A survey on strong KT structures - SSMR

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$Bull. Math. Soc. Sci. Math. Roumanie Tome 49(97) No. 4 ... - SSMR$

110 Anna Fino and Adriano Tomassini6 A 6-dimensi<strong>on</strong>al generalized Kähler structure solvmanifoldGeneralized Kähler structures have been introduced by Gualtieri in [24] in thec<strong>on</strong>text of generalized geometries as generalizati<strong>on</strong> of Kähler structures:Definiti<strong>on</strong> 6.1. A generalized Kähler structure <strong>on</strong> a 2n-dimensi<strong>on</strong>al manifoldM is a pair (J 1 , J 2 ) of generalized complex structures <strong>on</strong> M such that1. J 1 and J 2 commute;2. J 1 and J 2 are compatible with the indefinite metric (, ) <strong>on</strong> TM ⊕ T ∗ M;3. the bilinear form −(J 1 J 2 ·, ·) is positive definite.In terms of bi-Hermitian geometry, Apostolov and Gualtieri proved in [3] thata generalized Kähler structure <strong>on</strong> M is equivalent to a triple (g,J + ,J − ) where:1. g is a Riemannian metric <strong>on</strong> M;2. J + and J − are two complex structures <strong>on</strong> M compatible with g such thatd c +F + + d c −F − = 0, dd c +F + = dd c −F − = 0,where d c ± = i(∂ ± − ∂ ± ) and F ± is the fundamental form of the Hermitianstructure (J ± ,g).The 3-form d c +F + is called the torsi<strong>on</strong> form of the generalized Kähler structureand the generalized Kähler structure is said to be untwisted if the de Rhamcohomology class [d c +F + ] ∈ H 3 (M) vanishes and twisted if [d c +F + ] ≠ 0.C<strong>on</strong>structi<strong>on</strong>s of n<strong>on</strong>-trivial generalized Kähler structures are given for instancein [24, 3, 5, 31, 36, 35, 15]. For example in [36] the generalized Kählerquotient c<strong>on</strong>structi<strong>on</strong> is c<strong>on</strong>sidered in relati<strong>on</strong> with the hyperkähler quotient c<strong>on</strong>structi<strong>on</strong>and generalized Kähler structures are given <strong>on</strong> CP n , <strong>on</strong> some toric varietiesand <strong>on</strong> the complex Grassmannian.An interesting problem is thus to look for compact examples of generalizedKähler manifolds which do not admit any Kähler structure. A natural class ofmanifolds where to investigate the existence of these structures is provided byLie groups.By [24] any real compact semisimple Lie group G of even dimensi<strong>on</strong> admitsa twisted generalized Kähler structure. Indeed, by [42] G has left and rightinvariant complex structures J L and J R , which can be choosen to be Hermitianwith respect to the bi-invariant metric induced by the Killing form. Both (J L ,g)and (J R ,g) are str<strong>on</strong>g KT structures and the Bismut c<strong>on</strong>necti<strong>on</strong> ∇ is the flatc<strong>on</strong>necti<strong>on</strong> with skew-symmetric torsi<strong>on</strong> g(X,[Y,Z]). Moreover, (J L ,J R ,g) is ageneralized Kähler structure.In [10] Cavalcanti proved that there are no nilmanifolds, except tori, carryingan invariant generalized Kähler structure, since every generalized complex
A <str<strong>on</strong>g>survey</str<strong>on</strong>g> <strong>on</strong> str<strong>on</strong>g KT structures 111structure <strong>on</strong> a nilpotent Lie algebra has holomorphically trivial can<strong>on</strong>ical bundle.About solvmanifolds no general results are known for the existence of generalizedKähler (n<strong>on</strong> Kähler) structures.In this secti<strong>on</strong> we review the c<strong>on</strong>structi<strong>on</strong> of the 6-dimensi<strong>on</strong>al generalizedKähler solvmanifold obtained in [18] as T 2 -bundle over an Inoue surface of typeS M .Let s a,b be the 2-step solvable Lie algebra with structure equati<strong>on</strong>s:⎧de 1 = ae 1 ∧ e 2 ,de 2 = 0,⎪⎨ de 3 = 1 2 ae2 ∧ e 3 ,⎪⎩de 4 = 1 2 ae2 ∧ e 4 ,de 5 = be 2 ∧ e 6 ,de 6 = −be 2 ∧ e 5 ,where a and b are n<strong>on</strong>-zero real numbers.If we denote by S a,b the simply-c<strong>on</strong>nected solvable Lie group with Lie algebras a,b , then the product <strong>on</strong> the Lie group, in terms of the global coordinates(t,x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) <strong>on</strong> IR 6 , is given by:(t,x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) · (t ′ ,x ′ 1,x ′ 2,x ′ 3,x ′ 4,x ′ 5) = (t + t ′ , e −a t x ′ 1 + x 1 , e a 2 t x ′ 2 + x 2 ,e a 2 t x ′ 3 + x 3 ,x ′ 4 cos(bt) − x ′ 5 sin(bt) + x 4 ,x ′ 4 sin(bt) + x ′ 5 cos(bt) + x 5 ).(4)Since the trace of ad X vanishes for any X ∈ s a,b and ad e2 has complex eigenvalues,the Lie group S a,b is unimodular and it is n<strong>on</strong>-completely solvable. Moreover,S a,b is as a semi-direct product of the formIR ⋉ ϕ (IR × IR 2 × IR 2 ),where ϕ = (ϕ 1 ,ϕ 2 ) is the diag<strong>on</strong>al acti<strong>on</strong> of IR <strong>on</strong> (IR×IR 2 )×IR 2 described by (4).In c<strong>on</strong>trast with the nilpotent case, there are no existence theorems for uniformdiscrete subgroups of a solvable Lie group and, if the Lie group is n<strong>on</strong>-completelysolvable and admits a compact quotient, <strong>on</strong>e cannot apply Hattori’s theorem [30]to compute the de Rham cohomology of the compact quotient.In [18] we showed that S a,b admits a compact quotient. Indeed <strong>on</strong>e has thefollowingTheorem 6.2 ([18]). Let S 1, π2Lie algebra s 1, π . Then 21. S 1, π has a compact quotient M6 = Γ\S2 1, π2Γ.be the simply-c<strong>on</strong>nected solvable Lie group withby a uniform discrete subgroup
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