of the moduli spae of stable GL(p+q, C)- Higgs bundles. Remark 3.17. As pointed out previously, if GCD(p + q, a + b) = 1, then there are no stritly semistable elements in M(a, b) and hene M(a, b) is smooth. 3.4 Bounds on the topologial invariants In this setion we shall show how the Higgs bundle point of view provides an easy proof of a result of Domi and Toledo [11] whih allows us to bound the topologial invariants deg(v ) and deg(w ) for whih U(p, q)-higgs bundles may exist. The lemma is a slight variation on the results of [19, Setion 3] (f. also Lemma 3.6 of Markman and Xia [26]). Lemma 3.18. Let (E, Φ) be a semistable U(p, q)-higgs bundle. Then p(µ(v ) µ(e)) rk(γ)(g 1), (3.14) q(µ(w ) µ(e)) rk(β)(g 1). (3.15) If equality ours in (3.14) then either (E, Φ) is stritly semistable or p = q and γ is an isomorphism. If equality ours in (3.15) then either (E, Φ) is stritly semistable or p = q and β is an isomorphism. Proof. If γ = 0 then V is Φ-invariant and so, by stability, µ(v ) µ(e) where equality an only our if (E, Φ) is stritly semistable. This proves (3.14) in the ase γ = 0 and we may, therefore, assume that γ 0. Let N V be 21
the vetor bundle assoiated to ker(γ) and let I W be the vetor bundle assoiated to im(γ) K 1. Then rk(n) + rk(i) = p (3.16) and, sine γ indues a non-zero setion of det((v/n) I K), deg(n) + deg(i) + rk(i)(2g 2) deg(v ). (3.17) The bundles N and V I are Φ-invariant subbundles of E and hene we obtain by semistability that µ(n) µ(e) and µ(v I) µ(e) or, equivalently, that deg(n) µ(e) rk(n), (3.18) deg(i) µ(e)(p + rk(i)) deg(v ). (3.19) Adding (3.18) and (3.19) and using (3.16) we obtain deg(n) + deg(i) 2µ(E)p deg(v ). (3.20) Finally, ombining (3.17) and (3.20) we get deg(v ) rk(i)(2g 2) 2µ(E)p deg(v ), whih is equivalent to (3.14) sine rk(γ) = rk(i). Note that equality an only our if we have equality in (3.18) and (3.19) and thus either (E, Φ) is stritly semistable or neither of the subbundles N and V I is proper and non-zero. In the latter ase, learly N = 0 and I = W and therefore p = q; furthermore we must also have equality in (3.17) implying that γ is an isomorphism. An analogous argument applied to β proves (3.15). Remark 3.19. The proof also shows that if we have equality in, say, (3.14) then γ : V/N I K is an isomorphism. In partiular, if p < q and µ(v ) µ(e) = g 1 then γ : V = I K. We an reformulate Lemma 3.18 to obtain the following orollary. Corollary 3.20. Let (E, Φ) be a semistable U(p, q)-higgs bundle. Then q(µ(e) µ(w )) rk(γ)(g 1), (3.21) p(µ(e) µ(v )) rk(β)(g 1). (3.22) Proof. To see that (3.21) is equivalent to (3.14) one simply notes that µ(w ) µ(e) = p q ( µ(e) µ(v ) ). Similarly (3.22) is equivalent to (3.15). 22
An important orollary of the lemma above is the following Milnor Wood type inequality for U(p, q)-higgs bundles (due to Domi and Toledo [11], improving on a bound obtained by Dupont [13] in the ase G = SU(p, q)). This result gives bounds on the possible values of the topologial invariants deg(v ) and deg(w ). Corollary 3.21. Let (E, Φ) be a semistable U(p, q)-higgs bundle. Then Proof. Sine µ(e) = µ(w ) and therefore (3.14) gives pq µ(v ) µ(w ) min{p, q}(g 1). (3.23) p + q p p+q µ(v )+ q p+q µ(w ) we have µ(v ) µ(e) = q pq (µ(v ) µ(w )) rk(γ)(g 1). p + q A similar argument using (3.15) shows that pq (µ(w ) µ(v )) rk(β)(g 1). p + q (µ(v ) p+q But, obviously, rk(β) and rk(γ) are both less than or equal to min{p, q} and the result follows. Definition 3.22. Let a = deg(v ) and b = deg(w ). The number qa pb τ = τ(a, b) = 2 p + q (3.24) is known as the Toledo invariant of the representation orresponding to (E, Φ). Remark 3.23. Sine τ = 2 pq (µ(v ) µ(w )), p + q the inequality (3.23) an thus be written We denote τ M = min{p, q}(2g 2). τ min{p, q}(2g 2). 23
3.5 Moduli spae for p = q and τ = τ M Suppose p = q. Then τ = τ(p, p, a, b) = a b. In this setion we give an alternative (more expliit) desription of the moduli spae M(a, b) in the ase in whih the Toledo invariant is maximal, i.e. τ = a b = τ M = p(2g 2). Before doing this, we need to review briefly the notion of L-twisted Higgs pairs. Let L be a line bundle. An L-twisted Higgs pair (V, θ) onsists of a holomorphi vetor bundle V and an L-twisted homomorphism θ : V V L. The notions of stability, semistability and polystability are defined as for Higgs bundles. The moduli spae of semistable L-twisted Higgs pairs has been onstruted by Nitsure using GIT [29]. Let M L (n, d) be the moduli spae of polystable L-twisted Higgs pairs of rank n and degree d. Proposition 3.24. Let p = q and a b = p(2g 2). Then M(a, b) = M K 2(p, a) = M K 2(p, b). Proof. Let (E = V W, Φ) M(a, b). Suppose for definiteness that b a = p(2g 2). From (3.15) it follows that γ : V W K is an isomorphism. We an then ompose β : W V K with γ Id K : V K W K 2 to obtain a K 2 -twisted Higgs pair θ W : W W K 2. Similarly, twisting β : W V K with K and omposing with γ, we obtain a K 2 -twisted Higgs pair θ V : V V K 2. Conversely, given an isomorphism γ : V W K, we an reover β from θ V as well as from θ W. It is lear that the (poly)stability of (E, Φ) is equivalent to the (poly)stability of (V, θ V ) and to the (poly)stability of (W, θ W ), proving the laim. Remark 3.25. The moduli spae M K 2(p, a) ontains an open (irreduible) subset onsisting of a rank N vetor bundle over M s (p, a). This is beause the stability of V implies the stability of any K 2 -twisted Higgs pair (V, θ V ), and H 1 (End V K 2 ) = 0. The rank N is determined by the Riemann Roh Theorem. 3.6 Rigidity for extreme values of the Toledo invariant From the bounds in Setion 3.4 it follows that if p < q (a similar result holds for p > q) and (a, b) suh that τ = τ M there are no stable U(p, q)-higgs bundles and every element in M(a, b) is in fat reduible. In partiular the moduli spae has smaller dimension than expeted exhibiting a ertain kind of rigidity. This phenomenon (for large Toledo invariant) has been studied from the point of view of representations of the fundamental group by D. 24