Strong Kaehler metrics with torsion and generalized Kaehler structures on torus bundles Workshop on ”Special Geometries in Mathematical Physics” K¨uhlungsborn, April 2nd 2008

Strong Kaehler metrics with torsion and

generalized Kaehler structures on torus bundles

Workshop on

”Special Geometries in Mathematical Physics”

K¨uhlungsborn, April 2nd 2008

M²ⁿ 2n-dimensional (compact) manifold

• F non-degenerate 2-form on M²ⁿ

• J almost complex structure on M²ⁿ, i.e.

J ∈ End(T M²ⁿ) s.t. J² = −id_{T M2n}

• F is symplectic if dF = 0

• J is integrable if it is induced by a complex structure.

Newlander-Nirenberg

J is integrable ⇐⇒ N_J = 0 where

N_J = [JX, JY ] − [X, Y ] − J[JX, Y ] − J[X, JY ]

∀X, Y vector fields on M.

• A Riemannian metric g on (M²ⁿ, J) is said to be J-Hermitian if

g(JX, JY ) := g(X, Y ) , ∀X, Y .

F symplectic form; J almost complex structure M is said to be F-calibrated if

g_J[x](X, Y ) := F[x](X, JY ) is a J-Hermitian metric on M.

• (M, J, F, g_J) K¨ahler, if F is symplectic, J is complex and F-calibrated.

Weaker conditions

(I) dF = 0, J non-integrable.

(II) dF 6= 0, J integrable.

(I)

• Special symplectic manifolds,

• Geometry of Lagrangian submanifolds.

(II)

• Geometry with Torsion,

• Generalized K¨ahler Geometry,

• Bi-Hermitian Structures,

• Special metrics on Complex manifolds e.g.

balanced, strong KT, astheno-K¨ahler

Special Symplectic Manifolds

Def 1. A special symplectic Calabi-Yau mani- fold (SSCY) is the datum of (M⁶, F, J, ψ) where

• F is a symplectic structure

• J is a F-calibrated almost complex structure

• g_J(·,·) := F(·, J·)

• ψ ∈ ∧^3,0(M), ψ 6= 0, s.t.

d<

e

3i F³ Rem.

• If d<

e

m

• <

e

Theorem (P. de Bartolomeis,—) There ex- ists a compact complex manifold M such that

• M has a symplectic structure satisfying the Hard Lefschetz Condition;

• M admits a SSCY structure;

• M has no K¨ahler structures.

M = (

C

t(z₁, z₂, z₃) ∗^t (w₁, w₂, w₃) =

t(z₁ + w₁, e^−w1z₂ + w₂, e^w1z₃ + w₃)

and Γ is a certain closed subgroup of (

C

• In [Conti,—](Quarterly J. ’07) nilmanifolds carrying SSCY-structures are classified.

• For other results in higher dimensions [de Bartolomeis,—](Inter. J. Math. ’06).

Generalized Complex Geometry Indefinite metric

V real vector space of dimension n.

(,) : V ⊕ V ^∗ →

R

2(ξ(w) + η(v))

(,) is the natural indefinite metric on V ⊕ V ^∗ with signature (n, n).

Twisted Courant bracket

M manifold, H closed 3-form.

Def.

[,] : Γ(T M⊕T^∗M)×Γ(T M⊕T^∗M) → Γ(T M⊕T^∗M) [X + ξ, Y + η] = [X, Y ] + L_Xη − L_Y ξ +

− 1

2d(ι_Xη − ι_Y ξ) + ι_Xι_Y H ,

L_X Lie derivative along X, ι_X contractions along X.

Generalized Complex Structures

M 2n-dimensional manifold, (,) indefinite met- ric on T M ⊕ T^∗M.

Def. A generalized complex structure on M (GC structure) is the datum of a subbundle E ⊂ (T M ⊕ T^∗M) ⊗

C

• E ⊕ E = (T M ⊕ T^∗M) ⊗

C

• the space of sections of E is closed with respect to the Courant bracket

• E is isotropic.

Basic examples

• M complex manifold. Then

E = T^0,1M ⊕ T^1,0∗M defines a GC structure on M.

i) E ⊕ E = (T M ⊕ T^∗M) ⊗

C

(Z + ϕ, W + ψ) = 1

2(ϕ(W) + ψ(Z)) = 0. iii)

[Z + ϕ, W + ψ] = [Z, W] + L_Zψ − L_Wϕ+

−¹₂d (ι_Zψ − ι_Wϕ)

= [Z, W] + L_Zψ − L_Wϕ . Since L_Zψ, L_Wϕ ∈ T^1,0∗M, then E is involu- tive.

• (M, ω) symplectic manifold. Then E = {Z − i Z

y

C

• Equivalently, a GC structure can be viewed as an almost complex structure

J ∈ End(T M ⊕ T^∗M), which is (,)-orthogonal and integrable with respect to the Courant bracket.

The previous examples shows that

Generalized K¨ahler structures

M 2n-dimensional manifold.

Def. A generalized K¨ahler structure on M

(GK structure) is a pair of generalized complex structures (J₁, J₂) on M such that

• J₁ e J₂ commute

• J₁ and J₂ are compatible with the natural pairing (, ) on T M ⊕ T^∗M

• −(J₁J₂ ·, ·) is positive definite

¤

Theorem (Apostolov and Gualtieri, Comm.

Math. Phys. ’07)

A GK structure on M is equivalent to assign a triple (g, J₊, J₋) where:

• g is a Riemannian metric on M

• J₊ and J₋ are two complex structures on M, compatible with g and such that

d^c₊F₊ + d^c₋F₌0, dd^c₊F₊ = 0 , dd^c₋F₋ = 0, F₊, F₋ fundamental forms of (g, J₊), (g, J₋),

d^c₊ = i(∂₊ − ∂₊), d^c₋ = i(∂₋ − ∂₋) .

¤

Example (M, g, J) K¨ahler

J₊ = J , J₋ = ±J

⇒ (g, J₊, J₋) GK structure on M.

¤

Pb. When does a compact complex manifold (M, J) admit a GK structure (g, J₊, J₋) with J = J₊?

Interesting case: J₊ 6= ±J₋, i.e. the GK structure is not induced by a K¨ahler metric on (M, J).

Strong KT geometry

(M, J, g) Hermitian manifold

∇ Bismut connection

∇g = 0 , ∇J = 0 ,

g(X, T^∇(Y, Z)) totally skew-symmetric The torsion form

T(X, Y, Z) = g(X, T^∇(Y, Z))

is JdF, where F is the fundamental form of g.

Def. A Hermitian metric g on a complex man- ifold (M, J) is said to be strong K¨ahler with torsion (strong KT) if the fundamental form F is ∂∂-closed, i.e.

∂∂ F = 0 .

Rem. (M, J) GK =⇒ (M, J) has a strong KT metric.

• (M, J) compact complex surface ⇒ any con- formal class of a Hermitian metric has a strong KT representative (Gauduchon, Math. Ann.

’84).

• dim_R M > 4 compact examples of strong KT metrics on nilmanifolds (Fino, Parton, Sala- mon, Comm. Math. Helv. ’04).

Existence results

• (M, J) compact complex surface.

Classification theorem of generalized K¨ahler structures

(Apostolov and Gualtieri, Comm. Math. Phys.

’07)

• dim_RM = 6.

By [Cavalcanti and Gualtieri, J. of Sympl.

Geom. ’05]

any nilmanifold carries a GC structure

• dim_RM = 2n

there are no nilmanifolds (different from Tori) admitting an invariant GK structure.

(Cavalcanti, Topol. and its Applic. ’06)

Compact example

•

s



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

































de¹ = a e¹ ∧ e² , de² = 0,

de³ = ¹₂a e² ∧ e³ , de⁴ = ¹₂a e² ∧ e⁴ , de⁵ = b e² ∧ e⁶ , de⁶ = −b e² ∧ e⁵ ,

(1)

a, b real parameters different from zero.

• S_a,b simply-connected Lie group whose Lie algebra is

s

(t, x₁, x₂, x₃, x₄, x₅) global coordinates on

R

• Product on S_a,b

(t, x₁, x₂, x₃, x₄, x₅) · (t⁰, x⁰₁, x⁰₂, x⁰₃, x⁰₄, x⁰₅) = (t + t⁰, e^{a t}x⁰₁ + x₁, e^2atx⁰₂ + x₂, e^2atx⁰₃ + x₃, x⁰₄ cos(b t) − x⁰₅ sin(b t) + x₄,

x⁰₄ sin(b t) + x⁰₅ cos(b t) + x₅).

• S_a,b unimodular semidirect product

R n

R

R

R

ϕ = (ϕ₁, ϕ₂) diagonal action of

R

R

R

R

Theorem (A. Fino, —)(to appear in J. of Sympl. Geom.)

• S_1,^π

2 has a compact quotient M⁶ = S_1,^π

2/Γ.

• M⁶ is the total space of a

T

• M⁶ = S_1,^π

2/Γ has a non-trivial left invariant GK structure.

• b₁(M⁶) = 1 ⇒ M⁶ has no K¨ahler metrics.

¤

• GK structure on M⁶ = S_1,^π

2/Γ

ϕ¹₊ = e¹ + ie², ϕ²₊ = e³ + ie⁴, ϕ³₊ = e⁵ + ie⁶, ϕ¹₋ = e¹ − ie², ϕ²₋ = e³ + ie⁴, ϕ³₋ = e⁵ + ie⁶.

(ϕ¹_±, ϕ²_±, ϕ³_±) (1,0)-forms associated with J_±.

• J_± integrable.

• g =

X6

α=1

e^α ⊗ e^α J_±-Hermitian.

Then

d^c₊F₊ + d^c₋F₌0, dd^c₊F₊ = 0 , dd^c₋F₋ = 0, (g, J₊, J₋) defines a left-invariant GK structure on M⁶.

d^c₊F₊ = e¹ ∧ e³ ∧ e⁴ closed non-exact Uniform subgroup

• S_1,^π

2 is isomorphic to (

R

R n

R

C

C

(t, u, z, w) ∗ (t⁰, u⁰, z⁰, w⁰) = (t + t⁰, c^tu⁰ + u,

α^tz⁰ + z, e^iπ2tw⁰ + w),

∀t, t⁰, u, u⁰ ∈

R

C

• Γ is isomorphic to

Z n

Z

Z

g_j : (t, u, z, w) 7→ (t, u + c_j, z + α_j, w), j = 1,2,3, g₄ : (t, u, z, w) 7→ (t, u, z, w + 1),

g₅ : (t, u, z, w) 7→ (t, u, z, w + i).

It can be checked that

i) Γ acts freely and in a properly discontinuos way on S_1,^π

2

ii) S_1,^π

2/Γ is compact. Furthermore

π :

R n

R

C

C

R n

R

C

M⁶ is a

T