Kahler metric on the space of convex real projective structures on surface
aa r X i v : . [ m a t h . G T ] J un K ¨AHLER METRIC ON THE SPACE OF CONVEX REAL PROJECTIVESTRUCTURES ON SURFACE
INKANG KIM AND GENKAI ZHANGA
BSTRACT . We prove that the space of convex real projective structures on a surface ofgenus g ≥ admits a mapping class group invariant K¨ahler metric where Teichm¨ullerspace with Weil-Petersson metric is a totally geodesic complex submanifold.
1. I
NTRODUCTION
Recently, the character variety χ ( π ( M ) , G ) of representations of π ( M ) in a real al-gebraic group G , has drawn many attentions from different branches of mathematics. The G -character variety χ ( π ( M ) , G ) is the geometric quotient of Hom ( π ( M ) , G ) by innerautomorphisms of G . Often, some components of the character variety correspond tosome geometric structures on M . Hitchin [16] introduced Hitchin component in the char-acter variety of a closed surface group in P SL ( n + 1 , R ) generalizing Teichm ¨uller spacein P SL (2 , R ) . More precisely, he showed that for the adjoint group G of the split realform of a complex simple Lie group G c , the quotient by the conjugation action of G ofthe set of homomorphisms, from the fundamental group Γ of a closed surface S of genus g ≥ to G , which acts completely reducibly on the Lie algebra of G , has a connectedcomponent homeomorphic to Euclidean space of dimension (2 g − G . His methodis the use of Higgs bundle theory developed by himself, K. Corlette, S. Donaldson, C.Simpson and many others [15, 31]. A homomorphism from Γ to G c defines a flat prin-cipal G c -bundle. Given a complex structure on S , denoted by Σ , a theorem of Corletteand Donaldson associates a natural G u -connection A , where G u is the maximal compactsubgroup of G c , and a Higgs field Ψ ∈ H (Σ , ad P ⊗ K ) which satisfy the equation F A + [Ψ , Ψ ∗ ] = 0 . Here K is the canonical line bundle over Σ , P is a principal G u -bundle and ad P is the Lie algebra bundle associated to the adjoint representation of G u .Solutions to these equations provide a holomorphic parametrization of the equivalenceclasses of homomorphisms from Γ to G c . For appropriately chosen Ψ the solutions arestable under the complex conjugation in G c and reduces to G -connection correspondingto elements in the Hitchin component. Keywordsandphrases. Hitchin component, real projective structure, K¨ahler metric. Research partially supported by STINT-NRF grant (2011-0031291). Research by G. Zhang is supportedpartially by the Swedish Science Council (VR). I. Kim gratefully acknowledges the partial support of grant(NRF-2014R1A2A2A01005574) and a warm support of Chalmers University of Technology during hisstay.
For the real linear group SL ( n + 1 , R ) , Hitchin showed that the Hitchin componentis homeomorphic to L nj =1 H (Σ , K j +1 ) . We need to mention that this homeomorphismdepends on a priori fixed complex structure on S , and hence it is not mapping classgroup equivariant. After Hitchin’s work, many people pursued to clarify this componentin many different ways. Notably Labourie [21] introduced a notion of Anosov repre-sentations and proved that Hitchin representations are exactly Anosov representations in SL ( n + 1 , R ) . In [22], he also suggested a mapping class group equivariant parametriza-tion using Hitchin map and an adaptation of an energy functional over Teichm ¨uller space.We will review his interpretation of Hitchin map in Section 3.It has been conjectured for a long time that the Hitchin component admits a mappingclass group invariant K¨ahler metric. There have been many evidences for this, see [13, 24,6]. In the last section, we prove the existence of a mapping class group invariant K¨ahlermetric on the Hitchin component for n = 2 . Theorem 1.1.
The Hitchin component of the character variety χ ( π ( M ) , SL (3 , R )) canbe equipped with a mapping class group invariant K¨ahler metric where M is a closedsurface of genus ≥ . Furthermore Teichm¨uller space equipped with the Weil-Peterssonmetric is a totally geodesic complex submanifold. This K¨ahler metric is constructed using certain L -metric. Intuitively, we glue Weil-Petersson metric on the base and L -metric along vertical fibers using Griffith negativity.Indeed, we need the dual of a holomorphic vector bundle over Teichm ¨uller space whosefibres are cubic holomorphic forms. The geometric properties of this metric such asvarious curvatures, geodesics will be explored in a near future. We hope that this newnatural K¨ahler metric will help us to better understand the moduli space of real projectivestructures and the Teichm ¨uller space as a byproduct.This particular Hitchin component has been intensively studied by many people. Choi-Goldman showed that the corresponding geometric structure is the convex real projectivestructure [7] and the bundle structure is verified by Labourie [20] and Loftin [27] inde-pendently using Monge- `Ampere equations relying on the seminal work of Cheng-Yau [8].The symplectic structure on the Hitchin component has been studied by Goldman [13].Recently Li [24] constructed a mapping class group invariant metric using explicit con-structions over sl (3 , R ) -bundles and Cheng-Yau metric over the cone. It would be inter-esting to compare this construction with ours. Bridgeman-Canary-Labourie-Sambarino[6] constructed a pressure metric on Hitchin component of SL ( n +1 , R ) for every n usingdifferent method. Acknowledgement
We would like to thank Bo Berndtsson for a few helpful discus-sions on complex vector bundles, in particular for K¨ahler property based on Griffith’spositivity. We are grateful for the anonymous referee for valuable suggestions and for thecareful reading of an earlier version of this paper.
ITCHIN COMPONENT 3
2. C
ONVEX PROJECTIVE STRUCTURES ON A MANIFOLD M A flat projective structure on an n -dimensional manifold M is a ( RP n , P SL ( n +1 , R )) -structure, i.e., there exists a maximal atlas on M whose transition maps are restrictionsto open sets in RP n of elements in P SL ( n + 1 , R ) . Then there exist a natural holonomymap ρ : π ( M ) → P SL ( n + 1 , R ) and a developing map from the universal cover ˜ M , f : ˜ M → RP n such that ∀ x ∈ ˜ M , ∀ γ ∈ π ( M ) , f ( γx ) = ρ ( γ ) f ( x ) . We will consider projective structures deformed from hyperbolic structures, and all ho-lonomy representations will lift to SL ( n + 1 , R ) . An RP n -structure is convex if thedeveloping map is a homeomorphism onto a convex domain in RP n . It is properly convex if the domain is included in a compact convex set of an affine chart, strictly convex if theconvex set is strictly convex.When M = S is a closed Riemann surface of genus at least 2, a huge amount of litera-ture for the set of marked strictly convex real projective structures on S exist concerningits parametrization [14], its identification with Hitchin component [7], degeneration of theprojective structures [18], entropy of geodesic flow [12], the marked length rigidity [17],its Zariski tangent space at Fuchsian locus of the character variety in SL ( n, R ) [19] andmany more. For a recent generalization to finite volume convex real projective structures,see [3, 10].In this paper, we utilize the holomorphic vector bundle structure of the space of themarked strictly convex real projective structures on a closed surface of genus at least 2,[20, 27], where Monge- `Ampere equation type argument is used. The method is initiatedby Cheng and Yau [8, 9].3. H ITCHIN MAP AND BUNDLE STRUCTURE
In this section we collect known results to introduce the Hitchin map. See [20, 22]for details. If ρ is a reductive representation from π ( S ) to a semisimple Lie group G ,Corlette proved [11] the following claim: There exists a unique, up to G , ρ -equivariantharmonic map f ρ,J : ˜ S → G/K where
G/K is the symmetric space of G and J is acomplex structure on S . If f is a ρ -equivariant map from the universal cover ˜ S to G/K ,then one can define the energy E ( J, f ) of f with respect to a complex structure J on S .Hence given ρ , one can define an energy functional e ρ ( J ) on Teichm ¨uller space by theinfimum of energy of ρ -equivariant functions with respect to a complex structure J . Thisharmonic map minimizes the energy, i.e. E ( J, f ρ,J ) = e ρ ( J ) . The minimum area of ρ is defined to be inf J e ρ ( J ) . Then it is known [28, 29] that a harmonic map realizing theminimum area of ρ is conformal.For a map f from S to a Riemannian manifold ( M, g ) , T f can be viewed as a 1-form on S with values in f ∗ T M . Let T C f ( u ) = T f ( u ) − iT f ( J u ) be a complexified tangent map. INKANG KIM AND GENKAI ZHANG
Then f is harmonic if and only if T C f is holomorphic. Furthermore g C ( T C f, T C f ) = 0 ifand only if f is minimal.Every G -invariant symmetric multilinear form P on g gives rise to a parallel polyno-mial function P , with the same notation, on G/K . Hence for any complex structure J on S , and for every symmetric G -invariant multilinear form P of degree k on g , any reductiverepresentation ρ gives rise to an element in Q ( k, J ) by P ( T C f, · · · , T C f ) where f is a ρ -equivariant harmonic map. Here Q ( k, J ) = H (( S, J ) , K k ) is the space of holomorphic k -differentials. Denote this map by F P,J ( ρ ) = P ( T C f, · · · , T C f ) .For G = SL ( n, R ) , we can use the symmetric polynomial P k of degree k . Then F P isa metric on SL ( n, R ) /SO ( n ) . Set Ψ J = k = n M k =2 F P k ,J . Hitchin proved that the map Ψ J is a homeomorphism from the Hitchin componentto Q (2 , J ) ⊕ · · · ⊕ Q ( n, J ) . Set ǫ ( n ) to be the bundle over Teichm ¨uller space with fibres Q ( k, J ) , k ≥ . Labourie introduced the Hitchin map from ǫ ( n ) to the Hitchin component H ( J, ω ) = Ψ − J ( ω ) for ω ∈ L nk =3 Q (3 , J ) ⊕ · · · ⊕ Q ( n, J ) . He showed that Theorem 3.1.
The Hitchin map is surjective.
For n = 3 , this Hitchin map is injective also. Let T be the Teichm ¨uller space ofcomplex structures Σ t on the surface S with the holomorphic tangent space given by H (0 , (Σ t , K − ) at each t ∈ T . Following [20] we consider the space V = { ( v, t ); v ∈ V t := H ( K t ) , t ∈ T } . Labourie [20] showed that
Theorem 3.2.
There exists a mapping class group equivariant diffeomorphism betweenthe moduli space of convex structures on S and the moduli space of pairs ( J, Q ) where J is a complex structure on S and Q is a cubic holomorphic differential on S with respectto J , i.e. diffeomorphic to V . Hence the moduli space of convex projective structures can be treated as the holomor-phic vector bundle V over T .The vector bundle V (sometimes called Hodge bundle) can be treated as in [4]. Con-sider the tautological bundle [1] ˆ T = { ( t, w ); t ∈ T , w ∈ Σ t } over the Techm ¨uller space T equipped with the canonical complex structure; see [1, § ˆ T is then a K¨ahler manifold with the K¨ahler metric being locally the productmetric. ITCHIN COMPONENT 5
The Hodge vector bundle V is then a complex vector bundle and in particular it is acomplex manifold, see e.g. [4]. We let V ∗ be the dual bundle of V . This can be realizedas V ∗ = { ( v, t ); v ∈ V t := H , ( K − t ) , t ∈ T } via the natural mapping class group invariant paring ( f, g ) = R Σ t g ( f ) , f ∈ H , ( K − t ) , g ∈ H ( K t ) = H ( K t ) .4. V ARIOUS NOTIONS OF CURVATURE POSITIVITY
We shall need some results on the positivity for the curvature of the Hodge bundleabove. Recall that generally a complex manifold M equipped with a Hermitian metric issaid to have a nonpositive bisectional curvature if R ( X, Y, ¯ X, ¯ Y ) ≤ for all X, Y ∈ T M ⊗ C where R is the curvature tensor extended complex linearlyto complexified bundle. When we deal with the holomorphic vector bundles, there aresimilar notions of positivity (negativity). Let E be a holomorphic vector bundle over aK¨ahler manifold M and h a Hermitian metric on E . Let ∇ be a Chern connection whichis compatible with the metric h and complex structure on E . If we write ∇ = D + ¯ ∂ , thenits curvature F is equal to D ¯ ∂ − ¯ ∂ D when acting on local holomorphic sections. Moreconcretely, for any section s and (complexified) vector fields X, Y , F ( X, Y )( s ) = ∇ X ∇ Y s − ∇ Y ∇ X s − ∇ [ X,Y ] s.F is of type (1 , , real and satisfies h ( F ( s ) , s ) + h ( s , F ( s )) = 0 for any sections s and s .If z i are local holomorphic coordinates on M and e α is a local orthogonal frame on E ,then the curvature F can be written by √− F = X c ¯ αβj ¯ k dz j d ¯ z k ⊗ e ∗ α ⊗ e β , where c ¯ αβj ¯ k = c ¯ βαkj . This curvature gives rise to a Hermitian (sesqui-linear) form Θ on T M ⊗ E , given locally by Θ = X c ¯ αβj ¯ k ( dz j ⊗ e ∗ α ) ⊗ ( dz k ⊗ e ∗ β ) . In tensorial notation, let e α be a local holomorphic frame of E and e α the dual frame,then the curvature tensor R ∈ Γ( M, ∧ T ∗ M ⊗ E ∗ ⊗ E ) of ∇ has the form R = √− π X R γi ¯ jα dz i d ¯ z j ⊗ e α ⊗ e γ where R γi ¯ jα = h γ ¯ β R i ¯ jα ¯ β and R i ¯ jα ¯ β = − ∂ h α ¯ β ∂z i ∂ ¯ z j + X h γ ¯ δ ∂h α ¯ δ ∂z i ∂h γ ¯ β ∂ ¯ z j . INKANG KIM AND GENKAI ZHANG
Then the Hermitian vector bundle ( E, h ) is said to be(1) Griffith positive if for any nonzero vectors u = P u i ∂∂z i and v = P v α e α , X R i ¯ jα ¯ β u i ¯ u j v α ¯ v β > , i.e., Θ( u ⊗ v, u ⊗ v ) = h ( F ( u, ¯ u )( v ) , v ) > for any section v = 0 and non-zero holomorphic tangent vector field u . In otherwords, Θ is positive definite on nonzero simple tensors of the form u ⊗ v .(2) Nakano positive if for any nonzero vector u = P u iα ∂∂z i ⊗ e α , X R i ¯ jα ¯ β u iα ¯ u jβ > , i.e., the associated sesqui-linear form Θ is a positive definite Hermitian form.(3) dual Nakano positive if for any nonzero vector u = P u iα ∂∂z i ⊗ e α , X R i ¯ jα ¯ β u iβ ¯ u jα > . It is obvious from the definition that Nakano positivity implies Griffith positivity. Also ( E, h ) is dual Nakano positive if and only if ( E ∗ , h ∗ ) is Nakano negative. If E is Griffithpositive then its dual E ∗ is Griffith negative. We also remark that Griffith positivityimplies that the bundle E is ample.In particular if E is a line bundle, then the above notions of positivity all agree. Thecurvature in this case is computed by ¯ ∂∂ log k w k for any local holomorphic frame (i.e.non zero-section) w . We will use these notions of positivity to prove the existence ofK¨ahler metric on the Hitchin component of χ ( π ( S ) , SL (3 , R )) .5. K ¨ AHLER PROPERTY
The following result is a corollary of a general theorem of Berndtsson [4] applied to thevector bundle V ; when we replace the fiber of the bundle V by the spaces H ( K t ) viewedas the dual space of the holomorphic tangent space of Teichm ¨uller space, it is a classi-cal result of Ahlfors [2] that the Teichm ¨uller space has negative holomorphic sectionalcurvature equipped with the Weil-Petersson metric. Theorem 5.1.
The bundle V is Griffith positive.Proof. The tangent line bundle K − t on each Riemann surface Σ t is equipped with aunique K¨ahler-Einstein metric of negative curvature − , in other words, the canonicalline bundle K t is K¨ahler-Einstein of positive curvature . Let L on ˆ T be the pull-back ofthe line bundle K t to ˆ T under the projection ˆ T → T . It follows from [32, Theorem 5.5,Lemma 5.8] that the bundle L is positive; see also [30, Main Theorem, Theorem 1] forgeneralization. Thus L is positive since L is a line bundle. In [4], it is proved that theHermitian vector bundle over T , H (Σ t , L ⊗ K ˆ T / T ) t ITCHIN COMPONENT 7 the fiber being the spaces of global sections, endowed with L -metric, is Nakano positive,where L is any positive line bundle over the K¨ahler manifold ˆ T . In particular taking L = L we have H (Σ t , L ⊗ K ˆ T / T ) = H (Σ t , K t ⊗ K ˆ T / T ) = H (Σ t , K t ) we see that the bundle with fiber H ( K t ) over T , i.e., the bundle V is Nakano positive,hence Griffith positive. (cid:3) Remark 5.2.
The Nakano positivity of the bundle H (Σ t , K t ) can also be directly provedby using the result of Berndtsson in [5]. It is proved there whenever the metric on L ispositive fiberwise then the Nakano positivity still holds for H (Σ t , L ⊗ K ˆ T / T ) . In ourcase the fiber metric is dual to the hyperbolic metric on the Riemann surfaces and hasconstant positive curvature. See also [25, 26] for related works. Corollary 5.3.
The bundle V ∗ is Griffith negative. The first statement below can be proved for general bundles with the Griffith negativity,and here we are only interested in the special case of V ∗ . Theorem 5.4.
The bundle V ∗ is a K¨ahler manifold. In particular the Hitchin componentof the character variety χ ( π ( S ) , SL (3 , R )) has a mapping class group invariant K¨ahlermetric.Proof. Let w = 0 be a fixed point in V ∗ with z = π ( w ) , where π : V ∗ → T is thedefining projection. We choose a local trivializing holomorphic frame { e α = e α ( z ) } in a coordinate neighborhood U of z , and write, with some abuse of notation, z =( z , · · · , z n ) as the coordinate of U . The local holomorphic coordinates near w will be w = P α x α e α ( z ) → ( z i , x α ) , and the holomorphic tangent vectors are T = ( u, v ) with u = P u i ∂∂z i and v = P v α ∂∂x α .We let ψ be a local K¨ahler potential for T near z , T being equipped with the Weil-Petersson metric. Thus π ∗ ψ ( w ) is defined in a neighborhood of w . We let φ ( w ) = k w k + ψ ( π ( w )) = k w k + π ∗ ψ ( w ) to be defined in a neighborhood of w .Let ∇ = D + ¯ ∂ be the Chern connection acting on sections of V ∗ and D e α = P θ βα e β where θ βα is type (1 , . We fix T = ( u, v ) = 0 at w and perform the differentiation ¯ ∂ T ∂ T φ with ∂ T = ∂ u + ∂ v . The vectors w = P α x α e α ( z ) will be viewed as holomorphicsections for fixed x α . The curvature R ( u, ¯ u ) of Chern connection ∇ is R ( u, ¯ u ) w = ∇ u ∇ ¯ u w − ∇ ¯ u ∇ u w − ∇ [ u, ¯ u ] w = ∇ u ¯ ∂ u w − ¯ ∂ u D u w = − ¯ ∂ u D u w using ¯ ∂ u w = 0 and [ u, ¯ u ] = 0 . We have ∂ u k w k = ( D u w, w ) + ( w, ¯ ∂ u w ) = ( D u w, w ) , ¯ ∂ u ∂ u k w k = ¯ ∂ u ( D u w, w ) = ( ¯ ∂ u D u w, w )+( D u w, D u w ) = − ( R ( u, ¯ u ) w, w )+( D u w, D u w ) . INKANG KIM AND GENKAI ZHANG
Since D u w = X D u x α e α + X x α D u e α = X x α θ βα ( u ) e β ( z ) ,θ βα ( u ) e β ( z ) depends smoothly only on z , and hence ¯ ∂ v D u w = 0 . Also ∂ v w = v , hence ¯ ∂ v ∂ u k w k = ( D u w, v ) , ¯ ∂ u ∂ v k w k = ( v, D u w ) and ¯ ∂ v ∂ v k w k = ( v, v ) . Thus ¯ ∂ T ∂ T k w k = − ( R ( u, ¯ u ) w, w ) + ( D u w, D u w ) + ( D u w, v ) + ( v, D u w ) + ( v, v )= − ( R ( u, ¯ u ) w, w ) + ( D u w + v, D u w + v ) ≥ . Here we use Griffith negativity of V ∗ to have ( R ( u, ¯ u ) w, w ) < . We have also ¯ ∂ T ∂ T π ∗ ψ = ¯ ∂ u ∂ u ψ, which is positive definite in u . Thus ¯ ∂ T ∂ T φ = − ( R ( u, ¯ u ) w, w ) + ( D u w + v, D u w + v ) + ¯ ∂ u ∂ u ψ = I + II + III, a sum of two semi-positive Hermitian forms. We prove that it is positive at w = w .Suppose the quadratic form vanishes. Then I = II = III = 0 . If w = 0 , then I = 0 implies that u = 0 by Griffith negativity, and the second term is II = ( v, v ) , whichimplies further that v = 0 . Suppose w = 0 , i.e., in the zero section. Then III = 0 implies that u = 0 , which in turn implies II = ( v, v ) and v = 0 . In either cases, T = (0 , , a contradiction to the choice of T = ( u, v ) = 0 . (cid:3) It follows from the definition that the map w → − w is an isometry and its fixed pointset is the space of zero sections, namely the Teichm ¨uller space identified as a submanifold.We have thus Corollary 5.5.
Let Teichm¨uller space T be equipped with the mapping class group in-variant Weil-Petersson metric and the Hitchin component χ H ( π ( S ) , SL (3 , R )) of thecharacter variety χ ( π ( S ) , SL (3 , R )) be equipped with the mapping class group in-variant K¨ahler metric as in Theorem 5.4. Then T is a totally geodesic submanifold of χ H ( π ( S ) , SL (3 , R )) . Recenlty Labourie [23] generalized this theorem to the Hitchin components associatedto all real split simple Lie groups of rank 2.
Remark 5.6.
If we consider the bundle W over Teichm ¨uller space whose fiber over t equal to P j ≥ H (Σ t , K jt ) , then it is Griffith positive and the same method applies toshow that the total space W ∗ has a mapping class group invariant K¨ahler metric. Forexample, the space of the marked complex projective structures on a closed surface ofgenus at least 2 is a holomorphic vector bundle over Teichm ¨uller space with fibers being ITCHIN COMPONENT 9 the space of holomorphic quadratic differentials. Hence this space has a natural mappingclass group invariant K¨ahler metric. R
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