The Action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures
aa r X i v : . [ m a t h . S G ] F e b The Action homomorphism, quasimorphisms andmoment maps on the space of compatible almostcomplex structures
Egor Shelukhin
School of Mathematical SciencesTel Aviv University69978 Tel Aviv, Israel [email protected]
Abstract
We extend the definition of Weinstein’s Action homomorphism to Hamiltonian actions withequivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds,and show that under conditions including a uniform bound on the symplectic areas of geodesictriangles the resulting homomorphism extends to a quasimorphism on the universal cover of thegroup. We apply these principles to finite dimensional Hermitian Lie groups like the linear symplec-tic group, reinterpreting the Guichardet-Wigner quasimorphisms, and to the infinite dimensionalgroups of Hamiltonian diffeomorphisms of closed symplectic manifolds, that act on the space ofcompatible almost complex structures with an equivariant moment map given by the theory ofDonaldson and Fujiki. We show that the quasimorphism on the universal cover of the Hamiltoniangroup obtained in the second case is symplectically conjugation-invariant and compute its restric-tions to the fundamental group via a homomorphism introduced by Lalonde-McDuff-Polterovich,answering a question of Polterovich; to the subgroup Hamiltonian biholomorphisms via the Futakiinvariant; and to subgroups of diffeomorphisms supported in an embedded ball via the Barge-Ghysaverage Maslov quasimorphism, the Calabi homomorphism and the average Hermitian scalar cur-vature. We show that when the first Chern class vanishes this quasimorphism is proportional toa quasimorphism of Entov and when the symplectic manifold is monotone, it is proportional toa quasimorphism due to Py. As an application we show that a Sobolev distance on the universalcover of the Hamiltonian group is unbounded, similarly to the results of Eliashberg-Ratiu.
Contents L -distance on g Ham ( M, ω ) . . . . . . . . . . . . . . . . . . . . . . . 141.9 Finite dimensional examples: Guichardet-Wigner quasimorphisms . . . . . . . . . . . . . 14 A and I c on π ( Ham ( M, ω )) . . . . . . . . . . . . 202.5 The finite-dimensional case G = Sp (2 n, R ) and the Maslov quasimorphism . . . . . . . . 212.6 The local type of the quasimorphism on g Ham ( M, ω ) . . . . . . . . . . . . . . . . . . . . 232.7 The restriction to the Py quasimorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.8 The restriction to the Entov quasimorphism . . . . . . . . . . . . . . . . . . . . . . . . . 302.9 Calibrating the L norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 In [6] Barge and Ghys have introduced a quasimorphism on the fundamental groups Γ of surfacesof genus g ≥ H . Indeed, choosing a Γ-invariant one-form α on H whose differential is bounded in the way | dα | ≤ C α | σ H | for a constant C α with respect to the hyperbolicKahler form σ H on H , the quasimorhism is given by integrating α over the geodesic l ( x, γ · x ) between afixed base-point x and its image γ · x under the action of an element γ ∈ Γ. Using these quasimorphismsBarge and Ghys have obtained results on the second bounded cohomology H b (Γ) of such groups Γ.Further results on the second bounded cohomology of discrete groups following from their actionsupon certain spaces with ”negative enough” curvature - e.g. Gromov-hyperbolic groups - were studiedextensively in [37, 42, 53, 54, 67] to name a few works in such a direction. The second boundedcohomology of finite dimensional Lie groups was also studied extensively. For example, in the works[52, 31] and others, the action of simple Hermitian symmetric Lie groups G upon their symmetricspace X = G/K of non-compact type was utilized to construct bounded 2-cocycles on G . The basicconstruction of such cocycles similarly uses the integration of the natural Kahler form σ X on X onsimplices with geodesic boundaries.We shall first formulate a general setting in terms of the action of a group G on a space X for constructions related to integration on geodesic simplices to yield bounded 2-cocycles. Then weformulate a general principle, again in terms of such actions, for the construction of primitives to suchcocycles in the (unbounded) group cohomology, to wit - quasimorphisms - functions that satisfy thehomomorphism property up to a uniformly bounded error. For one, our construction gives a symplecticformula for the quasimorphisms on the universal covers e G of simple Hermitian symmetric Lie groupswhose differentials equal the Guichardet-Wigner cocycles (cf. [52, 31, 22, 82, 14]). A key notion inour construction is the use of equivariant moment maps for the Hamiltonian action of a group G ona space X with a symplectic form Ω. Another key notion is that of the Action homomorphism ofA.Weinstein [92] that generalizes to general Hamiltonian actions with equivariant moment maps. Asour construction is rather formal, or ”soft” in the terminology of Gromov [51] in that it does not requirethe solution of partial differential equations or the convergence of certain series, it readily applies tothe infinite dimensional case.Indeed there have been many constructions of equivariant moment maps for actions of infinitedimensional Lie groups on infinite dimensional symplectic spaces ( X , Ω). Starting with the work ofAtiyah and Bott [4, 3] - for the action of gauge groups of principal bundles over Riemann surfaces onthe corresponding spaces of connections, with numerous later developments including an extension tohigher dimensions - a general framework for the Hitchin-Kobayashi correspondence [25, 88, 26], theworks of Donaldson [29, 28, 27] and Fujiki [41] for actions of diffeomorphism groups upon spaces ofmappings (submanifolds or sections of bundles), and more recent advances e.g. [44, 38] this has beenan active and fruitful area of research for over three decades, with many applications - for example to2ahler geometry. Of these the Donaldson-Fujiki [27, 41] framework of the scalar curvature as a momentmap for the action of the Hamiltonian group on the space of compatible almost complex structures fitsthe setting of our construction. We shall, therefore, apply this framework to build new quasimorphismson the Hamiltonian group, or its universal cover, of an arbitrary symplectic manifold of finite volume(and of an arbitrary closed symplectic manifold in particular). Similarly to the finite-dimensional case,our quasimorphism provides a group-cohomological primitive for the restriction to the Hamiltoniangroup of a certain 2-cocycle that was constructed using the natural notion of geodesic simplices inspaces of almost complex structures by Reznikov [78, 77, 79] in his studies of the cohomology of thegroup of symplectomorphisms.The intriguing topic of the study of quasimorphisms on groups of (Hamiltonian) symplectomor-phisms has a long history. A very early work of Eugenio Calabi [17] constructs a homomorphism on thegroup of compactly supported symplectomorphisms of the symplectic ball of arbitrary dimension 2 n .An early example of a quasimorphism on a symplectomorphism group that is not a homomorphismwas constructed by Ruelle [80] on the group of compactly supported volume preserving diffeomor-phisms of the two-dimensional disk, as a certain average asymptotic rotation number. This result wasgeneralized using the Maslov quasimorphism on the universal cover f Sp (2 n, R ) of the linear symplec-tic group by Barge and Ghys [7] to the group of compactly supported symplectomorphisms of thesymplectic ball of arbitrary dimension 2 n . A quasimorphism on the universal cover ^ Symp ( M, ω ) ofclosed symplectic manifolds (
M, ω ) with c ( T M, ω ) = 0 was rather recently constructed by Entov [33],generalizing the previous quasimorphism in the sense that it equals the Barge-Ghys average Maslovquasimorphism when restricted to each subgroup of diffeomorphisms supported in an embedded ball- we shall say that it has the Maslov local type. In a recent work of Py [75, 76] a quasimorphism on ] Ham ( M, ω ) for closed symplectic manifolds (
M, ω ) with c ( T M, ω ) = κ [ ω ] for κ = 0 was constructedas a rotation number using the notion of a prequantization of an integral symplectic manifold. Thelocal type of the Py quasimorphism is Calabi-Maslov - it equals a certain linear combination of theCalabi homomorphism and the Barge-Ghys average Maslov quasimorphism when evaluated on diffeo-morphisms supported in a given embedded ball. A compelling discovery of quasimorphisms of Calabilocal type was made by Entov and Polterovich in [34] - one distinctive feature of which is that theembedded balls should be small enough - using ”hard” methods of Hamiltonian Floer homology andthe algebraic properties of quantum homology. These methods were since generalized and extendedto a large class of manifolds [69, 35, 70, 91, 89], a very recent result due to Usher [90] showing e.g.the existence of Calabi quasimorphisms on ] Ham of every one-point blowup of a closed symplecticmanifold. The sequent question of constructing a ”soft” quasimorphism of Calabi local type on theHamiltonian group of a closed symplectic manifold was recently solved for the two-torus and for sur-faces of genus g ≥ X of non-compact type by discrete groups of isometries [76].Another quasimorphism on ] Ham ( M, ω ) for (
M, ω ) the complex projective space ( C P n , ω F S ) with thenatural Fubini-Study Kahler form can be derived from the work of Givental [49] that uses methodsof generating functions, which also has the Calabi property by the work of Ben Simon [8] and caneasily be shown to descend to
Ham ( M, ω ) itself by results from [81]. In fact necessary and sufficientconditions for the above quasimorphisms on a group e G to descend to G are given by the vanishing ofcertain homomorphisms π ( G ) → R . This happens automatically for sufaces where the fundamentalgroup of G = Ham ( M, ω ) is finite, which is also known to be the case for certain four-dimensionalsymplectic manifolds - e.g. ( C P , ω F S ) , ( C P × C P , ω F S ⊕ ω F S ) [50] (cf. [64]). Remarkably, for allmonotone examples - (
M, ω ) such that c ( T M, ω ) = κ [ ω ] for κ = 0 - the homomorphism is the sameone [36] - the Action-Maslov homomorphism of Polterovich [71] (cf. [81]).3he quasimorphism we construct has Calabi-Maslov local type - it restricts to the difference ofsuitable multiples of the Calabi homomorphism [17, 63] and of the Barge Ghys average Maslov quasi-morphism on the subgroup of Hamiltonian diffeomorphisms supported in a small ball. Its restrictionto the fundamental group of G is equal by construction to the generalized Action homomorphism, in-volving in this case the Hermitian scalar curvature, and is also computed via a homomorphism earlierintroduced in [60] using a Hamiltonian fiber bundle obtained by the clutching construction. A previouswork that applies the theory of the Hermitian scalar curvature as a moment map to the study of thetopology of the Hamiltonian group is [1, 2].Furthermore, our quasimorphism agrees with the quasimorphisms of Py and Entov whenever thesequasimorphisms are defined. While, having a Maslov component in the local type, our quasimorphismcan at best be continuous in the C -topology, it is rather easily seen to be coarse-Lipschitz in theSobolev L -metric, using the isoperimetric propery of Kahler manifolds with a bounded primitive ofthe Kahler form. This allows us to prove that the Sobolev L -metric is unbounded on e G of everysymplectic manifold of finite volume, extending a consequence from previous works of Eliashberg-Ratiu [32] on the L -metric in the case when the symplectic manifold is exact. Moreover, we showthat on manifolds like the blowup Bl ( C P ), where the restriction of the quasimorphism to π G doesnot vanish, the metric is not bounded on π G either. We conclude with some questions and discussionrelated to the topics presented in the paper.As an aside, it is curious to note that this paper touches upon two directions that both have theirorigins with Eugenio Calabi - the study of canonical metrics on Kahler manifolds (e.g. [16, 19, 18])and the theory of the Calabi homomorphism ([17]). Assume that a Lie group G acts G × X → X , ( g, x ) g · x on a symplectic manifold ( X , Ω) in aHamiltonian fashion. Here both the group and the manifold can be infinite dimensional. The actiongives a homomorphism
G →
Dif f ( X ), φ φ with the property that to each element X ∈ Lie ( G )there corresponds an element µ ( X ) ∈ C ∞ ( X , R ), such that1. the equation ι Ξ Ω = − dµ ( X ) holds for Ξ ∈ V.F. ( X ) - the vector field on X corresponding to X
2. the resulting map
Lie ( G ) → C ∞ ( X , R ) is a homomorphism of Lie algebras (the Lie structure onthe latter is given by the Poisson bracket of the symplectic form Ω).The second condition is equivalent to the linearity and equivariance of the map X µ ( X ) - for all X ∈ Lie ( G ) and φ ∈ G we have µ ( Ad φ X ) = µ ( X ) ◦ φ − . In one direction one differentiates this equality and the other can be found in [63] Lemma 5.16.Note that the map X µ ( X ) gives us a pairing µ : Lie ( G ) × X → R that is linear in the firstvariable, and therefore a map x µ ( − )( x ) : X → ( Lie ( G )) ∗ . The equivariance condition correspondsto the invariance of the pairing with respect to the diagonal action of G - for all X ∈ Lie ( G ) , x ∈ X and φ ∈ G we have µ ( Ad φ X )( φ · x ) = µ ( X )( x ) . We call µ in any one of these three equivalent formulations a moment map for the Hamiltonian actionof G on X . Remark . For infinite-dimensional Lie groups we use the approach of regular Fr´echet Lie groups(cf. [65] and references therein), while one could also use the inverse limit (ILH or ILB) approach ofOmori [68]. In any case, as we are interested only in the soft features of the theory of Lie groups and ourinfinite-dimensional example is a diffeomorphism group where all computations can be carried out asexplicit differential-geometric formulae, the foundational theory of infinite-dimensional Lie groups canfor the most part be ignored. The same remark applies to infinite-dimensional symplectic manifolds.4 .3 The action homomorphism
Assume that π ( X ) = 0. Denote by P Ω ⊂ R the spherical period group h Ω , π ( X ) i of Ω. FollowingWeinstein [92], we define the Action homomorphism π ( G ) → R / P Ω as follows.Suppose a class a ∈ π ( G ) is represented by a path { φ t } based at the identity element Id . Pick apoint x ∈ X . Consider its trace φ x = { φ t · x } t =0 under the action of the loop. Pick a disk D that spans φ x - that is D : D → X is a smooth map from D = {| z | ≤ } ⊂ C to X that satisfies D ( e πit ) = φ t · x for all t ∈ S = R / Z . Then the Action homomorphism is defined as A µ ( a ) = Z D Ω − Z µ ( X t )( φ t · x ) dt. It is independent of x ∈ X by the first property of µ and of { φ t } in the homotopy class a ∈ π ( G , Id )by the second property of µ . It does depend on the spanning disk D , however the ambiguity lies in P Ω . At last, the homomorphism property follows by a short concatenation argument. Detailed proofscan be found in Section 2. Remark . Note that when π ( X ) = 0, the Action homomorphism takes values in R , since P Ω = 0. Remark . This definition extends the original definition because given a closed symplectic manifold(
M, ω ) , the group G = Ham ( M, ω ) acts on (
M, ω ) in a Hamiltonian fashion with the equivariantmoment map µ ( X ) = H X where H X ∈ C ∞ ( M, R ) is the zero-mean normalized Hamiltonian functionof X . On an open symplectic manifold ( M, ω ) the group G = Ham c ( M, ω ) of compactly supportedHamiltonian diffeomorphisms acts in a Hamiltonian fashion with the equivariant moment map µ ( X ) = H X where H X is the compact-support normalized Hamiltonian function of X . To ensure the existenceof a contracting disk, we assume that the manifold is simply connected in the open case. In the closedcase the contracting disk always exists by Floer theory, by the existence of the Seidel element or by adirect geometric degeneration argument [62]. A quasimorphism ν on a group G is a function ν : G → R that satisfies the additivity property upto a uniformly bounded error. That is for all x ∈ G and y ∈ G we have ν ( xy ) = ν ( x ) + ν ( y ) + b ( x, y ) , where | b ( x, y ) | ≤ C ν for a constant C ν depending only on ν (and not on x, y ). In such cases the limit ν ( x ) := lim k →∞ k ν ( x k )exists by Fekete’s lemma on subadditive sequences and is also a quasimorphism. Moreover, it is homogenous that is ν ( x k ) = k ν ( x )for all x ∈ G and k ∈ Z and satisfies ν ≃ ν, where for any two functions a, b : G m → R we write a ≃ b (1)if they differ by a uniformly bounded function d : G m → R - that is | d ( x , ..., x m ) | ≤ C d for a constant C d independent of x , ..., x m . We refer to the book [20] by Calegari for these statements and foradditional information about quasimorphisms.We will use the following simple fact. 5 emma 1. For every quasimorphism ν : G → R we have ν ( x ) ≃ − ν ( x − ) as functions G → R .Proof. Indeed ν ( x ) ≃ ν ( x ) = − ν ( x − ) ≃ − ν ( x − ).Explicit constructions of quasimorphisms on Lie groups often use rotation numbers. For thispurpose we require the notion of the variation of angle of a continuous path δ : [0 , → S . Definition 1.4.1.
We define the full variation of angle of δ : [0 , → S as varangle ( δ ) = e δ (1) − e δ (0)for any continuous lift e δ : [0 , → R of δ to the universal cover R Z −→ S . The general principle says that when groups act well enough on spaces of negative enough curvature,then they have quasimorphisms and non-trivial bounded (or bounded-continuous) cohomology. Whileusually this principle is applied to proper discontinuous actions of discrete groups, we propose a versionof this principle for smooth actions of (possibly infinite dimensional) Lie groups. Firstly, we proposea version of ”negative enough curvature” - (possibly infinite-dimensional) symplectic manifolds ( X , Ω)with bounded Gromov norm of Ω. We make, more specifically, the following definition.
Definition 1.5.1. (Domic-Toledo space ( X , Ω , K )) Assume that X has π ( X ) = 0 (as before) and π ( X ) = 0 also. Moreover assume that there is a system K of paths [ x, y ] := γ ( x, y ) for all x ∈ X and y ∈ X , such that for all x, y, z ∈ X | Z ∆( x,y,z ) Ω | < C X , for a constant C X that does not depend on x, y, z . Here ∆ = ∆( x, y, z ), which we will call a geodesictriangle is any disk with boundary ∂ ∆ = [ x, y ] ∪ [ y, z ] ∪ [ z, x ]. We call the triple ( X , Ω , K ) a Domic-Toledo space.Next we propose a version for ”act well enough” - by ”isometries” with an equivariant momentmap. More exactly, we make the following definition.
Definition 1.5.2. (Hamiltonian-Hermitian group G ) We call a (possibly infinite dimensional) Liegroup G Hamiltonian-Hermitian if it acts on a Domic-Toledo space ( X , Ω , K ) - preserving K and Ω -with an equivariant moment map µ : X × Lie ( G ) → R . We say that the action of G on ( X , Ω , K ) preserves K if for every two points x ∈ X and y ∈ X andevery g ∈ G we have g · [ x, y ] = [ g · x, g · y ] . Remark . All examples of Domic-Toledo spaces known to the author are (possibly infinite-dimensional) Kahler manifolds ( X , Ω , J ) with [ x, y ] being the geodesic segment between x ∈ X and y ∈ X . A first set of examples is given by Hermitian symmetric spaces D of non-compact type(bounded Hermitian domains) [24, 23]. The second one (trivially containing the first) is given byspaces of global sections of bundles with fiber D over a manifold ( M, φ ) with a volume form φ of finitevolume. Remark . Examples of finite dimensional Hamiltonian-Hermitian groups are given by Hermitiansymmetric Lie groups - like Sp (2 n, R ) - since they act by Hamiltonian biholomorphisms on the corre-sponding symmetric spaces of non compact type equipped with the Bergman Kahler structure, whichis Kahler-Einstein. Therefore, the natural lift (by use of the differential) of these diffeomorphisms tothe top exterior power of the tangent bundle furnishes the action with an equivariant moment map(note that the Kahler-Einstein condition implies that ( − i ) times the curvature of the Chern connection6n these bundles, given by the Hermitian metric, equals to the Kahler form on one hand, and on theother hand the corresponding connection form is surely preserved by the lifts). Details are presentedin Section 1.9.Infinite dimensional examples are given by groups Ham ( M, ω ) of closed symplectic manifolds(
M, ω ) since these act of the spaces J of compatible almost complex structures, which is a Domic-Toledo space - since it is the space of global sections of a bundle over ( M, ω ) with fiber the Siegelupper half-plane. This class of examples can be extended to arbitrary symplectic manifolds of finitevolume. Details are presented in Section 1.7.We now construct a quasimorphism on the universal cover of a Hamiltonian-Hermitian group G with an equivariant moment map µ and Domic-Toledo space ( X , Ω , K ). Given a path { g t } t =0 in G with g = Id , g = g representing a class e g in e G , consider the loop { g t · x } t =0 g · x, x ] for a fixed basepoint x ∈ X . Fill it by any disk D = D { g t } t =0 . Then define ν x ( e g ) = Z D Ω − Z µ ( X t )( g t · x ) dt, (2)where { X t } t =0 is the path in Lie ( G ) corresponding to the path { g t } t =0 . In Section 2.2 we show thatthis value is well-defined and gives a real-valued quasimorphism ν x : e G → R on the universal cover of G . Theorem 1.
Any Hamiltonian-Hermitian group G acting with an equivariant moment map µ on thecorresponding Domic-Toledo space ( X , Ω , K ) admits a real-valued quasimorphism ν x : e G → R on itsuniversal cover for each point x ∈ X , given by Equation 2. Moreover, the homogeneization ν of ν x doesnot depend on the basepoint x . By construction, the quasimorphism ν restricts to the homomorphism A µ on π ( G ) .Remark . If we assume additionally that the loop [ x, x ] ∈ K is the constant path at x , then thequasimorphism ν x also restricts to A µ on π ( G ).Note that this theorem does not state that the homogenous quasimorphism ν is necessarily not ahomomorphism, or even not trivial. It can in principle be identically equal to zero. However, in allthe known examples it turns out to be non-trivial and not a homomorphism.The key feature of the proof which we defer to Section 2.2 is that the differential of ν x in groupcohomology satisfies b ( g, h ) = ν x ( e g e h ) − ν x ( e g ) − ν x ( e h ) = Z ∆( x,g · x,gh · x ) Ω (3)for e g, e h ∈ e G with endpoints g, h ∈ G . The latter is a bounded cocycle by the properties of Domic-Toledospaces and ”isometric” actions upon them. Remark . From Equation 3, given that for all x ∈ X , [ x, x ] is the constant path at x , it followsthat for all e φ ∈ e G we have ν x ( e φ − ) = − ν x ( e φ ) . Indeed the difference equals R ∆( x,φ · x,x ) Ω = 0, since we can choose a degenerate filling disk.Furthermore, we would like to explore the invariance of the quasimorphism with respect to largergroups extending a given action of a Hamiltonian-Hermitian group G on a Domic-Toledo space. Forthis we have the following proposition, which we prove in Section 2.2. Proposition 1.1.
Assume
G ⊂ H is a normal subgroup, G is a Hamiltonian-Hermitian group actingwith an equiavariant moment map µ on the Domic-Toledo space ( X , Ω , K ) , and H is a (possibly infinite-dimensional) Lie group that acts on ( X , Ω , K ) preserving Ω and K and extending the action of G (however not necessarily with a moment map). Assume moreover, that the moment map µ : Lie ( G ) × X → R is equivariant with respect to the action of H (note that as G ⊂ H is normal, H acts on Lie ( G )7 y the adjoint representation). Then ν x ( h e gh − ) = ν h − x ( e g ) for all e g ∈ e G and h ∈ H . Consequently, bythe independence of the homogeneization upon the basepoint, we have ν ( h e gh − ) = ν ( e g ) , for all e g ∈ e G and h ∈ H . Equivalently ν ( e h e g e h − ) = ν ( e g ) , for all e g ∈ e G and e h ∈ e H . Given a compact symplectic manifold (
M, ω ) consider the space J of ω -compatible almost complexstructures. This space can be given the structure of an infinite-dimensional Kahler manifold ( J , Ω , J ) asfollows. Consider the bundle S → M , the general fibre of which over x ∈ M is the space J c ( T x M, ω x ) ∼ = Sp (2 n ) /U ( n ) of ω x -compatible complex structures on T x M . As J c posses a canonical Sp (2 n )-invariantKahler form σ = σ trace we have a fiberwise-Kahler form σ on S . Note now that J = Γ( M, S ) - thespace of global sections of the bundle S → M . Now define Ω( A, B ) := R M σ x ( A x , B x ) ω n ( x ). Thecomplex structure J on J is defined as J J A = JA for A ∈ T J J . Surely Ω and J are compatible.Note that the group G = Ham ( M, ω ) of Hamiltonian diffeomorphisms acts on J by φ · J := φ ∗ J .This action can be shown to be Hamiltonian [27, 41] with respect to the form Ω. The moment map isgiven as follows.First note that the Lie algebra of G is isomorphic to the space C ∞ ( M, R ) / R ∼ = C ∞ ( M, R ). Thelatter space consists of smooth functions F on M with integral zero: R M F ω n = 0. For an element φ ∈ G , the adjoint action is given in these conventions by Ad φ H = ( φ − ) ∗ H. (4)To a function H ∈ Lie ( G ) ∼ = C ∞ ( M, R ) there corresponds the function µ ( H ) on J given [27, 41]by the formula: µ ( H )( J ) = Z M S ( J ) Hω n , (5)where S ( J ) ∈ C ∞ ( M, R ) is the Hermitian scalar curvature of the Hermitian metric h ( J ) = g ( J ) − iω defined as follows. Consider the Hermitian line bundle L = Λ n C ( T M, J, h ( J )). It has a naturalconnection ∇ n induced from the canonical connection ∇ on ( T M, J, h ( J )) (cf. [48] Section 2.6, [58]and [87] Section 2, and references therein) defined by the properties ∇ J = 0 , ∇ h = 0 , T (1 , ∇ = 0 . This connection can also be equivalently (by [48] Section 2) defined by use of ∂ -operators, as in [27].The connection ∇ n has curvature i e ρ for the lift e ρ of a real valued closed two form ρ ∈ Ω ( M, R ) on M by the natural projection L → M . We define S ( J ) ∈ C ∞ ( M, R ) by S ( J ) ω n = nρ ∧ ω n − . (6)Whenever J is integrable S ( J ) coincides with the scalar curvature of the Riemannian metric g ( J ). Inthe above g ( J ) is the Riemannian metric corresponding to J given by g ( J )( ξ, η ) = ω ( ξ, Jη ). Note that g ( φ ∗ J ) = ( φ − ) ∗ g ( J ) and consequently the same is true for h ( J ). Hence S ( φ ∗ J ) = ( φ − ) ∗ S ( J ) (7)for all φ ∈ G .From the equalities 4,5 and 6 we obtain µ ( H )( φ − ∗ J ) = R M S ( φ − ∗ J ) Hω n = R M φ ∗ S ( J ) Hω n = R M S ( J )( φ − ) ∗ Hω n = µ (( φ − ) ∗ H )( J ) = µ ( Ad φ H )( J ). Therefore the moment map is equivariant.We remark that the action of G on J can be extended to the action of H = Symp ( M, ω ) thatpreserves Ω and J . Moreover G ⊂ H is a normal subgroup and (by the same computation as above)the moment for map µ : Lie ( G ) × J → R for the action of G on J is equivariant with respect to theaction of H (which acts on Lie ( G ) by the adjoint action Ad ψ H = ( ψ − ) ∗ H , ψ ∈ H ).8 .7 Quasimorphisms on the Hamiltonian groups of symplectic manifolds Here we apply the general principle for constructing quasimorphisms to the group G = Ham ( M, ω )acting on ( J , Ω , J ) and study the resulting object to obtain the main results of this paper. Corollary1 and Theorem 3 are of special note.First, it is rather easy to prove that the space ( J , Ω , K ) for the system K of paths consisting of thefiberwise geodesics is a Domic-Toledo space. In more detail for every two almost complex structures J , J ∈ J = Γ( S ; M ) we define [ J , J ] to be the fiberwise geodesic path [ J , J ]( t ) that restricts ineach fiber S x over a point x ∈ M to the unique geodesic [( J ) x , ( J ) x ]( t ) in ( S x , σ x , j x ) joining ( J ) x and ( J ) x . Moreover, for any three elements J , J , J ∈ J we choose ∆ = ∆( J , J , J ) to be thefiberwise geodesic convex hull of J , J , J so that in each fiber S x over x ∈ M , ∆ restricts to a geodesic2-simplex ∆ x with respect to σ x with vertices ( J ) x , ( J ) x , ( J ) x . Then since Z ∆( J ,J ,J ) Ω = Z M ( Z ∆ x σ x ) ω n ( x ) , we estimate | Z ∆( J ,J ,J ) Ω | ≤ Z M | Z ∆ x σ x | ω n ( x ) ≤ V ol ( M, ω n ) C S n , as ( S x , σ x , j x ) is a Domic-Toledo space (with geodesics for the system of paths) with the constant C S n .And surely, J is contractible so the the conditions π ( J ) = 0 and π ( J ) = 0 are satisfied.Second, we show that G = Ham ( M, ω ) is Hamiltonian-Hermitian with its action on ( J , Ω , K ).First, as explained above it acts on J preserving Ω with an equivariant moment map. It is also easyto deduce from the fact that the action preserves J that it also preserves K - though we give a directproof. Indeed this follows immediately from the fact that for every diffeomorphism f ∈ G and for all x ∈ M the map S f − x → S x given by J f − x ( f ∗ x ) J f − x ( f ∗ x ) − is an isometry of the Siegel upperhalf-spaces. The canonical metric ρ y on S y for y ∈ M is given by ( ρ y ) J y ( A y , B y ) = const · trace( A y B y )for A y , B y ∈ T J y S y ( J y ∈ S y ), and surely, trace is preserved by congugation with a linear isormophism.Therefore by Theorem 1 the group e G admits a homogenous quasimorphism, which we show to benon-trivial by computing its local type in Theorem 3. Corollary 1.
The universal cover e G of the group of Hamiltonian diffeomorphisms G = Ham ( M, ω ) of an arbitrary closed symplectic manifold ( M, ω ) admits a non-trivial homogenous quasimorphism S : e G → R . By construction the restriction S | π ( G ) equals A µ : π ( Ham ( M, ω )) → R . In more detail, for anelement φ = [ { φ t } ] ∈ π ( Ham ( M, ω )) with mean-normalized Hamiltonian H t ∈ C ∞ ( M, R ) , we have A µ ( φ ) = Z D Ω − Z dt Z M S (( φ t ) ∗ J ) H t ( x ) ω n , where J ∈ J is an arbitrary element and D is a disk in J spanning the loop { ( φ t ) ∗ J } t ∈ R / Z . We nowcompute the homomorphism A µ in terms of a previously known homomorphism on π ( Ham ( M, ω ))[60].
Definition 1.7.1. (The homomophism I c : π ( Ham ( M, ω )) → R ) As usual with topological groups,there is a bijective correspondence between π ( Ham ( M, ω )) and the isomorphism classes of bundles P M −→ S over the 2-sphere with fiber M , such that their structure group is contained in Ham ( M, ω )[60]. Such bundles are called
Hamiltonian fiber bundles (or fibrations) over the 2-sphere. Over sucha bundle, the vertical tangent bundle T V P is naturally endowed with the structure of a symplecticvector bundle. Hence it has Chern classes, called the vertical Chern classes , of which we shall use thefirst c V := c ( T V P ). There is also a natural characteristic class u ∈ H ( P, R ) of such bundles withthe defining properties u | fiber = [ ω ] and R fiber u n +1 = 0 (or in the case when the base is 2-dimensional9 n +1 = 0) - cf. [60, 71] and references therein. It is called the coupling class of the Hamiltonianfibration. With these two characteristic classes we compose the monomial c V u n , where n = dimM and integrate over P . This yields a homomorphism π ( Ham ( M, ω )) → R that we denote I c . Theformula for I c ( γ ) for a loop γ in Ham ( M, ω ) based at Id is therefore I c ( γ ) = Z P γ c V u n , where P γ is the Hamiltonian fibration corresponding to γ . Theorem 2.
The two homomorphisms A µ and I c from π ( G ) to the reals are equal.Remark . Assume now that the almost complex structure J is integrable - that is ( M, ω, J ) is aKahler manifold. Note that the restriction ι ∗ A µ of A µ to the π of the finite dimensional compact Liesubgroup K := G J of G consisting of Hamiltonian biholomorphisms satisfies ι ∗ A µ = − F , for the Futakiinvariant F [43] since the filling disk D can be chosen to be trivial. The equality is understood via theisomorphism π ( K ) ⊗ Z R ∼ = Lie ( K ) / [ Lie ( K ) , Lie ( K )] which holds by a classical result of Chevalleyand Eilenberg [21] (a short account can be found in [10]). The consequence of Theorem 2 that I c restricts to the (Bando-)Futaki invariant on G J has previously been shown in [81] using methods ofequivariant characteristic classes.As a corollary we answer a question of Polterovich (cf. [81], Discussion and Questions, 2). Corollary 2.
We have the equality S | π ( G ) ≡ I c on π ( Ham ( M, ω )) . By Proposition 1.1 we have that S is Symp ( M, ω )-invariant.
Corollary 3.
The quasimorphism S : e G → R is invariant with respect to conjugation by elements of ^ Symp ( M, ω ) or equivalently by elements of Symp ( M, ω ) . Moreover we compute the local type of the quasimorphism S . To state the result of our computationwe would first like to make two definitions of the more classical invariants in terms of which we expressthe answer. Definition 1.7.2. (Calabi homomorphism on G B = Ham c ( B n , ω B ) [17], cf. [63, 76]) Given a Hamil-tonian isotopy { φ t } t =0 ⊂ Ham c ( B n , ω B ) starting at φ = Id with endpoint φ = φ with generatingpath of vector fields { X t } t =0 , define H t (for each t ∈ [0 , ∂B and satisfies i X t ω = − dH t . Then the Calabi homomorphism is defined as
Cal B ( { φ t } t =0 ) = Z Z B H t ω n dt. It is, as can be verified using the differential homotopy and the cocycle formulas, a well-defined homo-morphism e G B → R . Moreover it vanishes on loops in G B hence descending from e G B to G B itself. Remark . We present a short proof that
Cal B vanishes on loops in G B that differs slightly fromthe one usually found in the literature. It is well-known cf. [63, 76] that the Calabi homomorphismcan be reinterpreted as Cal B ( { φ t } t =0 ) = − n · Z Z B ( i X t λ ) ω n dt, for a primitive λ of ω B in B . Hence Cal B ( { φ t } t =0 ) = − n + 1 · Z Z B ( i X t λ − H t ) ω n dt. { φ t } t =0 this is proportional to Z B ( Z { φ t x } t =0 λ − Z H t ( φ t x ) dt ) ω n ( x )wherein the integrand is independent of x , as it is the Hamiltonian Action of the periodic orbit { φ t x } t =0 of { φ t } t =0 . Consequently the integral localizes (up to a multiplicative constant) to the value of theintegrand at each point x ∈ B that vanishes Z { φ t x } t =0 λ − Z H t ( φ t x ) dt = 0for x close enough to ∂B . Definition 1.7.3. (The Barge-Ghys average Maslov quasimorphism on G B = Ham c ( B n , ω B ) [7])Given a Hamiltonian isotopy { φ t } t =0 ⊂ Ham c ( B n , ω B ) starting at φ = Id , choosing a trivializationΘ of the tangent bundle ( T B, ω B ) ∼ = B × ( V, ω ) over B as a symplectic vector bundle (here ( V, ω ) is acertain symplectic vector space e.g. ( T b B, ( ω B ) b )) for some b ∈ B ), we obtain from the family of paths ofdifferentials { φ t ∗ x : T x B → T φ t x B } t =0 (as x ranges over B ) a family { A ( x, t ) ∈ Sp ( V, ω ) } t =0 of pathsof symplectic linear automorphisms of ( V, ω ). For each x ∈ B we compute the value τ Lin ( { A ( x, t ) } t =0 )on the path { A ( x, t ) } t =0 of the Maslov quasimorphism on the universal cover of the symplectic lineargroup. Then the map τ Θ ,B : { φ t } t =0 Z B τ Lin ( { A ( x, t ) } t =0 )( ω B ) n ( x )does not depend upon homotopies of { φ t } t =0 with fixed endpoints and yields a quasimorphism τ Θ ,B : e G B → R . The Barge-Ghys average Maslov quasimorphism τ B : e G → R is its homogeneization τ B ( e φ ) = lim k →∞ k τ Θ ,B ( e φ k ) . It does not depend on the choice of the symplectic trivialization Θ. Both τ Θ ,B and τ B vanish on loopsin G B and therefore descend to quasimorphisms G B → R . Remark . The vanishing of τ Θ ,B on loops can be shown by a similar localization argument asfor the Calabi homomorphism. Indeed for a loop { φ t } t =0 in G B the value τ Lin ( { A ( x, t ) } t =0 ) equalsthe Maslov index of the loop { A ( x, t ) } t =0 which by the homotopy invariance of the Maslov index isindependent of x , and for x near ∂B the loop { A ( x, t ) } t =0 is trivial. Hence the integrand vanishes forall x ∈ B , wherefrom τ Θ ,B ( { φ t } t =0 ) = 0. Theorem 3.
Let c = n R M c ω n − / R M ω n = R S ( J ) ω n /V ol ( M, ω n ) be the average Hermitian scalarcurvature. Then the restriction of S to the subgroup G B = Ham c ( B, ω | B ) ⊂ G of Hamiltonian diffeo-morphisms supported in an embedded ball B in M satisfies S | G B = 12 τ B − c Cal B , where τ B is the Barge-Ghys Maslov quasimorphism on G B = Ham c ( B n , ω std ) and Cal B is the Calabihomomorphism. We describe the relation of the quasimorphism S to the quasimorphisms S P y and S En introducedby Py [75, 76] for closed manifolds ( M, ω ) with c ( T M, ω ) = κ [ ω ] for κ = 0 and Entov [33] for closedmanifolds ( M, ω ) with c ( T M, ω ) = 0. First we state briefly the definitions of the quasimorphisms S P y and S En . The detailed definitions appear in the proofs section.11 efinition 1.7.4. (A sketch of a definition of S P y [75, 76]) Endow the unit frame bundle P S −−→ M of L = Λ n C ( T M, J, ω ) for a compatible complex structure J ∈ J with the structure α of a prequantizationof ( M, − ω ). Note that there is a natural map det : L ( T M, ω ) → P from the Lagrangian Grassman-nian bundle L ( T M, ω ) to the unitary frame bundle P of L ⊗ , since L ( T M ) x = U ( T M x , ω x , J x ) /O ( n ).Note that α induces a structure α of prequantization of ( M, − ω ) on P . Given a path −→ φ = { φ t } t =0 in G with φ = Id , choosing a point L ∈ L ( T M, ω ) x we have the curve { φ t ∗ x ( L ) } ≤ t ≤ in L ( T M, ω )and considering −→ φ as a path of Hamiltonian isotopies of ( M, − ω ) we have the canonical lifting { b φ t } ≤ t ≤ , b φ = Id of −→ φ to the identity component Q = Quant ( P , α ) of the group of diffeomor-phisms of P that preserve α . Consequently, one considers the two curves { det ( φ t ∗ x ( L )) } ≤ t ≤ and { b φ t ( det ( L )) } ≤ t ≤ in P . Both these curves in P start at det ( L ) and cover the path { φ t ( x ) } ≤ t ≤ in M and hence differby an angle: det ( φ t ∗ ( L )) = e i πϑ ( t ) b φ t ( det ( L )) , for a continuous function ϑ : [0 , → R . Define a continuous function on L ( T M, ω ) by angle ( L, −→ φ ) := ϑ (1) − ϑ (0) . Then the function angle ( x, −→ φ ) = inf L ∈L ( T M,ω ) x angle ( L, −→ φ ) on M is measurable, bounded and definesthe quasimorphism S ( −→ φ ) = − Z M angle ( x, −→ φ ) ω n ( x )that does not depend upon homotopies of −→ φ with fixed endpoints and is thus defined as a real-valuedfunction on e G . Its homogeneization S P y : e G → R , defined by S P y ( e φ ) := lim k →∞ S ( e φ k ) k is a homogenousquasimorhism on e G that is independent of the non-canonical structure on P of a prequantization of( M, − ω ), of the prequantization form α on it and of the almost complex structure J . Definition 1.7.5. (A sketch of a definition of S En [33]) Given a symplectic manifold ( M, ω ) with c ( T M, ω ) = 0 one first trivializes (
T M, ω, J ) for J ∈ J as a Hermitian vector bundle over thecomplement U = M \ Z of a compact triangulated subset Z of codim( Z ) ≥
3, where the differentialof the trivialization, appropriately defined, is uniformly bounded. For a path −→ φ = { φ t } t =0 in G byrelaxing Z to be a countable union Z −→ φ = S j ∈ Z Z j depending on −→ φ of sets Z j of codim( Z j ) ≥ U is invariant with respect to φ t for all t . Then from the path { φ t ∗ x } t =0 for x ∈ U one obtains a continuous path { A ( x, t ) } t =0 with A ( x,
0) = Id in Sp (2 n, R ) and proceeds to define angle ( x, −→ φ ) = varangle ( { det ( A ( x, t )) } t =0 ) . One then shows that this function extended by 0 on Z is integrable on M and that T ( −→ φ ) = Z M angle ( x, −→ φ ) ω n ( x )does not depend on homotopies of −→ φ with fixed endpoints (by relaxing Z to be a countable union ofsets Z ′ j of codim( Z ′ j ) ≥ T : e G → R . S En : e G → R , defined by S En ( e φ ) := lim k →∞ T ( e φ k ) k is a homogenous quasimorhismon e G that is independent of the non-canonical choices of trivialization, of the set U = M \ Z and ofthe almost complex structure J .We claim that the quasimorphism e G → R obtained from Corollary 1 agrees with these two quasi-morphisms in the settings of their definitions. Theorem 4.
1. On symplectic manifolds ( M, ω ) with c ( T M, ω ) = κ [ ω ] for κ = 0 we have S = − S P y .
2. On symplectic manifolds ( M, ω ) with c ( T M, ω ) = 0 we have S = S En . Remark . The analogues of Theorem 3 for the cases 1. and 2. above were shown in [33, 75].The analogue of Corollary 2 was shown in [76]. The agreement of our results with the ones shownin these papers is as follows. For analogues of Theorem 3 note that the average scalar curvature c satisfies c = nκ when c ( T M, ω ) = κ [ ω ], for every κ . For the analogue of Corollary 2, use the easyComputation 1 from [81], near the end of Section 1.2. Remark . We would also like to note that the general scheme of Theorem 1 applies to the con-struction of quasimorphisms e G → R for the group G = Ham c ( M, ω ) of Hamiltonian diffeomorphismswith compact support of symplectic manifolds (
M, ω ) of finite volume (without boundary) that are notcompact. Indeed, J here is also a Domic-Toledo space, since R M ω n is finite, and Donaldson’s theoryfor the scalar curvature as an equivariant moment map [27] applies here nearly verbatim. The onlydifference is that the symplectic form Ω is not defined on all the tangent space T J J - indeed given A, B ∈ T J J the function σ ( J ) x ( A x , B x ) may well be non-integrable with respect to ω n . However,since we compute for diffeomorphisms with compact support, all relevant computations happen in acompact subset of M where all functions that appear are integrable. Moreover, all functions, vectorfields, one-forms and sections of endomorphism bundles have compact support, therefore the only non-local part in Donaldson’s proof [27] - integration by parts to show the actual integral formulae - goesthrough (all the other arguments are local). At the same time, when the symplectic volume of M isnot finite, J stops being a Domic-Toledo space (at least with the natural definitions) and hence thisapproach does not seem to give quasimorphisms. It would be interesting to investigate the restrictionto π ( G ) of the quasimorphism in the finite volume case. The local type is obtained by nearly thesame computation as the one given for the closed case and is given by the Barge-Ghys average Maslovquasimorphism τ . The H = Symp c ( M, ω )-invariance holds as before.A corollary, as obtained in [33] for symplectic manifolds with c ( T M, ω ) = 0 , is that the commutatorlength of e G is unbounded. Corollary 4.
The diameter in the commutator length of the group e G for G = Ham ( M, ω ) of a closedsymplectic manifold ( M, ω ) and of the perfect ([5]) group Ker(
Cal : e G → R ) for G = Ham c ( M, ω ) ofan open finite volume symplectic manifold ( M, ω ) is infinite. In the closed case, under the additionalassumption I c ≡ , the same conclusion follows for G itself. We also note that for reasons of naturality of the constructions and normalizations of Hamiltonianswe have the following proposition.
Proposition 1.2. (Embedding functoriality) Given an open subset U ⊂ M of a closed symplectic man-ifold ( M, ω ) , denote by S M the quasimorphism obtained on G for ( M, ω ) and by S U the quasimorphismobtained on G U for ( U, ω | U ) . Then S M | G U = S U − c · Cal U , for the average Hermitian scalar curvature c . Similarly if M were an open symplectic manifold of finitevolume S M | G U = S U . .8 Application to the L -distance on ] Ham ( M, ω ) While it is not surprising that our quasimorphism is bounded by a multiple of the Sobolev L norm on ] Ham ( M, ω ), indeed S J is surely continuous in the C -topology induced to Ham ( M, ω ) from
Dif f ( M ), we present a proof for the sheer simplicity of the argument.For a Hamiltonian isotopy −→ φ = { φ t } t =0 of a symplectic manifold ( M, ω ) starting at the identitythat is generated by the zero-mean-normalized Hamiltonian H t put ||−→ φ || k,p = Z || H t || L pk ( M,ω n ) . Then define the norm of an element e φ ∈ e G by || e φ || k,p = inf [ −→ φ ]= e φ ||−→ φ || k,p . Finally define the norm of φ ∈ G by || φ || k,p = inf π ( e φ )= φ || e φ || k,p for the natural projection π : e G → G . Fortwo elements a, b of the above groups define the distance d p,k ( a, b ) = || a − b || k,p . The following facts are easy to check. • For k ≥ p, k )-norms and distances are equivalent to ( p, k − X t generating −→ φ . • For k ≥ S J calibrates the (2 , S J ( e φ ) ≤ C ( n, ω, J ) || e φ || , , (8)for a constant C ( n, ω, J ) that does not depend on e φ . As a corollary we obtain that the L distanceis unbounded on e G . Corollary 5.
The diameter of e G is infinite with respect to the L -distance for every symplectic manifold ( M, ω ) of finite volume.Remark . Given that A µ : π G → R vanishes, the same consequence holds for the L -distance onthe group G itself. For closed manifolds this condition is equivalent to the vanishing of I c .The unboundedness of the L -metric on compact exact symplectic manifolds was previously provenby Eliashberg and Ratiu [32] (their methods work even for the larger group H = Symp ( M, ω ) withappropriate definitions), while sharper topological bounds for the 2-disc were obtained by Gambaudoand Lagrange [47] (cf. [9], [12]).
The general principle outlined in Section 1.5 applies also to finite dimensional Hermitian Lie groupsacting on their corresponding Hermitian symmetric spaces of non-compact type. In this section wedescribe this application, in part for use in the proofs later.Let G be a simple Hermitian symmetric Lie group. Then the adjoint form of G belongs to thoseof the following list of Lie groups: SU ( p, q ) , SO (2 , q ) q = 2 , Sp (2 n, R ) , SO ∗ (2 n ) n ≥ E and E respectively. Let us assume that thecenter of G is finite, so that π ( G ) is infinite. Let K ⊂ G be the analytic subgroup corresponding14o the maximal compact Lie subalgebra k of g . In this situation there is a corresponding Hermitiansymmetric space X = G/K , endowed with a natural complex structure j X and a Kahler form σ X thatis invariant with respect to the transitive action of the group G (proportional to the Bergman Kahlerstructure when such a space is realized as a symmetric bounded domain in a complex affine space bythe Harish-Chandra embedding) cf. [55, 57, 66]. The works of Domic-Toledo and Ørsted [24, 23] showthat when we take the system of paths K to consist of the geodesics with respect to the invariantKahler metric, then ( X, σ X , K ) is a Domic-Toledo space in our terminology.Moreover, we note that by e.g. [66] these spaces ( X, σ X , j X ) are Kahler-Einstein manifolds (thatis - their Ricci forms are proportional to their Kahler forms: Ric ( σ X ) = λσ X , where for the Bergmanmetric we have λ = − Ric ( σ X ) is equal up to a universal constant to the curvature of theChern connection on the line bundle L X = Λ N C T X , with the holomorphic and Hermitian structuresinduced by j X and σ X .We now show that G is Hamiltonian-Hermitian with its action on X . Firstly the group G actson X by maps preserving j X and σ X (symplectic biholomorphisms) and hence preserving the systemof geodesics K . We now claim that the group G acts on X with an equivariant moment map µ X : Lie ( G ) × X → R . Note that as the Chern connection on T X is given canonically by ( σ X , j X ) and theaction preserves these structures, it will also preserve the Chern connection. Consider the natural liftof the action of G on X to an action of G on T X by taking differentials. This induces an action of G on L X = Λ N C T X . Note that this action preserves the Hermitian structure on L X , and hence it descendsto the circle bundle P X S −−→ X of unit vectors in L X (the unitary frame bundle of the Hermitianvector bundle L X ). The Chern connection on L X induces a real-valued connection one-form (cf. [81]Appendix A) α X on the principal S -bundle P X over X , that by the Kahler-Einstein property satisfiesthe relation dα X = f σ X (9)for the lift f σ X of σ X to P X by the natural projection P X → X , as follows from what is noted above.Now the action of the group G on P X covering the action of G on X preserves the one-form α X (bypreservation of the Chern connection). This is enough to give an equivariant moment map for theaction of G on X . Indeed, it is constructed as follows. A vector ξ ∈ Lie ( G ) induces the vector field ¯ ξ on X by the action of G on X and a vector field b ξ on P X that covers ¯ ξ , by the action of G on P X . Weclaim that the equivariant moment map is given by µ X ( ξ )( x ) = ( α X ) y ( b ξ y ) (10)for any y ∈ P X over x ∈ X (indeed b ξ is equivariant with respect to the natural circle action on P X asis α X and hence ( α X ) y ( b ξ y ) does not depend on the choice of y over x ). Firstly by relation (9) and thepreservation L b ξ α X = 0 of the connection by the infinitesimal action we have i ¯ ξ σ X = − dµ ( ξ )( x ) . Hence µ X is a moment map for the action of G on X . For the equivariance we note once again that theaction of G on P X preserves α X and that the vector field b ξ has a corresponding equivariance property.Namely for any g ∈ G and y ∈ P denoting by b g · y the action of G on P X and by b g ∗ y the correspondingdifferential T y P X → T b g · y P X we have the very general equivariance property for infinitesimal actionscorresponding to Lie group actions on spaces b ξ ( b g · y ) = b g ∗ y ( \ Ad g − ξ ( y )) . Now noting that for y ∈ P X over x ∈ X the point b g · y is over g · x , we obtain µ X ( Ad g ξ )( g · x ) = ( α X ) b g · y ( [ Ad g ξ ( b g · y )) = ( α X ) b g · y ( b g ∗ y b ξ ( y )) = ( α X ) y ( b ξ ( y )) = µ X ( ξ )( x ) , showing equivariance.Hence G is Hamiltonian-Hermitian with Domic-Toledo space ( X, σ X , j X ) and equivariant momentmap µ X , and therefore by Theorem 1 has a homogenous quasimorphism.15 orollary 6. Theorem 1 gives a homogenous quasimorphism ν G : e G → R for every simple Hermitiansymmetric Lie group G . It remains to show that it is non-trivial. In fact we show in Section 2.3 that it is equal to theGuichardet-Wigner [52, 31, 22, 82, 14] quasimorphism ̺ G on e G by comparing them on π ( G ) andarguing that a homogenous quasimorphism on e G is determined by its restriction to the fundamentalgroup. Proposition 1.3.
The quasimorphisms ν G and ̺ G on e G satisfy the equality ν G = − ̺ G . We would now like to give a reformulation of the construction of ν x in the finite dimensional caseas a certain rotation number. Indeed consider once again the principal S -bundle P X → X . Trivializeit by taking parallel transports Γ γ ( y,x ) : ( P X ) y → ( P X ) x along geodesics γ ( y, x ) for y ∈ X . Then givena path −→ g = { g t } t =0 with g = Id in G , the path of differentials ( g t ) ∗ x : T x X → T g t · x X gives us a pathΓ γ ( g t · x,x ) ◦ b g t | ( P X ) x : ( P X ) x → ( P X ) x which we consider as a path in U (1) ∼ = S . Then ν x ( −→ g ) = varangle( { Γ γ ( g t · x,x ) ◦ b g t | ( P X ) x } t =0 ) . (11)Indeed, denoting γ t := γ ( g t · x, x ) and β t = { g t ′ · x } tt ′ =0 we have varangle ( { Γ γ ( g t · x,x ) ◦ b g t | ( P X ) x } t =0 ) = varangle ( { Γ γ t ◦ Γ β t } t =0 ) + varangle ( { Γ β t ◦ b g t | ( P X ) x } t =0 ) == Z D −→ g σ X − Z ( α X ) b g t · y ( b ξ t ) b g t · y dt = Z D −→ g σ X − Z µ ( ξ t )( g t · x ) dt = ν x ( −→ g ) . It is interesting to note that taking this reformulation as a definition for the quasimorphism, itsindependence upon homotopies with fixed endpoints follows immediately by continuity.
Acknowledgements
First and foremost I thank my advisor Leonid Polterovich for his support and encouragement,for his continuous interest in this project and for many fruitful discussions. I have also benefitedfrom several stimulating conversations with Pierre Py. A decisive part of this project was carried outduring the author’s visit to the Mathematics Department at the University of Chicago. I thank LeonidPolterovich and the Mathematics Department for their hospitality and for a great research atmosphere.I thank Akira Fujiki for sending me a proof of his theorem (Equation 17). I also thank Marc Burgerand Danny Calegari for useful comments. Many thanks are due to the referee for remarks that haveimproved the exposition. This paper is partially supported by the Israel Science Foundation grant
We prove that the number A µ ( a ) defined in Section 1.3 is well-defined and determines a homo-morphism π ( G ) → R / P Ω . We refer to Sections 1.2 and 1.3 for the relevant notation and definitions.Let us first prove that it is well-defined. First of all, the value A µ ( a ) ∈ R / P Ω obviously does notdepend on the spanning disk. Let us prove that it does not depend on the point x ∈ X . Take anotherpoint x ′ ∈ X and choose a path β : [0 , → X between the two: β (0) = x, β (1) = x ′ . Consider thecylindric cycle C : R / Z × [0 , → X defined by C ( t, s ) = φ t · β ( s ). Note that C ( t,
0) = φ t · x and that C ( t,
1) = φ t · x ′ . Define a spanning disk D ′ for φ x ′ by D ′ = D ∪ φ x C . Then the equality Z D Ω − Z µ ( X t )( φ t · x ) dt = Z D ′ Ω − Z µ ( X t )( φ t · x ′ ) dt Z D ′ Ω − Z D Ω = Z µ ( X t )( φ t · x ′ ) dt − Z µ ( X t )( φ t · x ) dt, which is equivalent to Z C Ω = Z µ ( X t )( C ( t, dt − Z µ ( X t )( C ( t, dt. This equality is established by direct computation of the left hand side. Indeed Z C Ω = Z Z Ω( ∂ s C ( s, t ) , ∂ t C ( s, t )) dsdt = Z Z Ω( ∂ s C, Ξ t ( C ( t, s ))) dsdt == Z Z d C ( s,t ) µ ( X t )( ∂ s C ) dsdt = Z Z ∂ s µ ( X t )( C ( s, t )) dsdt == Z µ ( X t )( C (1 , t )) − µ ( X t )( C (0 , t )) dt = Z µ ( X t )( C ( t, dt − Z µ ( X t )( C ( t, dt, yielding the desired equality.Let us now proceed to prove that A µ ( { φ t } ) = R D Ω − R µ ( X t )( φ t x ) dt remains invariant when φ t is deformed homotopically with fixed endpoints. Let φ st , ≤ s ≤ φ s ≡ Id, φ s ≡ Id and ( s, t ) → φ st is a smooth map [0 , × [0 , → G . Surely, it is enough to provethat for all s the derivative ∂∂s | s A µ ( φ st ) vanishes. To this end we use the following lemma, which is adirect consequence of the standard differential homotopy formula. Lemma 2.
Let X st and Y st be the elements X st = ∂∂τ | τ = t φ sτ · ( φ st ) − , Y st = ∂∂σ | σ = s φ σt · ( φ st ) − of Lie ( G ) (note that Y s ≡ and Y s ≡ ). Then ∂∂s Ad ( φ st ) − X st = Ad ( φ st ) − ∂∂t Y st and ∂∂t Ad ( φ st ) − Y st = Ad ( φ st ) − ∂∂s X st . Proof.
The differential homotopy formula says ∂∂s X st = ∂∂t Y st + [ X st , Y st ]. Differentiating ∂∂s Ad ( φ st ) − X st we obtain Ad ( φ st ) − ([ Y st , X st ]+ ∂∂s X st ) , which by the differential homotopy formula equals Ad ( φ st ) − ∂∂t Y st .The other equality is obtained in the same way. Both are equivalent to the original differential homo-topy formula.Now − ∂∂s | s A µ ( φ st ) = ∂∂s Z µ ( X st )( φ st x ) dt − ∂∂s Z D st Ω == Z µ ( ∂∂s X st )( φ st x ) dt + Z ι Υ st ( φ st x ) d φ st x µ ( X st ) dt − Z ι Υ st ( φ st x ) Ω(Ξ st ( φ st x )) == Z µ ( Ad ( φ st ) − ∂∂s X st )( x ) dt + Z Ω(Ξ st ( φ st x ) , Υ st ( φ st x )) dt − Z Ω(Υ st ( φ st x ) , Ξ st ( φ st x )) == Z µ ( ∂∂t Ad ( φ st ) − Y st )( x ) dt = Z ∂∂t µ ( Ad ( φ st ) − Y st )( x ) dt == µ ( Ad ( φ s ) − Y s )( x ) − µ ( Ad ( φ s ) − Y s )( x ) = µ ( Y s )( x ) − µ ( Y s )( x ) = 0 . This yields the desired equality. Here D st is obtained by gluing D and C ( s, t ) = φ st x along φ x . Thevector fields Ξ st , Υ st are the infinitesimal actions of X st and Y st .At last, let us prove that A µ defines a homomorphism π ( G ) → R / P Ω . Indeed, take two loops φ = { φ t } , ψ = { ψ t } based at Id . Consider their concatenation χ = ψ ∗ φ . Then χ x = φ x ∗ ψ x .Moreover we can choose a spanning disk of χ x that factors through the topological wedge (we refer to1740] for the definition of wedge and for related notations) of the spanning disks D φ , D ψ of φ x , ψ x . Thatis D : D → X factors as D : D pr −→ D W D D φ W x D ψ −−−−−−−→ X . Hence A µ ( χ ) = R D φ Ω − R R µ ( X t )( φ t x ) + R D ψ Ω − R µ ( Y t )( ψ t φ x ) dt = R D φ Ω − R R µ ( X t )( φ t x ) + R D ψ Ω − R µ ( Y t )( ψ t x ) dt = A µ ( φ ) + A µ ( ψ ) . Here X t and Y t are the elements of Lie ( G ) corresponding to φ and ψ . The penultimate equality followsfrom the fact that φ = Id . We now prove Theorem 1 on the construction of quasimorphisms on Hamiltonian-Hermitian groups.
Proof.
First, the independence on the disk follows trivially, since π ( X ) = 0 and Ω is closed.We proceed to show that the map is independent upon homotopies of { g t } t =0 with fixed endpoints.Let g st be a homotopy with fixed endpoints g s ≡ Id , g s ≡ g of −→ g = { g t = g t } t =0 to −→ h = { g t = h t } t =0 .Note that in this situation the concatenation q = { g t } t =0 { h t } t =0 is a contactible loop in G basedat Id . Denote by C the disk C = { g st · x } ≤ s,t ≤ . Choose the disks of integration as follows. Whencomputing for { h t } t =0 choose an arbitrary disk D { h t } t =0 and for { g t } t =0 choose D { g t } t =0 = D { h t } t =0 ∪ C where the gluing is over the common path { h t · x } t =0 . Then ν x ( −→ g ) − ν x ( −→ h ) = Z C Ω − ( Z µ ( X t )( g t · x ) − Z µ ( Y t )( h t · x )) = A µ ( q ) = 0since A µ is a homomorphism on π ( G ) and q is a contractible loop. Here { X t } t =0 and { Y t } t =0 are thepaths in Lie ( G ) corresponding to { g t } t =0 and { h t } t =0 We now show the quasimorphism property of ν x . Take two paths −→ g = { g t } ≤ t ≤ , −→ h = { h t } ≤ t ≤ representing elements e g, e h of e G . Denote by g, h their endpoints. We would like to compare ν x ( e g e h ) with ν x ( e g ) + ν x ( e h ). Note that e g e h is represented by the path −→ g g −→ h , where g −→ h = { g h t } t =0 . Hence wewill compare ν x ( −→ g g −→ h ) to ν x ( −→ g ) + ν x ( −→ h ) . The definition of ν x involves two summands - one involving the symplectic area and one involving themoment map. We first show that the terms involving the moment maps are equal. And indeed Z µ ( X t )( g t · x ) dt + Z µ ( Ad g Y t )( g · h t · x ) dt = Z µ ( X t )( g t · x ) dt + Z µ ( Y t )( h t · x ) dt, (12)by the equivariance of the moment map.Now we show that the terms involving symplectic area agree up to the function Z ∆( x,g · x,gh · x ) Ω (13)which is bounded by a constant C X depending only on the Domic-Toledo space ( X , Ω , K ). Indeedchoosing arbitrary disks of integration D −→ g for −→ g and D −→ h for −→ h , choose −→ g g −→ h the disk D −→ g g −→ h =( D −→ g ∪ g · D −→ h ) ∪ ∆( x, g · x, gh · x ) where the gluing is over the common path [ x, g · x ] ∪ g [ x, h · x ], whichequals [ x, g · x ] ∪ [ g · x, gh · x ] by preservation of K . Hence Z D −→ g g −→ h Ω = Z D −→ g Ω + Z g · D −→ h Ω + Z ∆( x,g · x,gh · x ) Ω = Z D −→ g Ω + Z D −→ h Ω + Z ∆( x,g · x,gh · x ) Ω , (14)by preservation of Ω by the action. This finishes the proof of the quasimorphism property.Now we discuss the independence of the homogeneization ν ( e g ) = lim k →∞ k ν x ( e g k )18n the basepoint x . Take two basepoints x and x ′ and let { x s } s =0 , x = x , x = x ′ be a path in X connecting them. Note that it is enough for us to show that ν x and ν x ′ differ by a bounded function e G → R . Let us compare ν x ( −→ g ) and ν x ′ ( −→ g ). Let δ := −→ g · x g · x, x ], δ ′ := −→ g · x ′ g · x ′ , x ′ ] and let D, D ′ be their contracting discs. Define the disk C : [0 , × [0 , → X by C ( s, t ) = g t · x s . Moreoverdefine S be the contracting disk of { x s } x ′ , x ]. Then by preservation of K by the action S = g · S will be the contracting disk of { g · x s } g · x ′ , g · x ]. Note that g · x s ≡ C ( s, x, x ′ ] ∪ [ x, g · x ′ ] ∪ [ g · x, x ′ ] ∪ [ g · g, g · x ′ ] as the union Q = ∆ ∪ ∆ for the two geodesictriangles ∆ , ∆ on { x, x ′ , g · x } and on { g · x, x ′ , g · x ′ } . Note then that Σ = D ∪ D ∪ S ∪ S ∪ C ∪ Q where the gluings go along the overlapping paths, is a sphere. Therefore0 = Z Σ Ω = Z D ∪ D ∪ S ∪ S ∪ C ∪ Q Ω = Z D Ω − Z D Ω − Z S Ω + Z g · S Ω + Z Q Ω + Z C Ω == Z D Ω − Z D Ω+ Z Q Ω+ Z C Ω = ν x ( e φ ) − ν x ′ ( e φ )+ Z µ ( X t )( g t · x ) − Z µ ( X t )( g t · x ′ )+ Z C Ω+ Z Q Ω (15)since the action of e G on X preserves Ω. Wherefrom | ν x ( e g ) − ν x ′ ( e g ) | ≤ | Z | + 2 C X , (16)for Z = R µ ( X t )( g t · x ) − R µ ( X t )( g t · x ′ ) + R C Ω. We now show that Z equals zero, finishing theproof. And indeed letting { X t } t =0 be the path in Lie ( G ) corresponding to −→ g and Ξ t = X t , we have Z C Ω = Z Z Ω( ∂ s C ( s, t ) , ∂ t C ( s, t )) dsdt = Z Z Ω( ∂ s C, Ξ t ( C ( s, t ))) dsdt == Z Z d C ( s,t ) µ ( X t )( ∂ s C ) dsdt = Z Z ∂ s µ ( X t )( C ( s, t )) dsdt == Z µ ( X t )( C (1 , t )) − µ ( X t )( C (0 , t )) dt = Z µ ( X t )( g t · x ′ ) dt − Z µ ( X t )( g t · x ) dt. We also prove Proposition 1.1 on the transformation of ν x under conjugation with respect to asuitable normal extension. Proof.
Consider a path −→ g = { g t } t =0 representing e g ∈ e G . Then for an element h ∈ H the path h −→ g h − = { hg t h − } t =0 will represent h e gh − . By definition ν x ( h −→ g h − ) = Z D h −→ g h − Ω − Z µ ( Ad h X t )( hg t h − · x ) dt =for a disk D h −→ g h − with boundary δ x = −→ g · x g · x, x ], and noting that by preservation of K we havethe relation h · δ h − · x = δ x for δ h − · x = −→ g · ( h − · x ) g · h − · x, h − · x ] so that the disk D satisfying h · D = D h −→ g h − has boundary δ h − · x , so that by preservation of Ω and by equivariance of µ with respect to H we have= Z D Ω − Z µ ( X t )( g t h − · x ) dt = ν h − · x ( −→ g ) , which proves the proposition. Note that for every e h ∈ e H with endpoint h we have e h e g e h − = h e gh − since the paths { h t g t h − t } t =0 and h −→ g h − are homotopic with fixed endpoints.19 .3 Finite dimensional examples and Guichardet-Wigner quasimorphisms In this section we define the Guichardet-Wigner quasimorphisms and prove Proposition 1.3 onreconstructing these through moment maps.We remark that as we have assumed that G has finite center, there are no homogenous quasi-morphisms on G (cf. [82, 15] and [7] for the group Sp (2 n, R )). Moreover it is known that the allhomogenous quasimorphisms on e G are proportional to ̺ G (cf. [82, 15, 7]). From these two remarks itfollows that it is enough to show the equality of ν G and ̺ G on π ( G ) ∼ = π ( K ). In fact, ̺ G is definedas the unique homogenous quasimorphism e G → R such that its pullback ̺ G | e K : e K → R to e K bythe natural map e K → e G coincides with the lift e v : e K → R of the canonical (up to powers) character v : K → S , constructed in either one of several ways. The first way is as follows. The Lie algebra k of K satisfies k = z + [ k , k ] where z is the center of k (Corollaries 4.25 and 1.56 in [57]). In the case when G is a simple Hermitian symmetric Lie group, z is one-dimensional by [57] p.513. Hence the center Z of K is one dimensional. Take the identity component Z ∼ = S of Z . Then by Theorem 4.29 in [57] K = ( Z ) K ss , for K ss the analytic subgroup with Lie algebra [ k , k ]. The group K ss has a finite center,therefore by taking quotients by K ss we get a homomorphism v : K → Q ∼ = S from K to the quotient Q ∼ = S of Z ∼ = S by a finite subgroup. Example . For G = Sp (2 n, R ) we have K ∼ = U ( n ) and K ss ∼ = SU ( n ). Therefore the first constructiongives the homomorphism v : K → U ( n ) /SU ( n ) ∼ = S is simply v ( k ) = det C ( k ).The second way to construct v : K → S is by use of the action of G on the Hermitian symmetricspace X = G/K - it is shown in [52] that v equals the determinant of the linearization of the naturalaction of K ⊂ G at the fixed point x = [ Id ] ∈ X = G/K . Note that the two constructions of v agreeup to the power − C ( X ) / Z ∩ K ss ) since the determinant of a scalar matrix equals the scalarraised to the power of the dimension of the space (cf. [55] - proof of Theorem 6.1 and [52] - proof ofTh´eor`eme 2). Example . For G = Sp (2 n, R ) we have K ss ∼ = SU ( n ), Z ∼ = D - the subgroup of diagonal matrices in U ( n ) and Z ∩ K ss ) = n . As in this case dim C ( X ) = n ( n + 1) /
2, the second construction gives thehomomorphism v ( k ) = det − ( n +1) C ( k ).We use the second way to define ρ G now. Proposition 1.3 is then demonstrated as follows. Proof.
Consider the point x = [ Id ] ∈ X = G/K . It is a fixed point under the natural action of K ⊂ G . By the construction of ν x and of the equivariant moment map µ : Lie ( G ) × X → R for a path −→ k = { k t } t =0 in K with k = Id representing e k ∈ e K , we have ν x ( e k ) = − Z µ ( η t )( x ) dt = − Z ( α X ) y ( b η y ) dt = − i Z ddt ′ | t ′ = t det C (( k t ′ ) ∗ x ) det C (( k t ) ∗ x ) − dt = . = − varangle( { det C (( k t ) ∗ x ) } t =0 ) = − e v ( e k ) . Hence ν x equals e v on e K , and consequently ν x equals ̺ G on π ( G ) ∼ = π ( K ). Therefore the homo-geneization ν G of ν x equals − ̺ G on π ( G ) and this confers the equality ν G = − ̺ G on the whole group e G . A and I c on π ( Ham ( M, ω )) Now we prove Theorem 2.
Proof.
First we note the following equality due to Fujiki [41]. Given the bundle Z over J which has Z J := ( M, J ) for the fiber over J , denote by T Z , J the vertical bundle and take c ( K ) to be the Chern20orm of the vertical canonical bundle K relative to the Hermitian metric given by h ( J ) = g ( J ) − iω inthe fiber over J ∈ J , then Ω = Z fiber c ( K ) p ∗ ω n , (17)for p : Z → M the smooth projection map.The Hamiltonian fiber bundle over S corresponding to a loop γ = { φ t } t =0 in G based at theidentity can be described (cf. [71]) as P γ = M × D − ∪ Φ M × D + , where D − and D + are two copiesof the disk D and the gluing map Φ : ∂ ( M × D − ) ∼ = M × S → M × S ∼ = ∂ ( M × D + ) is given byΦ : ( x, t ) ( φ t x, t ).Note that given a Hamiltonian loop γ = { φ t } ≤ t ≤ the bundle P = P γ with a vertical compatiblecomplex structure is obtained by a map D : D → J representing a relative homotopy class in π ( J , G J )corresponding to the loop γ − = { ψ t = ( φ t ) − } ≤ t ≤ - that is ∂D : S → J is given by { ( ψ t ) ∗ J } t =0 .Note that P | D − with its fiberwise complex structure is equal to D ∗ Z . We denote by H t the zero-mean-normalized Hamiltonian for γ and by G t the zero-mean-normalized Hamiltonian for γ − . Thetwo are related by the formula G t ( x ) = − H t ( φ t x ).Moreover − I c ( γ ) = Z D Z fiber D ∗ c ( K ) u n . Since the coupling class u is represented by the form Υ := { ω on M × D + ; ω + d ( ψ ( r ) H t ( φ t x ) dt )) on M × D − } , we have − I c = Z D Z fiber D ∗ c ( K )( ω n + nd ( ψ ( r ) H t ( φ t x ) dt ω n − ))= Z D Z fiber D ∗ c ( K ) ω n + n Z D Z fiber D ∗ c ( K ) d ( ψ ( r ) H t ( φ t x ) dt ω n − ) . By the result of Fujiki the first summand equals R D D ∗ Ω. It is therefore enough to show that thesecond summand equals R dt R M S ( ψ t · J ) G t ( x ) ω n . The second summand satisfies n Z D Z fiber D ∗ c ( K ) d ( ψ ( r ) H t ( φ t x ) dt ω n − ) = n Z M × D d ( D ∗ c ( K ) ψ ( r ) H t ( φ t x ) dtω n − ) == n Z M × S H t ( φ t x ) D ∗ c ω n − dt =and by Equation 6 we have= Z Z M S ( ψ t · J ) H t ( φ t x ) ω n ( x ) dt = − Z Z M S ( ψ t · J ) G t ( x ) ω n ( x ) dt. Consequently we have I c ( γ ) = −A µ ( γ − ) = A ( γ ). G = Sp (2 n, R ) and the Maslov quasimor-phism In this section we would like to write out the finite-dimensional example more explicitly in the case G = Sp (2 n, R ) - for later use in particular. When G = Sp (2 n, R ) the maximal compact subgroup is K ∼ = U ( n ) and the space X = G/K has several guises. First it can be considered as the Siegel upperhalf-space [83] S n = { X + iY | X, Y ∈ Mat( n, R ) , X = X t , Y = Y t , Y > } ⊂ Mat( n, C ). Here there is21 natural Kahler form σ Siegel = trace( Y − dX ∧ Y − dY ) where the complex structure comes from theone on Mat( n, C ). This form is Kahler-Einstein with cosmological constant λ = − n +12 [83] - that is Ric ( σ Siegel ) = − n + 12 σ Siegel . (18)From which, since proportional metrics have equal Ricci forms, we have σ Siegel = 2 n + 1 σ Bergman , (19)for the Bergman Kahler form σ Bergman on X .Second, the space X = G/K can be considered as the space J c of ω std -compatible complex struc-tures on the symplectic vector space ( R n , ω std ). In this model, a natural symplectic form σ trace isgiven by ( σ trace ) J ( A, B ) = trace( JAB ) for J ∈ J c and A, B ∈ T J (J c ). A short computation based onthe fact that all G -invariant 2-forms on X are proportional gives σ trace = 12 σ Siegel , (20)under the natural isomorphisms J c ∼ = X , S n ∼ = X .By Examples 1 and 2 the Maslov quasimorphism τ Lin : e G → R restricting on e K to e v for v = det C on K ∼ = U ( n ) can be written as τ Lin = 2 n + 1 ν G,Bergman (21)in terms of ν G for σ X = σ Bergman . Therefore by Equation 19 τ Lin = ν G,Siegel (22)for σ X = σ Siegel , and by Equation 20 12 τ Lin = ν G, trace (23)for σ X = σ trace .Note that by [83] σ Siegel and consequently σ trace has non-positive sectional curvature. Moreover σ trace is Kahler-Einstein with cosmological constant − ( n + 1).Now consider X ∼ = J c with σ X = σ trace . By Equation 23, Lemma 1 and by the definition of ν x wehave − τ Lin = ν G ≃ ν x ( −→ g ) = R D −→ g σ X − R µ ( ξ t )( g t · x ) dt . Hence Z D −→ g σ X ≃ τ Lin ( −→ g ) + Z µ ( ξ t )( g t · x ) dt. (24)For later calculations we will want the moment map summand in this formula more explicit. Wewrite a formula for µ using the fact that it is an equivariant moment map for the action of thesemisimple Lie group Sp (2 n, R ) (cf. [57]) on S n . Note that an equivariant moment map for the actionof a semisimple Lie group on a symplectic manifold is unique [63]. Hence it is enough to show thefollowing. Lemma 3.
Consider the action of Sp (2 n, R ) on S n ∼ = J c with the invariant Kahler form σ trace . Thenit is Hamiltonian with the equivariant moment map µ S n : S n × sp (2 n, R ) → R given by µ S n ( J )(Ξ) = − trace (Ξ J ) .Proof. The symplectic form σ on S n can be described as σ ( A, B ) = trace( JAB ) using the isomorphism S n ∼ = J c - the space of complex structures on R n compatible with the standard symplectic form. Letus first compute the vector field Ξ generated by the infinitesimal action of Ξ. At a point J ∈ S n ,22enoting Φ t = exp ( t Ξ) ∈ Sp (2 n, R ) we have Ξ J = ddt | t =0 Φ t · J = ddt | t =0 Φ t J Φ − t = Ξ J − J Ξ = − [ J, Ξ].Then for B ∈ T J S n we compute d J (trace(Ξ J ))( B ) = trace(Ξ B ) . Finally, for B ∈ T J S n we have( i Ξ σ ) J ( B ) = σ J (Ξ J , B ) = − σ J ([ J, Ξ] , B ) = −
14 trace( J [ J, Ξ] B ) == −
14 trace( − J Ξ JB + J Ξ B ) = 14 trace(Ξ JBJ + Ξ B ) = 12 trace(Ξ B ) . The last expression equals − d J ( − trace(Ξ J ))( B ) as we have computed, and we’re done. ] Ham ( M, ω ) We shall now describe the local behaviour of the quasimorphism S - we compute its restriction tosubgroups G B ⊂ G of diffeomorphisms supported in embedded balls B in M , proving Theorem 3. Definition 2.6.1. (Embedded balls) We denote by U the set of embedded balls in M .Given a symplectic manifold ( M, ω ) with an almost complex structure J ∈ J with Hermitianscalar curvature S ( J ) of mean c = n R M c ( T M, ω ) ω n − / R M ω n , and B ∈ U an embedded ball in M ,we will show that the restriction ν B = S | G B of the quasimorphism S to G B = Ham c ( B, ω | B ) satisfies ν B = 12 τ B − c Cal , where τ B is the Barge-Ghys Maslov quasimorphism on G B = Ham c ( B n , ω std ) and Cal is the Calabihomomorphism.Since the quasimorphism S is homogenous and its distance from S J is bounded we can makecalculations with S J allowing for an error term that vanishes under homogenization. The proofconsists of writing ν B (using Section 2.5) as the sum of τ and a remainder term. Then we use somedifferential geometry to show that the remainder term equals a multiple of the Calabi homomorphism.For the differential geometry part we would like to use the canonical connection on the Hermitianmanifold ( M, ω, J ) that is defined by the following of its properties. It preserves ω and J and itstorsion has vanishing (1 , ∇ J = 0 , ∇ ω = 0 , T (1 , ∇ ≡ . (25)This connection has an equivalent definition in terms of ¯ ∂ -operators on complex vector bundles, whichis the one used in [27]. It is sometimes called ”the Chern connection”, and sometimes - ”the secondcanonical connection of Ehresmann-Liebermann” (cf. [48] Section 2, [58] and [87] Section 2).Consider B as a smooth embedding B : B n → M from the standard ball B n = { ( z , ..., z n ) | Σ nj =1 | z j | < } ⊂ C n to M . For purposes of trivialization and estimates choose for each two points x, y ∈ B a path γ x,y starting at x and ending at y , that depends continuously on ( x, y ) ∈ B × B where γ x,x is the constantpath at x for all x ∈ B . This can be achieved for example by taking linear segments in B n . Then wehave the following lemma. Lemma 4.
Let B − ⋐ B be any closed ball compactly contained in B . Then the following two statementshold by continuity and compactness of B − × B − . . For every one-form λ ∈ Ω ( B ) the function B − × B − → R given by ( x, y ) R γ x,y λ is boundedby a constant depending only on B, B − , λ .2. Given any connection ∇ ′ preserving ω and any fixed symplectic trivialization T B ∼ = V × B fora symplectic vector space ( V, ω V ) , the map B − × B − → Sp ( V, ω V ) obtained by means of thetrivialization by the parallel transport Γ γ x,y : T x B → T y B with respect to ∇ ′ has a compact imagein Sp ( V, ω V ) . Take a path { φ t } t =0 ⊂ G B with φ = Id . We shall now unwind the definition of ν B ( { φ t } t =0 ). Overeach x ∈ B we have the fiber S x of the bundle S → B . In S x we have the path ( φ t · J ) x . Now we shalldefine a path Φ( x ) t in Sp ( T x M, ω x ) associated to ( φ t ) ∗ x such that under the action of Sp ( T x M, ω x )on S x , we have Φ( x ) t · ( J ) x = ( φ t · J ) x = ( φ t ∗ φ − t x )( J ) φ − t x ( φ t ∗ φ − t x ) − .Indeed consider for each t ∈ [0 ,
1] the path γ x,φ − t x . The parallel transport along this path preserves J and maps Γ γ x,φ − t x : T x M → T φ − t x M . Then Φ( x ) t = ( φ t ∗ φ − t x ) ◦ Γ γ x,φ − t x : T x M → T x M is the re-quired map. Indeed Φ( x ) t · ( J ) x = Φ( x ) t ( J ) x Φ( x ) − t = ( φ t ∗ φ − t x )Γ γ x,φ − t x ( J ) x (Γ γ x,φ − t x ) − ( φ t ∗ φ − t x ) − =( φ t ∗ φ − t x )( J ) φ − t x ( φ t ∗ φ − t x ) − = ( φ t · J ) x , by preservation of J . Henceforth we omit the subscript z in ( J ) z whenever this is determined by the context.Then for all x ∈ B we have the loop δ ( x ) = { Φ( x ) t · J } t =0 J , Φ( x ) · J ]. We then for all x ∈ B choose a disk D ( x ) that bounds δ ( x ) - in fact one can construct D ( x ) as the geodesic join of { Φ( x ) t · J } t =0 with J - that is D ( x ) = S t [ J , Φ( x ) t · J ] properly parametrized. Denote γ t ( x ) =[ J , Φ( x ) t · J ]. Denote by β t ( x ) the path { Φ( x ) t ′ · J } t ′ = tt ′ =0 .Recall from Section 1.4 that a ≃ b denotes the equality of the functions a, b up to a function thatis bounded by a constant that does not depend on their arguments. Compute ν B ( { φ t } t =0 ) ≃ Z D Ω − Z µ ( X t )( φ t · J ) = Z B ( Z D ( x ) σ x ) ω n ( x ) − Z Z M S ( φ t · J ) H t ( x ) ω n ( x ) . (26)Now note that by Equation 24 and the definition of the moment map for the action of G = Sp (2 n ) on X = G/K Z D ( x ) σ x ≃ τ Lin ( { Φ( x ) t } t =0 ) + Z h ( x ) t (Φ( x ) t · ( J ) x ) dt, (27)where the function h ( x ) t ( · ) = µ S x (Ξ( x ) t )( · ), for Ξ( x ) t = ddτ | τ = t Φ( x ) τ ◦ Φ( x ) − t , is the contact Hamil-tonian for the canonical lifting of Φ( x ) t to the principal S -bundle of unit vectors in Λ N T S x simply byuse of the differential (cf. Equation 10). As a side remark it may be said, following [29], that this finite-dimensional moment map is the main reason for the existence of the corresponding infinite-dimensionalmoment map.Consequently, integrating over B with respect to the form ω n , we have ν B ( { φ t } t =0 ) ≃ · Z B τ Lin ( { Φ( x ) t } t =0 ) ω n ( x )+ Z B Z h ( x ) t (Φ( x ) t · J ) dt ω n ( x ) − Z Z M S ( φ t · J ) H t ( x ) ω n ( x ) . (28)By the definition of the Barge-Ghys Maslov quasimorphism on G and Lemma 4, the first term homog-enizes to τ B . Our goal is hence to compute the sum of the second and the third terms.By Lemma 3 we rewrite the second term in Equation 27 as Z h ( x ) t (Φ( x ) t · J ) = − Z trace (Ξ( x ) t (Φ( x ) t · J )) dt, (29)for Ξ( x ) t = ddτ | τ = t Φ( x ) τ ◦ Φ( x ) − t .Now note that instead of using the parallel transport along γ x,φ − t x to define Φ( x ) t : T x B → T x B wecould use the one along p x,t = { φ − t ′ x } tt ′ =0 to define the map Ψ( x ) t = ( φ t ∗ φ − t x ) ◦ Γ p x,t : T x B → T x B .Then we have Φ( x ) t = Ψ( x ) t U ( x, t ) , (30)24or the unitary map U ( x, t ) = Γ − p x,t ◦ Γ γ x,φ − t x : T x B → T x B . Form Υ( x ) t = ddτ | τ = t Ψ( x ) τ ◦ Ψ( x ) − t andΘ( x, t ) = ddτ | τ = t U ( x, τ ) ◦ U ( x, t ) − . Then by Equation 30 we haveΞ( x ) t = Ψ( x ) t Θ( x, t )Ψ( x ) − t + Υ( x ) t , (31)andΦ( x ) t · J = Φ( x ) t J Φ( x ) − t = Ψ( x ) t U ( x, t ) J U ( x, t ) − Ψ( x ) t − = Ψ( x ) t J Ψ( x ) t − = Ψ( x ) t · J , (32)because U ( x, t ) is J -linear.Therefore, by Equation (29) and noting thattrace(Ψ( x ) t Θ( x, t )Ψ( x ) − t (Ψ( x ) t · J )) = trace(Θ( x, t ) J )we have Z h ( x ) t (Φ( x ) t · J ) = − Z
12 trace(Θ( x, t ) · J ) dt − Z
12 trace(Υ( x ) t (Ψ( x ) t · J )) dt. (33)Note additionally, that 12 trace(Θ( x, t ) · J ) = − i trace C (Θ( x, t )) , considering Θ( x, t ) as a skew-Hermitian operator on the complex Hermitian space ( T x B, J , ω x ). More-over, trace C (Θ( x, t )) = Θ n ( x, t ) , where Θ n ( x, t ) = ddτ | τ = t U n ( x, τ ) ◦ U n ( x, t ) − , for U n ( x, t ) = (Γ np x,t ) − ◦ Γ nγ x,φ − t x : Λ n C T x B → Λ n C T x B, for the naturally induced parallel translations on the Hermitian complex line bundle Λ n C T B , endowing
T B with the Hermitian structure ( J , ω ) and the connection ∇ .Therefore Z h ( x ) t (Φ( x ) t · J ) = Z D B ( x ) ρ − Z trace(Υ( x ) t (Ψ( x ) t · J )) , (34)where iρ is the curvature two-form of the connection ∇ n on Λ n C ( T M, J ) naturally induced from ∇ on( T M, J ) and D B ( x ) is the disk spanned by S t =0 γ x,φ − t x . Note that ∂D B ( x ) = p x, γ x,φ − x . Now ρ | B ∈ Ω closed ( B, R ) has by the Poincar´e lemma a primitive α ∈ Ω ( B, R ) . Hence by Stokes’ formulawe have Z D B ( x ) ρ = Z p x, α − Z γ x,φ − x α. (35)Choosing B − ⋐ B such that supp ( φ t ) ⊂ B − for all t ∈ [0 , γ y,φ − y ≡ y for all y ∈ B \ B − ,hence by Lemma 4 we have the following uniform estimate for the second term | R γ x,φ − x α | ≤ C ( B − , B ) , for a constant C ( B − , B, α ) depending only on α , B − ⊃ S t =0 supp ( φ t ) and on B .Now denote ψ t = φ − t . Denote Y t the Hamiltonian vector generating ψ t . Recall that p x, = { φ − t x } t =0 = { ψ t x } t =0 . Hence the first term in Equation 35 satisfies Z p x, α = Z (( ψ t ) ∗ i Y t α ) x dt. Hence integrating Equation 34 over B we express Z B Z h ( x ) t (Φ( x ) t · J ) dt ω n ( x ) =25s = Z Z B ( ψ t ) ∗ i Y t α ω n dt − Z B Z trace(Υ( x ) t (Ψ( x ) t · J )) dtω n ( x ) + Bdd ( { φ t } t =0 ) (36)for a function Bdd ( { φ t } t =0 ) that satisfies | Bdd ( { φ t } t =0 ) | ≤ C ( B − , B, α )for a constant C ( B − , B ) depending only on α , B − ⊃ S t =0 supp ( φ t ) and on B .We shall now show that the first term in Equation 36 corrected by the moment map term R µ ( X t )( φ t · J ) dt in the definition of the quasimorhism is proportional to the Calabi homomorphism. After thatwe will show that the second term vanishes. Let G t (for each t ∈ [0 , ∂B and satisfies i Y t ω = − dG t . Then i Y t α ω n = nα i Y t ω ω n − = − nα dG t ω n − . Hence Z B ( ψ t ) ∗ i Y t α ω n = Z B i Y t α ω n = n Z B dG t α ω n − == − n Z B G t dαω n − = − n Z B G t dαω n − = − n Z B G t ρω n − =and by definition of the Hermitian scalar curvature we have= − Z B G t S ( J ) ω n =denoting H t (for each t ∈ [0 , ∂B and satisfies i X t ω = − dH t , andnoting that by the cocycle formula [72] G t ( x ) = − H t ( φ t x ) , we have= Z B S ( J ) H t ( φ t x ) ω n ( x ) . Hence Z dt Z B ( ψ t ) ∗ i Y t α ω n − Z dt Z M S ( φ t · J ) H t ( x ) ω n ( x ) = Z dt Z M S ( φ t · J )( H t − H t ) ω n =where we extend H t by zero from B to M , and noting that H t − H t depends on t only and equals themean R B H t ω n / R M ω n we have= − ( Z M S ( φ t · J ) ω n / Z M ω n ) Z dt Z B H t ω n = − c · Cal B ( { φ t } t =0 ) . (37)Now it remains to show that R R B trace(Υ( x ) t (Ψ( x ) t · J )) ω n ( x ) dt vanishes. First we would liketo note that since the (1 , T of ∇ vanishes, we have T ( X, J Y ) = T ( J X, Y ) (38)for all vector fields
X, Y on M . Moreover since ∇ preserves J we have J ∇ • X = ∇ • ( J X ) (39)for all vector fields X on M , where for a vector field Z on M , we denote by ∇ • Z the endomorphismof T M given by Y
7→ ∇ Y Z .For a vector field Z on M define then the endomorphism A Z of T M by A Z = L Z − ∇ Z . Then by[59], Vol 1, Appendix 6, page 292, we have A Z = −∇ • Z − T ( Z, · ) (40)26nd − trace A Z = div ω n ( Z ) , (41)where div ω n ( Z ) ∈ C ∞ ( M, R ) is defined bydiv ω n ( Z ) ω n = L Z ω n . Now we prove a formula relating the action of J on T M and the tensor A Z . We claim that for allvector fields X on M we have trace( A X J ) = trace( A J X ) . (42)Indeed − trace( A X J ) = trace( ∇ • X ◦ J + T ( X, J · )) =by Equation 40 = trace( J ∇ • X + T ( J X, · )) =by Equation 38 = trace( ∇ • J X + T ( J X, · )) = − trace A J X , by Equation 39.Let us now compute Υ( x ) t = ddτ | τ = t Ψ( x ) τ ◦ Ψ( x ) − t in terms of the connection and of the vectorfield X t generating the path of diffeomorphisms { φ t } t =0 . Recalling that Ψ( x ) t = ( φ t ∗ φ − t x ) ◦ Γ p x,t wehave ddτ | τ = t Ψ( x ) τ = ( φ t ∗ φ − t x )( L X t − ∇ X t ) φ − t ( x ) Γ p x,t . Consequently, Υ( x ) t = ( φ t ∗ φ − t x )( A X t ) φ − t ( x ) ( φ t ∗ φ − t x ) − (43)for the endomorphism A X t of T M . Thentrace(Υ( x ) t (Ψ( x ) t · J )) = trace(( φ t ∗ φ − t x )( A X t ) φ − t ( x ) ( φ t ∗ φ − t x ) − (( φ t ∗ φ − t x )Γ p x,t ( J ) x Γ − p x,t ( φ t ∗ φ − t x ) − )) == trace(( A X t ) φ − t ( x ) Γ p x,t ( J ) x Γ − p x,t ) = trace( A X t J )( φ − t ( x )) = trace( A J X t )( φ − t ( x )) , (44)by Equation 42. Hence Z Z B trace(Υ( x ) t (Ψ( x ) t · J )) ω n ( x ) dt = Z Z B trace( A J X t )( φ − t ( x )) ω n ( x ) dt == Z Z B trace( A J X t ) ω n dt = − Z Z B div( J X t ) ω n dt = 0 . (45)Therefore, assembling Equations 26,36,37, 45 and Definition 1.7.3 we have ν B ( { φ t } t =0 ) = 12 · τ B ( { φ t } t =0 ) − c · Cal B ( { φ t } t =0 ) + Bdd ( { φ t } t =0 ) , for a function Bdd ( { φ t } t =0 ) bounded by a constant C ( B − , B, α ) that depends only on B, α and B − ⊃ S t =0 supp ( φ t ). Noting that supp ( φ tk ) ⊂ supp ( φ t ) for every t ∈ [0 , , k ∈ Z and homogenizing,we finish the proof. 27 .7 The restriction to the Py quasimorphism In this section we prove the first point of Theorem 4 on the equality of the Py quasimorphism ofDefintion 1.7.4 and the general quasimorphism from Corollary 1 when the symplectic manifold (
M, ω )is monotone - that is c ( T M, ω ) = κ [ ω ] where κ = 0. The computation is somewhat similar to that ofthe local type - with the exception that there is no trivialization involved really.As in the computation of the local type, we use the parallel transport along p x,t = { φ − t ′ x } tt ′ =0 todefine the map Ψ( x ) t = ( φ t ∗ φ − t x ) ◦ Γ p x,t : T x B → T x B . Then Υ( x ) t = ddτ | τ = t Ψ( x ) τ ◦ Ψ( x ) − t willsatisfy Υ( x ) t = ( φ t ∗ φ − t x )( A X t ) φ − t ( x ) ( φ t ∗ φ − t x ) − for the endomorphism A X t of T M , for A X t = L X t − ∇ X t as in Equation 43. Thentrace(Υ( x ) t (Ψ( x ) t · J )) = trace( A J X t )( φ − t ( x )) , as before in Equation 44. Moreover, identically to Equation 45 we have Z M trace(Υ( x ) t (Ψ( x ) t · J )) ω n ( x ) = 0 . (46)We shall now rewrite S J ( { φ t } t =0 ) via Ψ t ( x ). For all x ∈ B we have the loop δ ( x ) = { Ψ( x ) t · J } t =0 J , Φ( x ) · J ]. We then for all x ∈ B choose a disk D ( x ) that bounds δ ( x ) - in fact onecan construct D ( x ) as the geodesic join of { Ψ( x ) t · J } t =0 with J - that is D ( x ) = S t [ J , Ψ( x ) t · J ]properly parametrized. Denote γ t ( x ) = [ J , Ψ( x ) t · J ]. Denote by β t ( x ) the path { Ψ( x ) t ′ · J } t ′ = tt ′ =0 .Compute S J ( { φ t } t =0 ) = Z D Ω − Z µ ( X t )( φ t · J ) = Z B ( Z D ( x ) σ x ) ω n ( x ) − Z Z M S ( φ t · J ) H t ( x ) ω n ( x ) . (47)Now as before by Equation 24 and the definition of the moment map for the action of G = Sp (2 n ) on X = G/K Z D ( x ) σ x ≃ τ Lin ( { Ψ( x ) t } t =0 ) − Z f ( x ) t (Ψ( x ) t · J ) dt (48)where the function f ( x ) t ( J ) = −
12 trace(Υ( x ) t J ) (49)is the contact Hamiltonian for the canonical lifting of Ψ( x ) t to the principal S -bundle of unit vectorsin Λ N C T S x , N = n ( n + 1) / S J ( { φ t } t =0 ) ≃ Z M τ Lin ( { Ψ( x ) t } t =0 ) ω n ( x ) − Z Z M S ( φ t · J ) H t ( x ) ω n ( x ) . (50)We shall now rewrite the Py quasimorphism S from Definition 1.7.4 via Ψ( x ) t . Then comparingthe effect of the difference in connections with the second term in Equation 50 we shall establish theequality.First we note that the connection ∇ gives us a parallel transport on L ( T M, ω ) and on P , since itpreserves J and ω . Moreover, since the map det : L ( T M, ω ) → P is defined using only J and ω the following diagram commutes. L ( T M, ω ) ( φ t ) − x Γ px,t −−−→ L ( T M, ω ) xdet ↓ det ↓ P φ t ) − x Γ px,t −−−→ P x (51)28n other words for L ∈ L ( φ t ) − x ( T M, ω ) we have det (Γ p x,t L ) = Γ p x,t det ( L ). It will be moreconvenient to compute S on the inverse path { ψ t = φ − t } t =0 . Indeed consider the paths det ( ψ t ∗ x L )and b ψ t ( det ( L )) in P for L ∈ L ( T M, ω ) x . These paths differ by an angle as follows det ( ψ t ∗ x ( L )) = e i πϑ ( t ) b ψ t ( det ( L ))Then the paths Γ p x,t det ( ψ t ∗ x L ) = det (Γ p x,t ψ t ∗ x L ) (here we use Equation 51) and Γ p x,t b ψ t ( det ( L ))in ( P ) x also differ by the same angle. And since these are paths in one fiber, we have angle ( L, { ψ t } t =0 ) = varangle ( { e i πϑ ( t ) } t =0 ) = varangle ( { det (Γ p x,t ψ t ∗ x L ) } t =0 ) − varangle ( { Γ p x,t b ψ t ( det ( L )) } t =0 ) . (52)Note that the second term in Equation 52 does not depend on the choice of L ∈ L ( T M, ω ) x , sinceboth Γ p x,t and b ψ t commute with rotations of the fibers. Therefore the function angle ( x, { ψ t } t =0 ) = inf L ∈L ( T M,ω ) x angle ( L, { ψ t } t =0 )satisfies angle ( x, { ψ t } t =0 ) = inf L ∈L ( T M,ω ) x ( varangle ( { det (Γ p x,t ψ t ∗ x L ) } t =0 )) − varangle ( { Γ p x,t b ψ t y } t =0 ) , (53)for any y ∈ ( P ) x . Note first that ψ t ∗ x = ( φ t ∗ ( φ t ) − x ) − and therefore Γ p x,t ψ t ∗ x = Ψ( x ) − t . Then notethat varangle ( { det (Γ p x,t ψ t ∗ x L ) } t =0 ) ≃ τ Lin ( { Ψ( x ) − t } t =0 ) = − τ Lin ( { Ψ( x ) t } t =0 ) (54)by the construction of the Maslov quasimorphism on the universal cover of the linear symplectic groupusing its action on the Lagrangian Grassmannian [7]. Thereforeinf L ∈L ( T M,ω ) x ( varangle ( { det (Γ p x,t ψ t ∗ x L ) } t =0 )) ≃ − τ Lin ( { Ψ( x ) t } t =0 ) . (55)It remains to interpret the integral over M with respect to ω n of the term varangle ( { Γ p x,t b ψ t y } t =0 )in Equation 53 via the Hermitian scalar curvature. For this purpose consider the two connection one-forms α and λ on P - where dα = 2 e ω and λ comes from the connection ∇ on T M and thereforesatisfies dλ = 2 e ρ (for a form η on M we denote by e η its lift by the natural projection P → M ). Theseconnection one-forms differ by e θ = α − λ for a one-form θ on M . Then denoting by Y t the Hamiltonianvector field generating { ψ t } with normalized Hamiltonian G t (by the zero mean condition), and by b Y t the vector field generating { b ψ t } we have varangle ( { Γ p x,t b ψ t y } t =0 ) = Z ( b ψ t ) ∗ i b Y t e θ ( x ) dt = Z ( ψ t ) ∗ i Y t θ ( x ) dt. (56)We now compute as follows Z M Z ( ψ t ) ∗ i Y t θ ( x ) dtω n ( x ) = Z Z M ( ψ t ) ∗ i Y t θω n dt = Z Z M i Y t θω n dt. (57)It is therefore sufficient to compute the integrand Z M i Y t θω n = n Z M θi Y t ωω n − = − n Z M θdG t ω n − = − n Z M dθG t ω n − == − n Z M ( ω − ρ ) G t ω n − = − n Z M G t ω n + 2 n Z M G t ρω n − = 2 n Z M G t ρω n − =29y the definition of the Hermitian scalar curvature= 2 Z M G t S ( J ) ω n =since G t ( x ) = − H t ( φ t x ) by the cocycle formula − Z M S ( J ) H t ( φ t x ) ω n ( x ) . (58)Therefore by Equations 53, 55, 56,58 we have from the definition of S (Definition 1.7.4) that − S ( { ψ t } t =0 ) ≃ − Z M τ Lin ( { Ψ( x ) t } t =0 ) ω n ( x ) + 2 Z M S ( J ) H t ( φ t x ) ω n ( x ) . (59)Therefore by Lemma 1 we have − S ( { φ t } t =0 ) ≃ S ( { ψ t } t =0 ) = Z M τ Lin ( { Ψ( x ) t } t =0 ) ω n ( x ) − Z M S ( J ) H t ( φ t x ) ω n ( x ) . (60)From Equations 50 and 60 we conclude that2 S J ≃ − S , which by homogenizing gives 2 S = − S P y finishing the proof.
Here we prove the second point of Theorem 4 on the agreement of the general quasimorphism ofCorollary 1 and the quasimorphism of Entov [33] from Definition 1.7.5. First we give an alternativedefinition of Entov’s quasimorphism along the lines of the definition of Py’s quasimorphism, which willmore easily be shown to agree with the general quasimorphism.
Definition 2.8.1. (A second definition of the quasimorphism S En ) Given a symplectic manifold( M, ω ) with c ( T M, ω ) = 0 one first trivializes the top exterior power Λ n C ( T M, J ) ∼ = C × M of( T M, ω, J ) for J ∈ J as a Hermitian line bundle. The square P of the unit frame bundle S × M ∼ = P S −−→ M of L = Λ n C ( T M, J, ω ) - that is the unitary frame bundle P of L ⊗ - admits a naturalmap det : L ( T M, ω ) → P from the Lagrangian Grassmannian bundle L ( T M, ω ), since L ( T M ) x = U ( T M x , ω x , J x ) /O ( n ). For a path −→ φ = { φ t } t =0 in G with φ = Id , choosing a point L ∈ L ( T M, ω ) x wehave the curve { φ t ∗ x ( L ) } ≤ t ≤ in L ( T M, ω ), and consequently the curve { det ( φ t ∗ x ( L )) } ≤ t ≤ in P .By means of the induced trivialization P ∼ = S × M this gives a continuous curve e i πϑ ( t ) : [0 , → S .Define angle ( L, −→ φ ) = varangle ( { e i πϑ ( t ) } t =0 ) = ϑ (1) − ϑ (0) , and then the function angle ( x, −→ φ ) = inf L ∈L ( T M,ω ) x angle ( L, −→ φ )is measurable and bounded on M and R ( −→ φ ) = Z M angle ( x, −→ φ ) ω n ( x )does not depend on homotopies of −→ φ with fixed endpoints, defining a quasimorphism R : e G → R . S En : e G → R , defined by S En ( e φ ) := lim k →∞ R ( e φ k ) k is a homogenous quasimorhismon e G that is independent of the non-canonical choices of trivialization, and of the almost complexstructure J . Proposition 2.1.
Definitions 1.7.5 and 2.8.1 for the Entov quasimorphism are equivalent.Proof (sketch).
Following Appendix C in [81] one notes that the trivialization of (
T M, ω, J ) over U = M \ Z can be chosen to agree with the restriction from M to U of a given trivialization ofΛ n C ( T M, J ). Then given a path −→ φ one immediately has ≃ equality of the two angle ( x, −→ φ ) functions on U −→ φ = M \ Z −→ φ by the construction of the Maslov quasimorphism on the universal cover of the linearsymplectic group using its action on the Lagrangian Grassmannian [7] and the commutativity of thediagram L ( T M, ω ) | U det −−−→ (Λ n C ( T M, J )) ⊗ U → b C | U ≀↓ ≀↓ q L ( b C n , ω std ) | U det −−−→ ( V n b C n ) ⊗ | U ∼ = b C | U , where b C is the trivial complex line bundle C × M over M , and all vector bundles are complex andHermitian.Now we turn to showing the equality S = S En . The proof is very similar to the one for the firstpoint of Theorem 4 and is even somewhat easier. Therefore we mostly outline the main steps andleave out details that are identical to those in Section 2.7.First we recall Equation 50 S J ( { φ t } t =0 ) ≃ Z M τ Lin ( { Ψ( x ) t } t =0 ) ω n ( x ) − Z Z M S ( φ t · J ) H t ( x ) ω n ( x ) . We also recall the commutation relation of Equation 51 L ( T M, ω ) ( φ t ) − x Γ px,t −−−→ L ( T M, ω ) xdet ↓ det ↓ P φ t ) − x Γ px,t −−−→ P x That is for L ∈ L ( φ t ) − x ( T M, ω ) we have det (Γ p x,t L ) = Γ p x,t det ( L ).It will be more convenient to compute R on the inverse path { ψ t = φ − t } t =0 . Indeed the path det ( ψ t ∗ x L ) in P gives by the trivialization a smooth angle function e i πϑ ( t ) : [0 , → S . The pathΓ p x,t : ( P ) ( φ t ) − x → ( P ) x also gives by the trivialization a smooth angle function e i πϕ ( x,t ) : [0 , → S . Noting the relation Γ p x,t det ( ψ t ∗ x L ) = det (Γ p x,t ψ t ∗ x L ) (by Equation 51), we have angle ( L, { ψ t } t =0 ) = varangle ( { e i πϑ ( t ) } t =0 ) = varangle ( { det (Γ p x,t ψ t ∗ x L ) } t =0 ) − varangle ( { e i πϕ ( x,t ) } t =0 ) . (61)Consequently, the function angle ( x, { ψ t } t =0 ) = inf L ∈L ( T M,ω ) x angle ( L, { ψ t } t =0 ) satisfies angle ( x, { ψ t } t =0 ) = inf L ∈L ( T M,ω ) x ( varangle ( { det (Γ p x,t ψ t ∗ x L ) } t =0 )) − varangle ( { e i πϕ ( x,t ) } t =0 ) . (62)Note first that ψ t ∗ x = ( φ t ∗ ( φ t ) − x ) − and therefore Γ p x,t ψ t ∗ x = Ψ( x ) − t . Then note that varangle ( { det (Γ p x,t ψ t ∗ x L ) } t =0 ) ≃ τ Lin ( { Ψ( x ) − t } t =0 ) = − τ Lin ( { Ψ( x ) t } t =0 ) (63)31y the construction of the Maslov quasimorphism on the universal cover of the linear symplectic groupusing its action on the Lagrangian Grassmannian [7]. Thereforeinf L ∈L ( T M,ω ) x ( varangle ( { det (Γ p x,t ψ t ∗ x L ) } t =0 )) ≃ − τ Lin ( { Ψ( x ) t } t =0 ) . (64)It remains to interpret the integral over M with respect to ω n of the term varangle ( { e i πϕ ( x,t ) } t =0 )in Equation 62 via the Hermitian scalar curvature. For this purpose note that the trivialization P ∼ = S × M is equivalent to a flat connection α on P without holonomy. Consider now the twoconnection one-forms α and λ on P - where in particular dα = 0 and λ comes from the connection ∇ on T M and therefore satisfies dλ = 2 e ρ (for a form η on M we denote by e η its lift by the naturalprojection P → M ). These connection one-forms differ by e θ = α − λ for a one-form θ on M .Then denoting by Y t the Hamiltonian vector field generating { ψ t } with Hamiltonian G t normalizedby the zero mean condition, and by b Y t the horizontal vector field that projects onto Y t generating thepath { b ψ t } of α -preserving diffeomorphism of P (in other words c ψ t = Id × ψ t in the trivialization P ∼ = S × M ) we have varangle ( { e i πϕ ( x,t ) } t =0 ) = Z ( b ψ t ) ∗ i b Y t e θ ( x ) dt = Z ( ψ t ) ∗ i Y t θ ( x ) dt. (65)We now compute as follows Z M Z ( ψ t ) ∗ i Y t θ ( x ) dtω n ( x ) = Z Z M ( ψ t ) ∗ i Y t θω n dt = Z Z M i Y t θω n dt. (66)It is therefore sufficient to compute the integrand Z M i Y t θω n = n Z M θi Y t ωω n − = − n Z M θdG t ω n − = − n Z M dθG t ω n − == 2 n Z M ρG t ω n − = 2 n Z M G t ρω n − = 2 n Z M G t ρω n − =by the definition of the Hermitian scalar curvature= 2 Z M G t S ( J ) ω n =since G t ( x ) = − H t ( φ t x ) by the cocycle formula= − Z M S ( J ) H t ( φ t x ) ω n ( x ) . (67)Therefore by Equations 62, 64, 65,67 we have from the definition of R that R ( { ψ t } t =0 ) ≃ − Z M τ Lin ( { Ψ( x ) t } t =0 ) ω n ( x ) + 2 Z M S ( J ) H t ( φ t x ) ω n ( x ) . (68)Therefore by Lemma 1 we have R ( { φ t } t =0 ) ≃ − R ( { ψ t } t =0 ) = Z M τ Lin ( { Ψ( x ) t } t =0 ) ω n ( x ) − Z M S ( J ) H t ( φ t x ) ω n ( x ) . (69)From Equations 50 and 69 we conclude that2 S J ≃ R , which by homogenizing gives 2 S = S En finishing the proof. 32 .9 Calibrating the L norm Here we derive Equation 8.Note that the second summand of S J ( −→ φ ) = R D −→ φ Ω − R S ( J ) H t ( φ t x ) ω n ( x ) dt satisfies | Z S ( J ) H t ( φ t x ) ω n ( x ) dt | ≤ || S ( J ) || L q ( M,ω n ) · Z || H t || L p ( M,ω n ) dt (70)where 1 ≤ p, q ≤ ∞ and 1 /p + 1 /q = 1 and is therefore bounded by C p ||−→ φ t || k,p for every k ≥ ≤ p ≤ ∞ .Let us turn to the first summand R D −→ φ Ω. First note that since on the Siegel upper half-space S n thenatural invariant Kahler form σ S n has a primitive λ S n that is bounded by a constant C ( n ) with respectto the metric induced by ( σ S n , j S n ) and vanishes on geodesics starting at iId , the infinite-dimensionalspace ( J , Ω , J ) also has a primitive Λ for Ω that is bounded with respect to the metric induced by(Ω , J ) by the constant C ( n, ω ) = C ( n )Vol( M, ω n ) / and vanishes on geodesics starting at J . That is | Λ(Υ) | ≤ C ( n )Vol( M, ω n ) / Ω(Υ , J Υ) / , for a vector Υ ∈ T J J . In that case R D −→ φ Ω = R −→ φ · J Λ = R Λ φ t · J (( φ t ) ∗ L X t J ) dt and consequently | Z D −→ φ Ω | ≤ C ( n, ω ) Z Ω φ t · J (( φ t ) ∗ L X t J , ( φ t · J )(( φ t ) ∗ L X t J )) / dt ≤≤ C ′ ( n, ω ) Z ( Z M trace (( φ t ) ∗ ( L X t J ) ) ω n ) / dt == C ′ ( n, ω ) Z ( Z M trace (( L X t J ) ) ω n ) / dt ≤≤ C ′′ ( n, ω, J ) Z ( Z M ( | X t | + |∇ X t | ) ω n ) / dt ≤ C (3) ( n, ω, J ) ||−→ φ || , . (71)Therefore by Equations 71 and 70 we have for all e φ ∈ e G the estimate S J ( e φ ) ≤ C ( n, ω, J ) || e φ || , . (72)
1. It was shown by Donaldson in [28, 29] that G acts in a Hamiltonian way on additional spaces(e.g. spaces of submanifolds/cycles). These may yield more homomorphisms π ( Ham ) → R bythe Action-homomorphism construction for equivariant moment maps, and perhaps new quasi-morphisms on e G . Moreover, Futaki shows in [44] that the space J int ⊂ J of integrable almostcomplex structures can be endowed with additional symplectic structures that give different mo-ment maps for the action of G , from which the Bando-Futaki invariants F c k are obtained whenrestricting to the subgroup G J . It would be interesting to extend the methods of Futaki to all J , taking care of the Nijenhuis tensor, and to check two things. First it is most likely that thecorresponding Action-homomorphisms on π ( G ) will coincide with the invariants I c k (cf. [61])obtained by integrating the k -th vertical Chern class times u n +1 − k in Definition 1.7.1. Second,it would be interesting to extend the perturbation of Futaki to incorporate such invariants as I c c corresponding to symmetric polynomials that are not elementary.33. It is interesting to note that the Entov quasimorphism (Defintions 1.7.5, 2.8.1) is defined on theextension H = Symp ( M, ω ) of the group G = Ham ( M, ω ), while the moment map picture iscurrently stated for the action of G on J only. It is therefore interesting to check whether in thecase c ( T M, ω ) = 0 the moment map for the action of G on J extends to a moment map for theaction of H on J - along the lines of [29] for example. It would also be interesting to investigatethe possibility of extending the moment map this way without conditions on c ( T M, ω ) - toprovide an extension when it is possible and to investigate the obstructions to extending whenthe extension is not possible. This may well be related to the Flux homomorphism.3. It is interesting to investigate the restriction A µ of S to π G for symplectic manifolds ( M, ω ) offinite volume that are not closed. Does this restriction have an interpretation like I c in terms ofcharacteristic numbers of the associated Hamiltonian vector bundle? It would also be interestingto say something new about the Entov quasimorphism in the new interpretation - can it becomputed for example for the new symplectic manifolds constructed by Fine and Panov ([39]and references therein)?4. It would be interesting to compare the general principle for generating quasimorphisms intro-duced in this paper with other general constructions of quasimorphisms. While the relation tothe Burger-Iozzi-Wienhard construction of the rotation number from [14] is at least intuitivelyrelatively simple to trace, the relation to the works of Ben Simon and Hartnick [86, 84, 85] (cf.Calegari [20]) is somewhat more mysterious, since there seems to be no straightforward analogueof the Shilov boundary for the space ( J , Ω , J ) of compatible almost complex structures on ( M, ω ).Hence it is an interesting question to exhibit a specific explicit invariant partial order or posetthat gives the quasimorphism S on ] Ham ( M, ω ).5. From a general philosophical point of view the action of G = Ham ( M, ω ) on J with Donaldson’sequivariant moment map allows one to consider G in its C -topology as a generalized HermitianLie group with a generalized Hermitian symmetric space of non-compact type - in a way it behavessimilarly to Sp (2 n, R ), which would be a ”Hermitian” feature of G . In comparison, the group G with the Hofer metric and related invariants is known to exhibit certain ”hyperbolic features”(cf. [73]) - shared with Gromov-hyperbolic finitely generated groups. This approach can beused to study the representations into G of fundamental groups of compact Kahler manifolds -e.g. Riemann surfaces of genus at least 2. It is easy to construct the analogue of the Toledoinvariant for representations of surface groups (using the bounded 2-cocycle of Reznikov [77, 78,79] that equals the differential of S J - which corresponds to the ”bounded Kahler class”) thatsatisfies a corresponding Milnor-Wood type inequality (this can for example be proven using thequasimorphism S J ). One could then check which values of the Toledo invariant can be attained- note that this value will be I c on a certain loop γ ρ associated with the representation ρ ,and hence for Kahler-Einstein manifolds is conjectured to vanish [81] - this holds for example on( C P n , ω F S ) [34, 36]. These methods could possibly be used to obtain restrictions on Hamiltonianactions of such groups, which would be complementary to those established by Polterovich (cf.[73]), since surface groups are undistorted. In particular the notion of maximal representations(following works of Burger-Iozzi-Wienhard and others cf. [13] for a survey) could be defined andtheir properties studied. The above-mentioned works of Ben Simon and Hartnick could again beof some use.Note also that while certain embeddings of right-angled Artin groups (and hence of most surfacegroups) into G of any symplectic manifold were constructed by Kapovich in [56] these repre-sentations will have zero Toledo invariant. Indeed these constructions either factor through thesubgroup G B of diffeomorphisms supported in a ball, where the restriction of the quasimorphismto π is trivial (c.f. definition 1.7.3 of the Barge-Ghys average Maslov quasimorphism) or takevalues in G of a surface of genus g , where the restriction I c vanishes since π ( G ) is trivial (ortorsion for the sphere). The surface can also have boundary - the Toledo invariant will stillvanish, by the embedding functoriality (Proposition 1.2). However, it is quite an easy fact that34ince ] Ham ( M, ω ) for closed M is perfect by a theorem of Banyaga [5], every element γ ∈ π Ham is of the form γ = γ ρ for some representation ρ : π (Σ g ) → Ham (one can take g to be the com-mutator length of γ ∈ ] Ham ). Hence for M = Bl ( C P ), say, there is a nonzero Toledo invariantrepresentation, the corresponding class in π represented by a toric loop. It would therefore beinteresting to write this class explicitly as a product of commutators in ] Ham .6. Another interesting computation to make is that of S on Hamiltonian paths generated by a time-independent (autonomous) Hamiltonian. This would give a quasi-state -type functional (cf. e.g.[35, 74]) on C ∞ ( M, R ) corresponding to the quasimorphism S . This functional would retain theproperties of linearity on Poisson-commutative subspaces and Symp ( M, ω )-invariance, howeverit would not be monotone (since this would imply continuity in the L ∞ -norm) or vanish onfunctions with supports displaceable by Hamiltonian isotopies. In particular, it would be curiousto find a formula for the value of this quasi-state on Morse functions on the manifold in termsof local data around the critical points, similarly to what was computed by Py in his thesis [76]for the case of the two-sphere S . Here Equation 50 could be very useful. 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