Topology of compact solvmanifolds
Sergio Console Topology of compact solvmanifolds Aims
Nilpotent and solvable Nilmanifolds joint work with A. Fino, M. Macrì, G. Ovando, M. Subils Solvmanifolds de Rham Cohomology Nilmanifolds Rosario (Argentina) – July 2012 Solmanifolds Main Theorem Proof of the Main Theorem
Applications Nakamura manifold Almost abelian 6 dim almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface The oscillator group
Sergio Console 1 Dipartimento di Matematica Università di Torino Topology of compact solvmanifolds 1 Aims Sergio Console
2 Nilpotent and solvable Lie groups Aims
Nilmanifolds Nilpotent and solvable Solvmanifolds Nilmanifolds Solvmanifolds
de Rham Cohomology 3 de Rham Cohomology Nilmanifolds de Rham cohomology of nilmanifolds Solmanifolds Main Theorem de Rham cohomology of solvmanifolds Proof of the Main Theorem Applications 4 Main Theorem Nakamura manifold Almost abelian Proof of the Main Theorem 6 dim almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface 5 Applications The oscillator group Nakamura manifold Almost abelian 6-dimensional almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface The oscillator group 2 Topology of compact M = G/Γ: solvmanifold solvmanifolds Sergio Console G: real simply connected solvable Lie group Γ: lattice (discrete cocompact subgroup) Aims Nilpotent and solvable Nilmanifolds Solvmanifolds
Aims: find lattices, compute de Rham cohomology, de Rham Cohomology study existence of symplectic structures, Nilmanifolds Solmanifolds hard Lefschetz property, formality Main Theorem Proof of the Main Theorem We will explain: Applications Nakamura manifold Almost abelian • the difference with nilmanifolds 6 dim almost abelian Kähler and symplectic structures on solvmanifolds • known results on the computation of Rham cohomology in Hyperelliptic surface special cases (completely solvable, Mostow condition) The oscillator group • a method to compute the de Rham cohomology in general (following results by Guan and Witte) • applications: Nakamura manifold, almost abelian solvable Lie groups, hyperelliptic surface, three families of lattices on the 3 oscillator group Topology of compact Nilpotent and solvable Lie groups solvmanifolds
Sergio Console
G is k-step nilpotent ⇐⇒ Aims
the descending chain of normal subgroups Nilpotent and solvable Nilmanifolds G = G ⊃ G = [G, G] ⊃ · · · ⊃ G = [G , G] ⊃ .... Solvmanifolds 0 1 i+1 i de Rham Cohomology Nilmanifolds Solmanifolds degenerates, i.e. Gi = {e} ∀i ≥ k,(e is the identity element). Main Theorem Proof of the Main Theorem
G is k-step solvable ⇐⇒ Applications Nakamura manifold the derived series of normal subgroups Almost abelian 6 dim almost abelian Kähler and symplectic G(0) = G ⊃ G(1) = [G, G] ⊃ · · · ⊃ G(i+1) = [G(i), G(i)] ⊃ .... structures on solvmanifolds Hyperelliptic surface The oscillator group degenerates.
In particular a solvable Lie group is completely solvable if every eigenvalue λ of every operator Ad g, g ∈ G, is real.
Note that a nilpotent Lie group is completely solvable. 4 Topology of compact Nilmanifolds solvmanifolds
Sergio Console G simply connected nilpotent Lie group Aims
Nilpotent and solvable Recall: exp : g → G is a diffeomorphism Nilmanifolds Solvmanifolds
de Rham Cohomology For nilpotent Lie groups there is a simple criterion for the Nilmanifolds existence of lattices: Solmanifolds Main Theorem Proof of the Main Theorem
Theorem (Malcev)ˇ Applications Nakamura manifold G simply connected nilpotent Lie group Almost abelian 6 dim almost abelian ∃ Γ lattice on G ⇐⇒ Kähler and symplectic structures on solvmanifolds the Lie algebra g of G has a basis such that the structure Hyperelliptic surface constants in this basis are rational ⇐⇒ The oscillator group
∃ gQ such that g = gQ ⊗ R
If g = gQ ⊗ R, one also says that g has a rational structure
A nilmanifold is the a quotient M = G/Γ, where G is a real 5 simply connected nilpotent Lie group and Γ is a lattice. Topology of compact Solvmanifolds solvmanifolds There is no simple criterion for the existence of a lattice in a Sergio Console connected and simply-connected solvable Lie group G. Aims Here are some necessary criteria. Nilpotent and solvable Nilmanifolds Solvmanifolds
Proposition (Milnor) de Rham Cohomology Nilmanifolds If G admits a lattice then it is unimodular. [tr adX = 0, ∀X ∈ g] Solmanifolds Main Theorem Proof of the Main Theorem
Applications The Mostow bundle Nakamura manifold Almost abelian Let G/Γ be a solvmanifold that is not a nilmanifold. 6 dim almost abelian Kähler and symplectic N = nilradical of G =largest connected nilpotent normal subgroup of G. structures on solvmanifolds Hyperelliptic surface Then ΓN := Γ ∩ N is a lattice in N, ΓN = NΓ is closed in G and The oscillator group G/(NΓ) =: Tk is a torus. =⇒ we have the fibration:
k N/ΓN = (NΓ)/Γ ,→ G/Γ −→ G/(NΓ) = T
6 Much of the rich structure of solvmanifolds is encoded in this bundle. The nilradical has an important rôle in the study of solvmanifolds. Topology of compact Solvmanifolds solvmanifolds
Sergio Console G connected and simply connected solvable Lie group
Aims diffeo n =⇒ G ' R Nilpotent and solvable Nilmanifolds (BUT exp : g → G is not necessarily injective or surjective. ) Solvmanifolds ∼ de Rham Cohomology =⇒ solvmanifolds G/Γ are aspherical and π1(G/Γ) = Γ. Nilmanifolds Solmanifolds
The fundamental group plays an important rôle: Main Theorem Proof of the Main Theorem Diffeomorphism Theorem Applications Nakamura manifold G /Γ and G /Γ solvmanifolds and Almost abelian 1 1 2 2 6 dim almost abelian ϕ :Γ → Γ isomorphism. Kähler and symplectic 1 2 structures on solvmanifolds =⇒ ∃ diffeomorphism Φ G1 → G2 such that Hyperelliptic surface The oscillator group
(i) Φ|Γ1 = ϕ,
(ii) ∀γ∈Γ1 ∀p∈G1 Φ(pγ) = Φ(p)ϕ(γ).
Corollary 7 Two solvmanifolds with isomorphic π1 are diffeomorphic. Topology of compact de Rham Cohomology solvmanifolds
Sergio Console
Aims
Nilpotent and solvable G/Γ solvmanifold (nilmanifold) Nilmanifolds Solvmanifolds
de Rham Cohomology General question: can one compute Dolbeault cohomology of Nilmanifolds M by invariant forms, Solmanifolds i.e., using the Chevalley-Eilenberg complex: Main Theorem Proof of the Main Theorem
Applications k−1 ∗ d k ∗ d k+1 ∗ Nakamura manifold · · · → Λ g → Λ g → Λ g → ... Almost abelian 6 dim almost abelian Kähler and symplectic X i+j ˆ ˆ structures on solvmanifolds dα(x1,..., xk+1) = (−1) α([xi , xj ], x1,... xi ,..., xj ,..., xk+1) Hyperelliptic surface i ∗ ∼ ∗ So, when is HdR(G/Γ) = H (g)? 8 Topology of compact de Rham Cohomology of nilmanifolds solvmanifolds Sergio Console Aims Nilpotent and solvable G/Γ nilmanifold Nilmanifolds Solvmanifolds de Rham Cohomology Theorem (Nomizu) Nilmanifolds Solmanifolds H∗ (G/Γ) =∼ H∗(g). Main Theorem dR Proof of the Main Theorem Applications Nakamura manifold V∗ Almost abelian g is a minimal model of G/Γ (in the sense of Sullivan). 6 dim almost abelian Kähler and symplectic structures on solvmanifolds ∗ V g is formal ⇐⇒ G is abelian and G/Γ is a torus. Hyperelliptic surface The oscillator group If a nilmanifold is Kählerian, then it is a torus. [Benson & Gordon, Hasegawa] 9 Topology of compact Idea of the proof of Nomizu’s Theorem solvmanifolds Sergio Console Aims Suppose G is k-step nilpotent. g` Lie algebra of G`. Nilpotent and solvable Nilmanifolds Let H := simply-connected Lie subgroup of G with Lie algebra Solvmanifolds ∼ n de Rham Cohomology h = gk−1 =⇒ H central and H = R . Nilmanifolds Solmanifolds We have the fibration: Main Theorem Proof of the Main Theorem π n = H/H ∩ Γ ,→ M = G/Γ → M = G/HΓ Applications T Nakamura manifold Almost abelian p,q 6 dim almost abelian E∗ := Leray-Serre spectral sequence associated with the Kähler and symplectic structures on solvmanifolds fibration: Hyperelliptic surface The oscillator group q p,q p q n ∼ p ^ n E2 = HdR(M, HdR(T )) = HdR(M) ⊗ R , p,q p+q E∞ ⇒ HdR (M) . 10 Topology of compact solvmanifolds Sergio Console Main idea: construct a second spectral sequence Aims ˜ p,q Nilpotent and solvable E∗ = Leray-Serre spectral sequence for the complex of V∗ ∗ Nilmanifolds G-invariant forms g Solvmanifolds de Rham Cohomology V∗ ∗ V∗ ˜ p,q p,q g subcomplex of M =⇒ E∗ ⊆ E∗ & Nilmanifolds Solmanifolds Main Theorem p,q ^q ˜ p n Proof of the Main Theorem E2 = H (g/h) ⊗ R , Applications Nakamura manifold E˜ p,q ⇒ Hp+q(g) . Almost abelian ∞ 6 dim almost abelian Kähler and symplectic structures on solvmanifolds induction on dim p ∼ p Hyperelliptic surface dim M < dim M, =⇒ HdR(M) = H (g/h) for any p. The oscillator group ˜ ˜ =⇒ E2 = E2 & E∞ = E∞. ` ∼ ` i.e., HdR(M) = H (g) for any `. 11 Topology of compact de Rham Cohomology of solvmanifolds solvmanifolds Sergio Console Aims Nilpotent and solvable G/Γ solvmanifold Nilmanifolds Solvmanifolds adX ∼ Ad G(G) = {e | X ∈ g} solvable and Aut (G) = Aut (g). de Rham Cohomology Nilmanifolds Solmanifolds A(Ad G(G)) and A(Ad G(Γ)): real algebraic closures of Ad G(G) Main Theorem and Ad G(Γ) (respectively) Proof of the Main Theorem Applications Nakamura manifold Theorem (Borel density theorem) Almost abelian 6 dim almost abelian Kähler and symplectic Let Γ be a lattice of a simply connected solvable Lie group G, structures on solvmanifolds =⇒ ∃ a maximal compact torus ⊂ A(Ad (G)), such that Hyperelliptic surface Tcpt G The oscillator group A(Ad G(G)) = Tcpt A(Ad G(Γ)). 12 Topology of compact When is the de Rham comology of a solvmanifold given by solvmanifolds the Chevalley-Eilenberg complex? Sergio Console There are 2 important cases: Aims Nilpotent and solvable Nilmanifolds 1 Hattori: If G is completely solvable, i.e., Solvmanifolds if the linear map adX : g → g has only real eigenvalues de Rham Cohomology Nilmanifolds 2 Mostow condition: If A(Ad G(G)) = A(Ad G(Γ)) Solmanifolds Main Theorem Proof of the Main Theorem Remarks Applications Nakamura manifold Almost abelian • (1) is a particular case of (2). 6 dim almost abelian Kähler and symplectic Indeed if all eigenvalues of Ad are real, then A(Ad G(G)) has no non structures on solvmanifolds Borel density Hyperelliptic surface trivial conn. compact subgroups =⇒ A(Ad G(G)) = A(Ad G(Γ)) The oscillator group • Recall that a nilpotent Lie group is completely solvable =⇒ (1) and (2) generalize Nomizu’s Theorem. • We will see that one can have the isomorphism ∗ ∼ ∗ H (g) = HdR (G/Γ) even if A(Ad G(Γ)) 6= A(Ad G(G)) (Example on hyperelliptic surface) hyperelliptic surface 13 Topology of compact Idea of the proof of (2) solvmanifolds Sergio Console We prove that if the Mostow condition holds, we still have a fibration of M = G/Γ over a smaller dimensional solvmani- Aims Nilpotent and solvable fold with a torus as fibre. Then one can proceed as in the Nilmanifolds proof of Nomizu’s Theorem. Solvmanifolds de Rham Cohomology Nilmanifolds Solmanifolds G = [G , G ] & Γ = [Γ , Γ ]: derived series (k) (k−1) (k−1) (k) (k−1) (k−1) Main Theorem Proof of the Main Theorem Remark that G is nilpotent for any k ≥ 1 (k) Applications Nakamura manifold Mostow condition + a gen. result on lattices in nilpotent Lie groups Almost abelian 6 dim almost abelian =⇒ G(k)/Γ(k) is compact for any k. Kähler and symplectic structures on solvmanifolds =⇒ G(k)/Γ ∩ G(k) is compact for any k. Hyperelliptic surface The oscillator group Let r be the last non-zero term in the derived series of G. Namely G(r+1) = (e) and G(r) =: A 6= (e). A is abelian =⇒ A/A ∩ Γ := Tm is a compact torus. Thus, M := G/AΓ is a compact solvmanifold with dimension π 14 smaller than M := G/Γ and Tm ,→ M → M is a fibration. Topology of compact solvmanifolds If A(Ad G(G)) 6= A(Ad G(Γ)) it is more difficult to compute the de Sergio Console Rham cohomology. We explain a method deriving from results of Guan and Witte Aims Nilpotent and solvable Nilmanifolds Main Theorem [ – , A. Fino] Solvmanifolds de Rham Cohomology Let M = G/Γ be a compact solvmanifold and Nilmanifolds Solmanifolds let Tcpt be a compact torus such that Main Theorem Proof of the Main Theorem cpt A(Ad G(Γ)) = A(Ad G(G)). Applications T Nakamura manifold Almost abelian ˜ 6 dim almost abelian Then there exists a subgroup Γ of finite index in Γ and a simply Kähler and symplectic ˜ structures on solvmanifolds connected normal subgroup G of Tcpt n G such that Hyperelliptic surface The oscillator group ˜ ˜ A(Ad G˜ (Γ)) = A(Ad G˜ (G)). ˜ ˜ ˜ ∗ ˜ ∼ ∗ =⇒ G/Γ is diffeomorphic to G/Γ and HdR(G/Γ) = H (g˜). ∗ ∼ ∗ ˜ Γ/Γ˜ Observe that HdR (G/Γ) = HdR (G/Γ) 15 (the invariants by the action of the finite group Γ/Γ˜). Topology of compact Proof of the Main Theorem solvmanifolds Sergio Console Aims It is not restrictive to suppose that A(AdG(Γ)) is connected. Nilpotent and solvable Otherwise we pass from Γ to a finite index subgroup Γ˜ Nilmanifolds Solvmanifolds = /Γ /Γ˜ [equivalently M G G finite-sheeted covering of M] de Rham Cohomology Nilmanifolds Let Tcpt be a maximal compact torus of A(Ad GG) which Solmanifolds contains a maximal compact torus of A(Ad (Γ))˜ . Main Theorem Scpt G Proof of the Main Theorem Applications Let Scpt be a subtorus of Tcpt complementary to Scpt so that Nakamura manifold cpt = cpt × cpt . Almost abelian T S S 6 dim almost abelian Kähler and symplectic Let σ be the composition of the homomorphisms: structures on solvmanifolds Hyperelliptic surface The oscillator group Ad proj proj x→x −1 σ : G −→ A(Ad G(G)) −→ Tcpt −→ Scpt −→ Scpt . The point now is to get rid of Scpt 16 Topology of compact solvmanifolds nilshadow map: Sergio Console ∆ : G → Scpt n G, g 7→ (σ(g), g), Aims Nilpotent and solvable Nilmanifolds [not a homomorphism (unless Scpt = {0} =⇒ σ = 0] One has: Solvmanifolds de Rham Cohomology −1 Nilmanifolds ∆(ab) = ∆(σ(b )aσ(b)) ∆(b), ∀a, b ∈ G Solmanifolds ˜ Main Theorem and ∆(γg) = γ∆(g), for every γ ∈ Γ, g ∈ G. Proof of the Main Theorem ∆ is a diffeomorphism onto it image =⇒ ∆(G) is simply connected Applications Nakamura manifold Almost abelian The product in ∆(G) is given by: 6 dim almost abelian Kähler and symplectic structures on solvmanifolds −1 ∆(a)∆(b) = (σ(a), a)(σ(b), b) = (σ(a)σ(b), σ(b )aσ(b) b), Hyperelliptic surface The oscillator group for any a, b ∈ G. ˜ By construction , A(Ad G(Γ)) projects trivially on Scpt and σ(Γ)˜ = {e}. =⇒ ˜ ˜ Γ = ∆(Γ) ⊂ ∆(G). 17 Topology of compact Let G˜ = ∆(G). solvmanifolds ˜ Sergio Console [Witte]: Scpt is a maximal compact subgroup of A(Ad G˜ (G)) and ˜ ˜ ˜ Scpt ⊂ A(Ad G˜ (Γ)) =⇒ A(Ad G˜ (G)) = A(Ad G˜ (Γ)) Aims Diffeomorphism Theorem Nilpotent and solvable ˜ ˜ ˜ Nilmanifolds =⇒ G/Γ is diffeomorphic to G/Γ. Solvmanifolds ∗ ˜ ∼ ∗ de Rham Cohomology Mostow condition holds =⇒ H (G/Γ) = H (g˜). Nilmanifolds Solmanifolds ˜ −1 By the diffeomorphism ∆ : G → G, ∆ induces a finite Main Theorem sheeted covering map ∆∗ : G˜ /Γ˜ → G/Γ. Proof of the Main Theorem Applications Nakamura manifold Corollary Almost abelian 6 dim almost abelian ˜ Kähler and symplectic The Lie algebra g˜ of G can be identified by structures on solvmanifolds Hyperelliptic surface The oscillator group g˜ = {(Xs, X) | X ∈ g} with Lie bracket: [(Xs, X), (Ys, Y )] = (0, [X, Y ] − ad(Xs)(Y ) + ad(Ys)(X)). 18 where Xs the image σ∗(X), for X ∈ g Topology of compact Applications solvmanifolds Sergio Console Now we obtain some applications of the Main Theorem, by computing explicitly the Lie group G˜ . Aims Nilpotent and solvable Nilmanifolds Example (Nakamura manifold – description) Solvmanifolds Consider the simply connected complex solvable Lie group G: de Rham Cohomology Nilmanifolds z Solmanifolds e 0 0 w1 −z Main Theorem 0 e 0 w2 G = , w1, w2, z ∈ C . Proof of the Main Theorem 0 0 1 z Applications 0 0 0 1 Nakamura manifold Almost abelian ∼ 2 G = C nϕ C , where 6 dim almost abelian ez 0 Kähler and symplectic ϕ(z) = . structures on solvmanifolds 0 e−z Hyperelliptic surface The oscillator group Let L1,2π = Z[t0, 2πi] = {t0k + 2πhi, h, k ∈ Z}, µ L = P , µ, α ∈ [i] . 2 α Z Then, by Yamada Γ = L1,2π nϕ L2 is a lattice of G. G/Γ: Nakamura manifold 19 Topology of compact solvmanifolds Example (Nakamura manifold – computation of cohomology) Sergio Console ∼ ∼ 2 4 G has trivial center =⇒ Ad G(G) = G = R n R . Moreover, Aims # 1 4 A(Ad GG) = (R × S ) n R , Nilpotent and solvable A(Ad Γ) = # 4, Nilmanifolds G R n R Solvmanifolds 1 de Rham Cohomology # 2 (z+z) where the split torus R corresponds to the action of e and Nilmanifolds 1 (z−z) Solmanifolds the compact torus S1 to the one of e 2 . Main Theorem 1 =⇒ A(Ad G(G)) = S A(Ad G(Γ)) and A(Ad G(Γ)) is connected. Proof of the Main Theorem Applications Main Theorem Nakamura manifold =⇒ ∃ a simply connected normal subgroup G˜ = ∆(G) of S1 G. n Almost abelian 6 dim almost abelian 1 (z−z) The new Lie group G˜ is obtained by killing the action of e 2 : Kähler and symplectic structures on solvmanifolds Hyperelliptic surface 1 (z+z) e 2 0 0 w1 The oscillator group − 1 (z+z) ˜ ∼ 0 e 2 0 w G = 2 , w1, w2, z ∈ C . 0 0 1 z 0 0 0 1 diffeo G/Γ ' G˜ /Γ was already shown by Yamada. =⇒ ∗ ∼ ∗ ˜ ∼ ∗ 20 HdR (G/Γ) = H (˜g),(˜g Lie algebra of G) and HdR (G/Γ) =6 H (g). Topology of compact solvmanifolds Sergio Console Example (A three dimensional example) 1 de = 0, Aims 2 2 1 3 G = R n R with structure equations de = 2πe ∧ e is Nilpotent and solvable 3 1 2 Nilmanifolds de = −2πe ∧ e Solvmanifolds non-completely solvable and admits a lattice Γ = 2. de Rham Cohomology Z n Z Nilmanifolds Indeed, Solmanifolds Main Theorem Proof of the Main Theorem cos(2πt) sin(2πt) 0 x − sin(2πt) cos(2πt) 0 y Applications 2 = Nakamura manifold R n R Almost abelian 0 0 1 t 6 dim almost abelian 0 0 0 1 Kähler and symplectic structures on solvmanifolds Hyperelliptic surface The oscillator group and Γ is generated by 1 in R and the standard lattice Z2. 1 2 2 Main Theorem A(Ad G(G)) = S n R and A(Ad G(Γ)) = R =⇒ ˜ 3 1 G =∼ R ⊂ S n G. Indeed, it is well known that G/Γ is diffeomorphic to a torus. 21 Topology of compact solvmanifolds 2 The previous example R n R is an almost abelian Lie group: Sergio Console A Lie algebra g is called almost abelian if it has an abelian Aims ideal of codimension 1, Nilpotent and solvable i.e. g =∼ b, where b =∼ n is an abelian ideal of g. Nilmanifolds R n R Solvmanifolds 1 de Rham Cohomology In this case the Mostow bundle is a torus bundle over S Nilmanifolds n Solmanifolds The action ϕ of R on R is represented by a family of ma- Main Theorem trices ϕ(t), which encode the monodromy or “twist” in the Proof of the Main Theorem Applications Mostow bundle. Nakamura manifold Almost abelian 6 dim almost abelian A nice feature of almost abelian solvable groups is that there is Kähler and symplectic structures on solvmanifolds a criterion on the existence of a lattice Hyperelliptic surface The oscillator group Proposition (Bock) n Let G = R nϕ R be almost abelian solvable Lie group. Then G admits a lattice if and only if there exists a t0 6= 0 for which ϕ(t0) can be conjugated to an integer matrix. 22 Topology of compact solvmanifolds Sergio Console The Lie algebra g of G has form Aims n Nilpotent and solvable ad , R n Xn+1 R Nilmanifolds Solvmanifolds n de Rham Cohomology where we consider generated by {X1, ..., Xn} and by Xn+1, R R Nilmanifolds tadX and ϕ(t) = e n+1 . Solmanifolds Main Theorem n Moreover, a lattice can be always represented as Γ = Z n Z Proof of the Main Theorem Applications Nakamura manifold For almost abelian solvmanifolds, Gorbatsevich found a Almost abelian 6 dim almost abelian criterion to decide whether the Mostow condition holds: Kähler and symplectic structures on solvmanifolds Hyperelliptic surface Proposition (Gorbatsevich) The oscillator group The Mostow condition is satisfied if and only if πi can not be written as linear combination in Q of the eigenvalues of t0adXn+1 , where Γ is generated by t0. 23 Topology of compact solvmanifolds • If iπ is not representable as a Q-linear combination of the Sergio Console Mostow condition ∗ ∼ ∗ numbers λk , =⇒ H (G/Γ) = H (g). dR Aims Nilpotent and solvable • Otherwise the only known result on cohomology [Bock] Nilmanifolds Solvmanifolds de Rham Cohomology b1(G/Γ) = n + 1 − rank(ϕ(1) − id). Nilmanifolds Solmanifolds Main Theorem By applying the Main Theorem one obtains a method to Proof of the Main Theorem compute the de Rham cohomology of G/Γ. Applications Nakamura manifold Almost abelian [ – , M. Macrì] construct lattices on six dimensional not completely 6 dim almost abelian Kähler and symplectic solvable almost abelian Lie groups, for which the Mostow condition structures on solvmanifolds does not hold. We compute Hyperelliptic surface The oscillator group • cohomology (does not agree with the one of g) • minimal model • show that some of these solvmanifolds admit not invariant symplectic structures and we study formality and Lefschetz properties 24 Topology of compact solvmanifolds Example (6-dimensional indecomposable almost abelian Sergio Console solvmanifolds not satisfying the Mostow condition) Aims Nilpotent and solvable Nilmanifolds Solvmanifolds de Rham Cohomology Nilmanifolds Solmanifolds Main Theorem Proof of the Main Theorem Applications Nakamura manifold Almost abelian 6 dim almost abelian × = for both the invariant and not invariant symplectic structures considered. Kähler and symplectic structures on solvmanifolds ∗ = for the invariant symplectic structures. Hyperelliptic surface The oscillator group F: formality IS: existence of invariant symplectic structures S: existence of symplectic structures (induced by ones on the modified Lie alg) HL: Hard Lefschetz property 25 Topology of compact Example (6-dimensional decomposable almost abelian solvmanifolds solvmanifolds not satisfying the Mostow condition) Sergio Console Aims Nilpotent and solvable Nilmanifolds Solvmanifolds de Rham Cohomology Nilmanifolds Solmanifolds Main Theorem Proof of the Main Theorem Applications Nakamura manifold Almost abelian 6 dim almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface The oscillator group 26 Topology of compact Example (6-dimensional decomposable almost abelian solvmanifolds solvmanifolds not satisfying the Mostow condition) Sergio Console Aims Nilpotent and solvable Nilmanifolds Solvmanifolds de Rham Cohomology Nilmanifolds Solmanifolds Main Theorem Proof of the Main Theorem Applications Nakamura manifold Almost abelian 6 dim almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface The oscillator group 27 Topology of compact solvmanifolds Sergio Console Example (6-dimensional decomposable almost abelian Aims solvmanifolds not satisfying the Mostow condition) Nilpotent and solvable Nilmanifolds Solvmanifolds de Rham Cohomology Nilmanifolds Solmanifolds Main Theorem Proof of the Main Theorem Applications Nakamura manifold Almost abelian 6 dim almost abelian Kähler and symplectic structures on solvmanifolds Hyperelliptic surface ∃ examples where the cohomology depends strongly on the lattice: The oscillator group ∗ ∼ ∗ ∼ ∗ 0 HdR(G/Γπ) =6 HdR(G/Γ2π) =6 H (g) , G = G5.18 × R . 28 Topology of compact Kähler structures on solvmanifolds solvmanifolds Sergio Console Benson-Gordon conjecture: a compact solvmanifold has a Aims Nilpotent and solvable Kähler structure if and only if it is a complex torus Nilmanifolds Solvmanifolds de Rham Cohomology Nilmanifolds Hasegawa (2006): Solmanifolds Main Theorem A solvmanifold carries a Kähler metric if and only if it is covered Proof of the Main Theorem Applications by a finite quotient of a complex torus, which has the structure Nakamura manifold of a complex torus bundle over a complex torus. Almost abelian 6 dim almost abelian Kähler and symplectic An example is provided by the hyperelliptic surface structures on solvmanifolds Hyperelliptic surface hyperelliptic surface The oscillator group in particular, a compact solvmanifold of completely solvable type has a Kähler structure if and only if it is a complex torus 29 Topology of compact Half-flat symplectic structures on solvmanifolds solvmanifolds Sergio Console Six dimensional almost abelian solvmanifolds were consider in string backgrounds where the internal compactification manifold is a Aims solvmanifold (see e.g. [Andriot, Goi, Minasian and Petrini]). Nilpotent and solvable Nilmanifolds Solvmanifolds They are related to solutions of the supersymmetry (SUSY) equations. de Rham Cohomology Nilmanifolds By [Fino-Ugarte], solution of the SUSY equations IIA possess a Solmanifolds symplectic half-flat structure, whereas solutions of the SUSY Main Theorem equations IIB admit a half-flat structure Proof of the Main Theorem Applications Nakamura manifold An SU(3) structure on a six-dimensional manifold M Almost abelian (i.e., an SU(3) reduction of the frame bundle of M) 6 dim almost abelian Kähler and symplectic structures on solvmanifolds • a non-degenerate 2-form Ω, Hyperelliptic surface The oscillator group • an almost-complex structure J, • a complex volume form Ψ. The SU(3) structure is called half-flat if Ω ∧ Ω and the real part of Ψ are closed [Chiossi-Salamon]. If in addition Ω is closed, the half-flat structure is called symplectic. 30 Topology of compact Half-flat symplectic structures on solvmanifolds solvmanifolds Sergio Console Aims Nilpotent and solvable Proposition ( – , M. Macrì) Nilmanifolds Solvmanifolds We have the following behavior concerning half flatness of de Rham Cohomology Nilmanifolds (invariant) symplectic structures for the above solvmanifolds: Solmanifolds • Ga=0/Γ and G0 × /Γ admit (not) invariant Main Theorem 6.10 2π 5.14 R 2π Proof of the Main Theorem symplectic forms which are not half flat. Applications Nakamura manifold p,−p,r r1 • G × /Γ2πr (r = ∈ ) admits an invariant Almost abelian 5.17 R 2 r2 Q 6 dim almost abelian symplectic form which is half flat only for p ≥ 0 and r = 1 Kähler and symplectic structures on solvmanifolds and it admits a not invariant symplectic form which is half Hyperelliptic surface flat. The oscillator group 0 0 3 • G5.18 × R/Γ2π and G3.5 × R /Γ2π admit (not) invariant symplectic forms which are half flat. 31 Topology of compact Example (Hyperelliptic surface) solvmanifolds G = R nϕ (C × R), with ϕ : R → Aut (C × R) defined by Sergio Console 2 1 1 Aims ϕ(t)(z, s) = (eiηt z, s), where η = π, π, π or π 3 2 3 Nilpotent and solvable Nilmanifolds Solvmanifolds 3 Hasegawa: G has 7 isomorphism classes of lattices Γ = Z nϕ Z , de Rham Cohomology 3 iη −iη Nilmanifolds where ϕ : Z → Aut (Z ) has matrix ϕ(1) with eigenvalues 1, e , e . Solmanifolds Gorbatsevich Main Theorem ϕ(1) has a pair of cx conj roots =⇒ A(Ad G(G)) 6= A(Ad G(Γ)). Proof of the Main Theorem Applications In this case A(Ad G(Γ)) is not connected, but Nakamura manifold ˜ ∼ 4 Almost abelian Γ contains as a finite index subgroup Γ = Z 6 dim almost abelian Kähler and symplectic =⇒ G/Γ is a finite covering of a torus structures on solvmanifolds Hyperelliptic surface 1 ∼ 1 The oscillator group Note: HdR(G/Γ) = H (g) even if A(Ad G(G)) 6= A(Ad G(Γ)) Indeed, G has structure equations: de1 = e2 ∧ e4 de2 = −e1 ∧ e4 and H1(g) = span < e3, e4 > . de3 = 0 32 de4 = 0 Topology of compact solvmanifolds Sergio Console Example (Three families of lattices in the oscillator group [ – , G. Ovando, M. Subils ] ) Aims Nilpotent and solvable Oscillator group: G = R nα H3(R) Nilmanifolds H3(R) (real) three dimensional Heisenberg group Solvmanifolds de Rham Cohomology cos(t) sin(t) 0 Nilmanifolds Solmanifolds α : R → Aut (h3) , t 7→ − sin(t) cos(t) 0 . Main Theorem 0 0 1 Proof of the Main Theorem Applications Nakamura manifold The oscillator group is an almost nilpotent solvable Lie group Almost abelian 6 dim almost abelian 3 Kähler and symplectic If we regard H3(R) as R endowed with the operation structures on solvmanifolds Hyperelliptic surface 1 (x, y, z) · (x0, y 0, z0) = (x + x0, y + y 0, z + z0 + (xy 0 − x0y)) The oscillator group 2 =⇒ H3(R) admists the co-compact subgroups Γk ⊂ H3(R) given by 1 Γk = Z × Z × Z . 2k 33 Topology of compact solvmanifolds Sergio Console Example (The oscillator group) Aims The lattice Γk (for any k) is invariant under the subgroups generated Nilpotent and solvable π by α(0) = α(2π), α(π) and α( ) =⇒ Nilmanifolds 2 Solvmanifolds we have three families of lattices in G = α H3( ): R n R de Rham Cohomology Nilmanifolds Λk,0 = 2πZ n Γk ⊂ G , Solmanifolds Λk,π = πZ n Γk ⊂ G , Main Theorem π Proof of the Main Theorem Λk,π/2 = Γk ⊂ G . 2 Z n Applications Nakamura manifold =⇒ Λk,0 . Λk,π . Λk,π/2 (.: “contains as a normal subgroup”), Almost abelian 6 dim almost abelian we have the solvmanifolds Kähler and symplectic structures on solvmanifolds Hyperelliptic surface Mk,0 = G/Λk,0 , The oscillator group Mk,π = G/Λk,π , Mk,π/2 = G/Λk,π/2 . All subgroups of the families Λk,i are not pairwise isomorphic =⇒ determine non-diffeomorphic solvmanifolds. 34 Topology of compact solvmanifolds Example (The oscillator group) Sergio Console The action of α(0) is trivial, so Λk,0 = 2πZ × Γk Aims Diffeomorphism Theorem ∼ 1 =⇒ Mk,0 = G/Λk,0 = S × H3(R)/Γk , Nilpotent and solvable a Kodaira–Thurston manifold. Nilmanifolds Solvmanifolds Moreover, for any fixed k, we have the finite coverings de Rham Cohomology Nilmanifolds Solmanifolds pπ : Mk,0 → Mk,π , pπ/2 : Mk,0 → Mk,π/2 , Main Theorem Proof of the Main Theorem Applications Nakamura manifold [ – , G. Ovando, M. Subils ] Almost abelian 6 dim almost abelian Kähler and symplectic The Betti numbers bi of the solvmanifolds Mk,∗ are given by structures on solvmanifolds Hyperelliptic surface The oscillator group b0 b1 b2 (clearly b3 = b1 and b4 = b0, Mk,0 1 3 4 . by Poincaré duality). Mk,π 1 1 0 Mk,π/2 1 1 0 There are many symplectic structures on M which are invariant by k,0 35 the group R × H3(R) but not under the oscillator group G.