A survey on strong KT structures - SSMR

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