Cohomologically symplectic solvmanifolds are symplectic
aa r X i v : . [ m a t h . S G ] M a r COHOMOLOGICALLY SYMPLECTIC SOLVMANIFOLDSARE SYMPLECTIC
HISASHI KASUYA
Abstract.
We consider aspherical manifolds with torsion-free virtuallypolycyclic fundamental groups, constructed by Baues. We prove that ifthose manifolds are cohomologically symplectic then they are symplectic.As a corollary we show that cohomologically symplectic solvmanifoldsare symplectic. Introduction
A 2 n -dimensional compact manifold M is called cohomologically symplec-tic (c-symplectic) if we have ω ∈ H ( M, R ) such that ω n = 0. A compactsymplectic manifold is c-symplectic but the converse is not true in general.For example C P C P is c-symplectic but not symplectic. But for someclass of manifolds these two conditions are equivalent. For examples, nil-manifolds i.e. compact homogeneous spaces of nilpotent simply connectedLie group. In [7], for a nilpotent simply connected Lie group G with a co-compact discrete subgroup Γ (such subgroup is called a lattice), Nomizushowed that the De Rham cohomology H ∗ ( G/ Γ , R ) of G/ Γ is isomorphicto the cohomology H ∗ ( g ) of the Lie algebra of G . By the application ofNomizu’s theorem, if G/ Γ is c-symplectic then G/ Γ is symplectic (see [3,p.191]). Every nilmanifold can be represented by such G/ Γ (see [6]).Consider Solvmanifolds i.e. compact homogeneous spaces of solvable sim-ply connected Lie groups. Let G be a solvable simply connected Lie groupwith a lattice Γ. We assume that for any g ∈ G the all eigenvalues of theadjoint operator Ad g are real. With this assumption, in [5] Hattori extendedNomizu’s theorem. By Hattori’s theorem, for such case, without difficulty,we can similarly show that if G/ Γ is c-symplectic, then G/ Γ is symplectic.But the isomorphism H ∗ ( G/ Γ , R ) ∼ = H ∗ ( g ) fails to hold for general solvableLie groups, and not all solvmanifolds can be represented by G/ Γ. Thus it is aconsiderable problem whether every c-symplectic solvmanifold is symplectic.Let Γ be a torsion-free virtually polycyclic group. In [1] Baues constructedthe compact aspherical manifold M Γ with π ( M Γ ) = Γ. Baues proved thatevery infra-solvmanifold (see [1] for the definition) is diffeomorphic to M Γ . Key words and phrases. cohomologically symplectic, solvmanifold, polycyclic group.
In particular the class of such aspherical manifolds contains the class ofsolvmanifolds. We prove that if M Γ is c-symplectic then M Γ is symplectic. Inother words, for a torsion-free virtually polycyclic group Γ with 2 n = rank Γ,if there exists ω ∈ H (Γ , R ) such that ω n = 0 then we have a symplecticaspherical manifold with the fundamental group Γ.2. Notation and conventions
Let k be a subfield of C . A group G is called a k -algebraic group if G is a Zariski-closed subgroup of GL n ( C ) which is defined by polynomialswith coefficients in k . Let G ( k ) denote the set of k -points of G and U ( G )the maximal Zariski-closed unipotent normal k -subgroup of G called theunipotent radical of G . Let U n ( k ) denote the n × n k -valued upper triangularunipotent matrix group.3. Aspherical manifolds with torsion-free virtually polycyclicfundamental groupsDefinition 3.1.
A group Γ is polycyclic if it admits a sequenceΓ = Γ ⊃ Γ ⊃ · · · ⊃ Γ k = { e } of subgroups such that each Γ i is normal in Γ i − and Γ i − / Γ i is cyclic. Wedenote rank Γ = P i = ki =1 rank Γ i − / Γ i . Proposition 3.2. ([8, Proposition 3.10])
The fundamental group of a solv-manifold is torsion-free polycyclic.
Let k be a subfield of C . Let Γ be a torsion-free virtually polycyclic group.For a finite index polycyclic subgroup ∆ ⊂ Γ, we denote rank Γ = rank ∆.
Definition 3.3.
We call a k -algebraic group H Γ a k -algebraic hull of Γif there exists an injective group homomorphism ψ : Γ → H Γ ( k ) and H Γ satisfies the following conditions:(1) ψ (Γ) is Zariski-dense in H Γ .(2) Z H Γ ( U ( H Γ )) ⊂ U ( H Γ ) where Z H Γ ( U ( H Γ )) is the centralizer of U ( H Γ ).(3) dim U ( H Γ )=rank Γ. Theorem 3.4. ([1, Theorem A.1])
There exists a k -algebraic hull of Γ anda k-algebraic hull of Γ is unique up to k -algebraic group isomorphism. Let Γ be a torsion-free virtually polycyclic group and H Γ the Q -algebraichull of Γ. Denote H Γ = H Γ ( R ). Let U Γ be the unipotent radical of H Γ and T a maximal reductive subgroup. Then H Γ decomposes as a semi-directproduct H Γ = T ⋉ U Γ . Let u be the Lie algebra of U Γ . Since the exponentialmap exp : u −→ U Γ is a diffeomorphism, U Γ is diffeomorphic to R n suchthat n = rank Γ. For the semi-direct product H Γ = T ⋉ U Γ , we denote OHOMOLOGICALLY SYMPLECTIC SOLVMANIFOLDS 3 φ : T → Aut( U Γ ) the action of T on U Γ . Then we have the homomorphism α : H Γ −→ Aut( U Γ ) ⋉ U Γ such that α ( t, u ) = ( φ ( t ) , u ) for ( t, u ) ∈ T ⋉ U Γ .By the property (2) in Definition 3.3, φ is injective and hence α is injective.In [1] Baues constructed a compact aspherical manifold M Γ = α (Γ) \ U Γ with π ( M Γ ) = Γ. We call M Γ a standard Γ-manifold. Theorem 3.5. ([1, Theorem 1.2, 1.4])
A standard Γ -manifold is unique upto diffeomorphism. A solvmanifold with the fundamental group Γ is diffeo-morphic to the standard Γ -manifold M Γ . Let A ∗ ( M Γ ) be the de Rham complex of M Γ . Then A ∗ ( M Γ ) is the set ofthe Γ-invariant differential forms A ∗ ( U Γ ) Γ on U Γ . Let ( V u ∗ ) T be the left-invariant forms on U Γ which are fixed by T . Since Γ ⊂ H Γ = U Γ · T , we havethe inclusion ( ^ u ∗ ) T = A ∗ ( U Γ ) H Γ ⊂ A ∗ ( U Γ ) Γ = A ∗ ( M Γ ) . Theorem 3.6. ([1, Theorem 1.8])
This inclusion induces an isomorphismon cohomology.
By the application of the above facts, we prove the main theorem of thispaper.
Theorem 3.7.
Suppose M Γ is c-symplectic. Then M Γ admits a symplecticstructure. In particular cohomologically symplectic solvmanifolds are sym-plectic.Proof. Since we have the isomorphism H ∗ ( M Γ , R ) ∼ = H ∗ (( V u ∗ ) T ), we have ω ∈ ( V u ∗ ) T such that 0 = [ ω ] n ∈ H n (( V u ∗ ) T ). This gives 0 = ω n ∈ ( V u ∗ ) T and hence 0 = ω n ∈ V u ∗ . Since ω n is a non-zero invariant 2 n -formon U Γ , we have ( ω n ) p = 0 for any p ∈ U Γ . Hence by the inclusion ( V u ∗ ) T ⊂ A ∗ ( U Γ ) T = A ∗ ( M Γ ), we have ( ω n ) Γ p = 0 for any Γ p ∈ Γ \ U Γ = M Γ . Thisimplies that ω is a symplectic form on M Γ . Hence we have the theorem. (cid:3) remarks Let G = R ⋉ φ U ( C ) such that φ ( t ) · x z y = e iπt · x z e − iπt · y , and D = Z ⋉ φ D ′ with D ′ = x + ix z + iz y + iy : x , y , z ∈ Z , x , y , z ∈ R . HISASHI KASUYA
Then D is not discrete and G/D is compact. We have
D/D ∼ = Z ⋉ ϕ U ( Z )such that ϕ ( t ) · x z y = − t x z − − t y , where D is the identity component of D . Denote Γ = D/D . We have thealgebraic hull H Γ = {± } ⋉ ψ ( U ( R ) × R ) such that ψ ( − · x z y , t = − x z − y , t . The dual of the Lie algebra u of U ( R ) × R is given by u ∗ = h α, β, γ, δ i suchthat the differential is given by dα = dβ = dδ = 0 ,dγ = − α ∧ β, and the action of {± } is given by( − · α = − α, ( − · β = − β, ( − · γ = γ, ( − · δ = δ. Then we have a diffeomorphism M Γ ∼ = G/D and an isomorphism H ∗ ( M Γ , R ) ∼ = H ∗ (( V u ∗ ) {± } ). By simple computations, H (( V u ∗ ) {± } ) = 0 and hencethe solvmanifold G/D is not symplectic.
Remark . The proof of the Theorem 3.7 contains a proof of the followingproposition.
Proposition . If M Γ admits a symplectic structure, then U Γ has an in-variant symplectic form. Otherwise for the above example, U Γ = U ( R ) × R has an invariant sym-plectic form but M Γ is not symplectic. Thus the converse of this propositionis not true. If Γ is nilpotent, then T is trivial and any invariant symplecticform on U Γ induces the symplectic form on M Γ . Hence for nilmanifolds theconverse of Proposition 4.1 is true. Remark . Γ is a finite extension of a lattice of U Γ = U ( R ) × R . Hence M Γ is finitely covered by a Kodaira-Thurston manifold (see [9], [3, p.192]). M Γ is an example of a non-symplectic manifold finitely covered by a symplecticmanifold. OHOMOLOGICALLY SYMPLECTIC SOLVMANIFOLDS 5
Let H = G × R . Then the dual of the Lie algebra h of H is given by h ∗ = h σ, τ, ζ , ζ , η , η , θ , θ i such that the differential is given by dσ = dτ = 0 ,dζ = τ ∧ ζ , dζ = − τ ∧ ζ ,dη = τ ∧ η , dη = − τ ∧ η ,dθ = − ζ ∧ η + ζ ∧ η , dθ = − ζ ∧ η − ζ ∧ η . By simple computations, any closed invariant 2-form ω ∈ V h ∗ satisfies ω = 0. Hence H has no invariant symplectic form. Otherwise we have alattice ∆ = 2 Z ⋉ U ( Z + i Z ) × Z which is also a lattice of R × U ( C ). Thus H/ ∆ is diffeomorphic to a direct product of a 2-dimensional torus and anIwasawa manifold (see [4]). Since an Iwasawa manifold is symplectic (see[4]), H/ ∆ is also symplectic. By this example we can say: Remark . For a simply connected nilpotent Lie group G with lattice Γ, ifthe nilmanifold G/ Γ is symplectic then G has an invariant symplectic form.But suppose G is solvable we have an example of a symplectic solvmanifold G/ Γ such that G has no invariant symplectic form. Acknowledgements.
The author would like to express his gratitude to Toshitake Kohno forhelpful suggestions and stimulating discussions. This research is supportedby JSPS Research Fellowships for Young Scientists.
References [1] O. Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraicgroups. Topology (2004), no. 4, 903–924.[2] A. Borel, Linear algebraic groups 2nd enl. ed Springer-verlag (1991).[3] Y. F´elix, J. Oprea and D. Tanr´e, Algebraic Models in Geometry, Oxford GraduateTexts in Mathematics 17, Oxford University Press 2008.[4] M. Fernandez, A. Gray, The Iwasawa manifold. Differential geometry, Peniscola 1985,157–159, Lecture Notes in Math., 1209, Springer, Berlin, 1986.[5] A. Hattori, Spectral sequence in the de Rham cohomology of fibre bundles. J. Fac.Sci. Univ. Tokyo Sect. I , (1949). 9–32.[7] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Liegroups. Ann. of Math. (2) , (1954). 531–538.[8] M. S. Raghunathan, Discrete subgroups of Lie Groups, Springer-Verlag, New York,1972.[9] W. P. Thurston, Some simple examples of symplectic manifolds. Proc. Amer. Math.Soc. (1976), no. 2, 467–468.(H.kasuya) Graduate school of mathematical science university of tokyojapan
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