aa r X i v : . [ m a t h . AG ] N ov STABLE HIGGS BUNDLES ON COMPACT GAUDUCHONMANIFOLDS
INDRANIL BISWAS
Abstract.
Let M be a compact complex manifold equipped with a Gauduchon metric.If T M is holomorphically trivial, and (
V , θ ) is a stable SL( r, C )–Higgs bundle on M ,then we show that θ = 0. We show that the correspondence between Higgs bundles andrepresentations of the fundamental group for a compact K¨ahler manifold does not extendto compact Gauduchon manifolds. This is done by applying the above result to Γ \ G ,where Γ is a discrete torsionfree cocompact subgroup of a complex semisimple group G . R´esum´e.
Les fibr´es de Higgs stables sur les vari´et´es de Gauduchon.
Soit M une vari´et´e complexe compacte muni d’une m´etrique de Gauduchon. Si T M est holomor-phiquement trivial, et (
V, θ ) est un fibr´e SL( r, C )–Higgs stable, alors on d´emontre que θ = 0. On d´emontre que la correspondance entre les fibr´es de Higgs et les repr´esentationsdu groupe fondamental pour une vari´et´e k¨ahlerienne compacte ne s’´etend pas aux vari´et´esde Gauduchon. Ceci est accompli en appliquant le r´esultat ci-dessus `a Γ \ G , o`u Γ est unsous-groupe discret, sans torsion et co-compact d’un groupe semi-simple complexe G . Introduction
Let (
M , g ) be a compact connected K¨ahler manifold. A theorem due to Uhlenbeckand Yau says that the isomorphism classes of stable vector bundles on M are in bijectivecorrespondence with the solutions of the Hermitian–Yang–Mills equation on M [12]. Thistheorem was later extended to compact complex manifold equipped with a Gauduchonmetric by Li–Yau and Buchdahl [8], [2] (the result of [2] is for complex surfaces). Gaudu-chon metrics are a generalization of K¨ahler metrics; their definition is recalled in Section2. Hitchin, Simpson, Donaldson and Corlette established a bijective correspondence be-tween the isomorphism classes of stable SL( r, C )–Higgs bundles on M , with vanishingrational Chern classes, and the equivalence classes of irreducible homomorphisms from π ( M ) to SL( r, C ) [6], [10], [4], [3]. It is natural to ask whether this correspondence ex-tends to compact complex Gauduchon manifolds. We show that the correspondence doesnot extend in general by constructing explicit examples of compact complex Gauduchonmanifolds for which this correspondence fails.Let ( M , g ) be a compact connected complex Gauduchon manifold. We prove the fol-lowing theorem:
Theorem 1.1. If T M is holomorphically trivial, and ( V , θ ) is a stable SL( r, C ) –Higgsbundle on M , then θ = 0 . Mathematics Subject Classification.
Key words and phrases.
Gauduchon metric, stable Higgs bundle, representation.
The above mentioned correspondence valid for K¨ahler manifolds implies that if (
M , g )is a compact connected K¨ahler manifold such that θ = 0 for any stable SL( r, C )–Higgsbundle ( V , θ ) on M , then any irreducible representation of π ( M ) into SL( r, C ) is unita-rizable (see Remark 2.3).Let G be a connected complex semisimple group defined over C . LetΓ ⊂ G be a torsionfree discrete subgroup such that the quotient Γ \ G is compact. This compactcomplex manifold Γ \ G is not K¨ahler, but it has explicit Gauduchon metrics. Also, T (Γ \ G )is trivial, so Theorem 1.1 applies to it. It turns out that the restriction of each nontrivialirreducible representation of G is a nonunitarizable irreducible representation of Γ.2. Stable Higgs bundles
Let M be a compact connected complex manifold of complex dimension d . Let g be a C ∞ Hermitian structure on the holomorphic tangent bundle
T M . Let ω g be the positive(1 , M given by g . We recall that g is called a Gauduchon metric if ∂∂ω d − g = 0 . A theorem due to P. Gauduchon says that given any C ∞ Hermitian structure g on T M , there is a positive smooth function f on M such that f g is a Gauduchon metric;furthermore, if n ≥
2, then f is unique up to a positive constant. (See [5, p. 502].)Fix a Gauduchon metric g on M . As before, the corresponding (1 , M willbe denoted by ω g .Let F be a coherent analytic sheaf on F . Consider the determinant line bundle det F on M ; see [7, Ch. V, §
6] for the construction of det F . Fix a Hermitian structure h F ondet F . Define the degree of F to bedegree( F ) := Z M c (det F , h F ) ∧ ω d − g ∈ R , where c (det F , h F ) is the Chern form of the Hermitian connection on det F . It shouldbe clarified that degree( F ) is independent of the choice of h F , but it depends on g ; see [2,p. 626], [8, p. 563].A holomorphic vector bundle V on M is called stable if for every coherent analyticsubsheaf F ⊂ V with 0 < rank( F ) < rank( V ), the inequalitydegree( F )rank( F ) < degree( V )rank( V )holds.Let Ω M be the holomorphic cotangent bundle of M . A Higgs field on a holomorphicvector bundle V on M is a section θ ∈ H ( M, End ( V ) ⊗ Ω M ) IGGS BUNDLES ON GAUDUCHON MANIFOLDS 3 such that θ V θ = 0. A Higgs vector bundle is a pair ( V , θ ), where V is a holomorphicvector bundle, and θ is a Higgs field on V ; see [6], [10].A Higgs vector bundle ( V , θ ) is called stable if for every coherent analytic subsheaf F ⊂ V satisfying the two conditions that θ ( F ) ⊂ F ⊗ Ω M and 0 < rank( F ) < rank( V ),the inequality degree( F )rank( F ) < degree( V )rank( V )holds.Let trace : End ( V ) ⊗ Ω M −→ Ω M be the homomorphism defined by trace ⊗ Id Ω M . Sofor a Higgs vector bundle ( V , θ ),trace( θ ) ∈ H ( M, Ω M ) . A SL( r, C )– Higgs bundle is a Higgs vector bundle (
V , θ ) of rank r such that det V = O M (the trivial line bundle), and trace( θ ) = 0. Theorem 2.1.
Assume that the tangent bundle
T M is holomorphically trivial. Let ( V , θ ) be a stable SL( r, C ) –Higgs bundle on M . Then θ = 0 .Proof. Fix a holomorphic trivialization of Ω M by choosing d linearly independent sections β i ∈ H ( M, Ω M )1 ≤ i ≤ d . Then θ = d X i =1 θ i ⊗ β i , where θ i ∈ H ( M, End ( V )). Since θ V θ = 0,(1) θ i ◦ θ j = θ j ◦ θ i for all i , j ∈ [1 , d ].Assume that θ = 0. Choose i such that θ i = 0.For any point x ∈ M , let λ ( x ) , · · · , λ n x ( x ) be the eigenvalues of θ i ( x ) ∈ End C ( V x );let m xj be the multiplicity of the eigenvalue λ j ( x ). Since all holomorphic functions on M are constants, the characteristic polynomial of θ i ( x ) is independent of x . Hencethe collection { ( λ ( x ) , m x ) , · · · , ( λ n x ( x ) , m xn x ) } is independent of x . Let V ⊂ V be thegeneralized eigenbundle for the eigenvalue λ ( x ) of θ i . So for each point x ∈ M , the fiberof V over x is generalized eigenspace of θ i ( x ) for the eigenvalue λ ( x ); it is a holomorphicsubbundle. Let V c ⊂ V be the holomorphic subbundle given by the direct sum of thegeneralized eigenbundles for all the eigenvalues of θ i different from λ ( x ). Therefore, wehave a decomposition V = V ⊕ V c . From (1) it follows immediately that θ j ( V ) ⊂ V and θ j ( V c ) ⊂ V c for all j . Hence θ ( V ) ⊂ V ⊗ Ω and θ ( V c ) ⊂ V c ⊗ Ω . Therefore, if both V and V c are nonzero, thenthe Higgs bundle ( V , θ ) decomposes. But a stable Higgs bundle is indecomposable. Since
I. BISWAS ( V , θ ) is stable, we conclude that θ i has exactly one eigenvalue. On the other hand,trace( θ ) = 0. Hence 0 is the only eigenvalue of θ i . So, θ i is nilpotent.Consider the short exact sequence of coherent analytic sheaves on M (2) 0 −→ kernel( θ i ) −→ V −→ image( θ i ) −→ . From (1) it follows that θ j (kernel( θ i )) ⊂ kernel( θ i ) and θ j (image( θ i )) ⊂ image( θ i )for all j . Hence(3) θ (kernel( θ i )) ⊂ kernel( θ i ) ⊗ Ω M and θ (image( θ i )) ⊂ image( θ i ) ⊗ Ω M . Since θ i is nonzero and nilpotent,0 < rank(kernel( θ i )) , rank(image( θ i )) < r . In view of (3), the stability condition for (
V , θ ) says that(4) degree(kernel( θ i ))rank(kernel( θ i )) , degree(image( θ i ))rank(image( θ i )) < degree( V )rank( V ) . On the other hand, from (2),degree(kernel( θ i )) + degree(image( θ i )) = degree( V )and rank(kernel( θ i ))+rank(image( θ i )) = rank( V ). But these contradict (4). Therefore, θ = 0. (cid:3) Theorem 2.1 has the following corollary:
Corollary 2.2. If T M is holomorphically trivial, and ( V , θ ) is a stable SL( r, C ) –Higgsbundle on M , then the vector bundle V is stable. Let (
M , g ) be a compact connected K¨ahler manifold. All the Chern classes will bewith rational coefficients. There is a bijective correspondence between the isomorphismclasses of stable SL( r, C )–Higgs bundles ( V , θ ) on M , with c i ( V ) = 0 for all i ≥
1, andthe equivalence classes of irreducible homomorphisms from π ( M ) to SL( r, C ) (see [11] forthe details of this correspondence). Also, there is a bijective correspondence between theisomorphism classes of stable vector bundles V on M of rank r and trivial determinant,with c i ( V ) = 0 for all i ≥
1, and the equivalence classes of irreducible homomorphismsfrom π ( M ) to SU( r ) (see [11]). The first correspondence is an extension of the secondcorrespondence: The inclusion of SU( r ) in SL( r, C ) gives a map of homomorphisms, anda stable vector bundles V on M of rank r and trivial determinant produces a stableSL( r, C )–Higgs bundle by assigning the zero Higgs field. Remark 2.3.
Assume that the K¨ahler manifold (
M , g ) has the following property: If(
V , θ ) is a stable SL( r, C )–Higgs bundle, then θ = 0. Take any irreducible homomorphism ρ : π ( M ) −→ SL( r, C ) . Let ( V ρ , θ ρ ) be the stable Higgs bundle corresponding to ρ . We have θ ρ = 0 by theassumption on M . Hence ρ is conjugate to a unitary representation, meaning there is anelement A ∈ SL( r, C ) such that A − ( ρ ( z )) A ∈ SU( r ) for all z ∈ π ( M ). IGGS BUNDLES ON GAUDUCHON MANIFOLDS 5 An example
Let G be a connected semisimple affine algebraic group defined over C ; we assume that G = e . Let d be the (complex) dimension of G . LetΓ ⊂ G be a torsionfree discrete subgroup such that Γ \ G is compact. Since Γ \ G is compact, thesubgroup Γ is Zariski dense in G [1]. (See [9] for such manifolds.)We note that there are explicit Gauduchon metrics on the complex manifold Γ \ G .Indeed, take any Hermitian metric h on Γ \ G given by some left translation invariantHermitian metric e h on G . Let ω h and ω e h be the corresponding (1 , \ G and G respectively. Since e h is left translation invariant, the top degree form ∂∂ω d − e h is alsoleft translation invariant. Hence ∂∂ω d − e h = c · µ G , where c ∈ R , and µ G is the Haarmeasure form on G . The form ∂∂ω d − h is closed because ∂∂ω d − h = d∂ω d − h . Hence usingStokes’ theorem, 0 = Z Γ \ G ∂∂ω d − h = Z Γ \ G c µ G = c Vol µ G (Γ \ G ) . Therefore, c = 0. Hence h is a Gauduchon metric.The holomorphic tangent bundle T (Γ \ G ) is trivial (a trivialization is given by any lefttranslation invariant trivialization of T G ).It can be shown that Γ \ G does not admit any K¨ahler metric. Indeed, any compactconnected K¨ahler manifold with trivial tangent bundle is isomorphic to a complex torus,implying that its fundamental group is abelian. But the fundamental group Γ of Γ \ G isnot abelian. (Since Γ is Zariski dense in G , if Γ is abelian, then G is abelian.)Take any nontrivial irreducible representation ρ ′ : G −→ SL( V ) . Let(5) ρ := ρ ′ | Γ be the restriction of ρ ′ to the subgroup Γ.We have π (Γ \ G ) = Γ. Since Γ is Zariski dense in G , the restriction ρ in (5) remainsirreducible. Since ρ (Γ) ⊂ SL( V )is an infinite, closed and discrete subgroup, it cannot be conjugated, by some element ofSL( V ), to a subgroup of a maximal compact subgroup of SL( V ) (every closed infinitesubgroup of a compact group has a limit point, hence it is not discrete). Compare thiswith Remark 2.3. References [1] A. Borel, Density properties for certain subgroups of semi-simple groups without compact compo-nents,
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