A survey on strong KT structures - SSMR

104 Anna Fino and Adriano TomassiniC<strong>on</strong>sider the inner product g defined byg =6∑e j ⊗ e j . (2)j=1Thus g is J-Hermitian. Denote by F the fundamental 2-form associated with theHermitian structure (J,g). By a direct computati<strong>on</strong> we haveJdF = −e 1 ∧ e 2 ∧ e 3 ,with e 1 ∧ e 2 ∧ e 3 a closed and n<strong>on</strong>-exact 3-form.Therefore g admits a str<strong>on</strong>g KT structure defined by (J,g). The Bismut c<strong>on</strong>necti<strong>on</strong>∇ B in this case is flat. Indeed, the torsi<strong>on</strong> of ∇ B is the 3-formJdF = −e 1 ∧ e 2 ∧ e 3 .Hence, the torsi<strong>on</strong> forms τ i ,i = 1,...,6 of ∇ B are given byτ 1 = e 2 ∧ e 3 , τ 2 = −e 2 ∧ e 3 , τ 3 = e 1 ∧ e 2 , τ 4 = 0, τ 5 = 0, τ 6 = 0.Denoting by ω i j the c<strong>on</strong>necti<strong>on</strong>s 1-forms of ∇B , by the Cartan structure equati<strong>on</strong>s6∑de i + ωj i ∧ e j = τ i , ωj i + ω j ij=1= 0, i,j = 1,...,6, (3)we immediately getω 5 6 = e 4 ,the other ω i j being zero. Hence ∇B is flat and c<strong>on</strong>sequently its hol<strong>on</strong>omy algebrais trivial. Moreover, the Lee form, given by e 4 is closed and then the Hermitianmetric is locally c<strong>on</strong>formally balanced. The simply c<strong>on</strong>nected Lie group H withLie algebra h is a semi-direct product of the form IR⋉IR 2 and by c<strong>on</strong>sidering thelattice Γ in H generated by 1 2 and Z2 , <strong>on</strong>e has that b 1 (Γ\H) = 1. Therefore, theabove Hermitian structure (J,g) <strong>on</strong> g induces a left-invariant Hermitian structure(J,g) <strong>on</strong> the compact manifold M = Γ\H ×S 3 /K, which can be viewed as a n<strong>on</strong>trivialT 2 -bundle over the Hopf surface. Note that∂ ( η 1 ∧ η 2 ∧ η 3) = 1 2 (1 − i) η1 ∧ η 1 ∧ η 2 ∧ η 3 ,and thus there are no left-invariant holomorphic (3,0)-forms <strong>on</strong> M. Since thehol<strong>on</strong>omy of the Bismut c<strong>on</strong>necti<strong>on</strong> is c<strong>on</strong>tained in SU(3), then by [16, Theorem4.1] (M,J) cannot admit any n<strong>on</strong>-vanishing holomorphic (3,0)-form. Indeed, if(M,J) admits such a form, then (J,g) has to be c<strong>on</strong>formally balanced, but thisis not possible since (J,g) is str<strong>on</strong>g KT.