A survey on strong KT structures - SSMR

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

112 Anna Fino and Adriano Tomassini2. The compact manifold M 6 is the total space of a T 2 -bundle over an Inouesurface S M .3. M 6 has first Betti number equal to 1, thus it has no Kähler structures.4. The solvmanifold M 6 carries a left-invariant (n<strong>on</strong>-trivial) twisted generalizedKähler structure.In order to obtain a uniform discrete subgroup Γ of S a,b , in [18] we showedthat the solvable Lie group S 1, π is isomorphic to (IR6 = IR ⋉ (IR × C × C), ∗)2with product ∗ given by(t,u,z,w) ∗ (t ′ ,u ′ ,z ′ ,w ′ ) = (t + t ′ ,c t u ′ + u,α t z ′ + z,e i π 2 t w ′ + w),for any t,t ′ ,u,u ′ ∈ IR and z,z ′ ,w,w ′ ∈ C.It turns out that the discrete subgroup Γ is isomorphic to Z ⋉( Z 3 × Z 2 ) andit is generated by the transformati<strong>on</strong>sMoreover, the mapg 0 : (t,u,z,w) ↦→ (t + 1,cu,αz,iw),g j : (t,u,z,w) ↦→ (t,u + c j ,z + α j ,w), j = 1,2,3,g 4 : (t,u,z,w) ↦→ (t,u,z,w + 1),g 5 : (t,u,z,w) ↦→ (t,u,z,w + i).π :IR ⋉ (IR × C × C) → IR ⋉ (IR × C),(t,u,z,w) ↦→ (t,u,z)inherits to M 6 the structure of a T 2 -bundle over an Inoue surface of type S M .In [18] we proved thatΓ/[Γ,Γ] ∼ = Zand c<strong>on</strong>sequently the first Betti number of M 6 is equal to 1.The left-invariant generalized Kähler structure is defined by the pair of Hermitianstructures (J ± ,g) <strong>on</strong> the Lie algebra s 1, π , given by the two integrable2complex structures J ± associated with the (1,0)-formsω 1 + = e 1 + ie 2 , ω 2 + = e 3 + ie 4 , ω 3 + = e 5 + ie 6 ,ω 1 − = e 1 − ie 2 , ω 2 − = e 3 + ie 4 , ω 3 − = e 5 + ie 6 .and compatible with the inner productg =6∑e α ⊗ e α .α=1
A <str<strong>on</strong>g>survey</str<strong>on</strong>g> <strong>on</strong> str<strong>on</strong>g KT structures 113By a direct computati<strong>on</strong> we get thatd c +F + = −d c −F − = e 1 ∧ e 3 ∧ e 4 ,which is a closed 3-form. Therefore, since the 3-form is n<strong>on</strong>-exact, the corresp<strong>on</strong>dingleft-invariant generalized Kähler structure <strong>on</strong> M 6 is twisted.References[1] E. Abbena, S. Garbiero, and S. Salam<strong>on</strong>, Almost Hermitian geometryof 6-dimensi<strong>on</strong>al nilmanifolds, Ann. Sc. Norm. Sup. 30 (2001), 147–170.[2] L. Alessandrini and G. Bassanelli, Plurisubharm<strong>on</strong>ic currents andtheir exstensi<strong>on</strong> across analytic subsets, Forum Math. 5 (1993), 291–316.[3] V. Apostolov and M. Gualtieri, Generalized Kähler manifolds withsplit tangent bundle, Comm. Math. Phys. 271 (2007), 561–575.[4] B. Alexandrov and S. Ivanov, Vanishing theorems <strong>on</strong> Hermitian manifolds,Diff. Geom. Appl. 14 (2001), 251–265.[5] H. Bursztyn, G. Cavalcanti and M. Gualtieri, Reducti<strong>on</strong> of Courantalgebroids and generalized complex structures, Adv. Math. 211 (2007), 726–765.[6] F.A. Belgun, On the metric structure of n<strong>on</strong>-Kähler complex surfaces,Math. Ann. 317 (2000), 1–40.[7] C. Bens<strong>on</strong> and C. Gord<strong>on</strong>: Kähler and symplectic structures <strong>on</strong> nilmanifolds,Topology 27 (1988), 513–518.[8] J.M. Bismut, A local index theorem of n<strong>on</strong>-Kähler manifolds, Math. Ann.284 (1989), 681–699.[9] A. Blanchard, Sur les variétés analtyques complexe, Ann. Scient. del’E.N.S. 73 (1956), 157–202.[10] G.R. Cavalcanti, Formality in generalized Kähler geometry, TopologyAppl. 154 (2007), 119–1125.[11] G. R. Cavalcanti, M. Fernández and V. Munoz, Symplectic resoluti<strong>on</strong>s,Lefschetz property and formality, Adv. Math. 218 (2008), 576–599.[12] G. R. Cavalcanti and M. Gualtieri, Generalized complex structures <strong>on</strong>nilmanifolds, J. Symplectic Geom. 2 (2004), 393–410.
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