A survey on strong KT structures - SSMR

102 Anna Fino and Adriano Tomassiniendowed with a left-invariant complex structure. In general, by [39] any simplyc<strong>on</strong>nected Lie group which admits a discrete subgroup with compact quotient isunimodular and in particular has a bi-invariant volume form dµ. For the balancedc<strong>on</strong>diti<strong>on</strong>, by using a “symmetrizati<strong>on</strong>” process, which is based <strong>on</strong> a previous ideaof Belgun [6], <strong>on</strong>e has the followingTheorem 2.2. [16] Let M be the compact quotient Γ\G of a 2n-dimensi<strong>on</strong>al Liegroup G by a discrete subgroup Γ. If M admits a left-invariant complex structureJ and F is the fundamental 2-form of a n<strong>on</strong>-invariant J-Hermitian metric g,then∫α(A 1 ,...,A 2n−2 ) = F n−1 | m (A 1 | m ,...A 2n−2 | m )dµ,Mis equal to ˜F n−1 for some fundamental 2-form ˜F of a left-invariant J-invariantHermitian metric ˜g. If dF n−1 = 0, then d ˜F n−1 = 0.Therefore, if g is balanced, then (M,J) admits a left-invariant balanced J-Hermitian metric ˜g. By using essentially the same argument, Ugarte showedin [46] that if the Hermitian metric g is str<strong>on</strong>g KT, then (M,J) admits a leftinvariantinvariant J-Hermitian metric ˜g, which is str<strong>on</strong>g KT.By using Theorem 2.1 and 2.2 the first author with Grantcharov c<strong>on</strong>structedin [16] a family of left-invariant complex structures <strong>on</strong> the Iwasawa manifold notadmitting balanced metrics, except for the natural bi-invariant complex structure.Example 2.3. The Iwasawa manifold is the compact quotient Γ\HC 3, where⎧⎛⎞ ⎫⎨ 1 z 1 z 3 ⎬H3 C = ⎝ 0 1 z 2 ⎠ : z j ∈ C⎩⎭ ,0 0 1is the complex Heisenberg group and Γ is the uniform discrete subgroup of HC3defined by z j , j = 1,2,3, Gaussian integers. Let g s,t be the family of 2-stepnilpotent Lie algebras defined by:⎧⎪⎨de j = 0, j = 1,...,4,de 5 = s(e 1 ∧ e 2 + 2e 3 ∧ e 4 ) + t(e 1 ∧ e 3 − e 2 ∧ e 4 ),⎪⎩de 6 = t(e 1 ∧ e 4 + e 2 ∧ e 3 ),where t ≠ 0 and s are real numbers. For any t ≠ 0 and s, the Lie algebra g t,sis isomorphic to the Lie algebra of H 3 C . Therefore, by c<strong>on</strong>sidering <strong>on</strong> g t,s thecomplex structure defined by the (1,0)-formsη 1 = e 1 + ie 2 , η 2 = e 3 + ie 4 , η 3 = e 5 + ie 6 ,<strong>on</strong>e gets a family of left-invariant complex structures J t,s <strong>on</strong> the Iwasawa manifold.In [16] it was proved that the Iwasawa manifold (Γ\H 3 C ,J t,s) admits ametric with vanishing Ricci tensor for the Bismut c<strong>on</strong>necti<strong>on</strong> if and <strong>on</strong>ly if s = 0.