A survey on strong KT structures - SSMR

108 Anna Fino and Adriano TomassiniKT structure, or alternatively, that any 6-dimensi<strong>on</strong>al compact simply-c<strong>on</strong>nectedspin manifold with torsi<strong>on</strong> free cohomology and free S 1 -acti<strong>on</strong> has a str<strong>on</strong>g KTmetric. The previous manifold has been c<strong>on</strong>structed as a real two-dimensi<strong>on</strong>altoric bundle over the complex surface obtained as blow up of the complex projectiveplane CP 2 at k (k ≥ 2) points <strong>on</strong> a smooth irreducible cubic.Recently Swann in [45] reproduced the previous examples via the twist c<strong>on</strong>structi<strong>on</strong>,extending them to higher dimensi<strong>on</strong>s, and finding further compactsimply-c<strong>on</strong>nected str<strong>on</strong>g KT manifolds. The basic idea of this c<strong>on</strong>structi<strong>on</strong> is toc<strong>on</strong>sider a manifold M with an acti<strong>on</strong> of a n-dimensi<strong>on</strong>al torus T n . If P → M is isa principal T n -bundle with c<strong>on</strong>necti<strong>on</strong> and if the T n -acti<strong>on</strong> lifts to P commutingwith the principal acti<strong>on</strong>, then <strong>on</strong>e can c<strong>on</strong>struct the quotient space (the “twist ”)P/T n . Moreover, if the lifted T n -acti<strong>on</strong> preserves the principal c<strong>on</strong>necti<strong>on</strong>, thentensors <strong>on</strong> M can be transferred to tensors <strong>on</strong> the quotient P/T n by requiringtheir pull-backs to P to coincide <strong>on</strong> horiz<strong>on</strong>tal vectors. In this way, an invariantgeometric structure <strong>on</strong> M determines a corresp<strong>on</strong>ding geometric structure <strong>on</strong> thetwist P/T n .5 6-dimensi<strong>on</strong>al str<strong>on</strong>g KT nilmanifoldsOther examples, besides the compact semisimple Lie groups, of homogeneousstr<strong>on</strong>g KT manifolds have been found in [17] and they are 6-dimensi<strong>on</strong>al nilmanifoldsΓ\G, i.e. compact quotients of 6-dimensi<strong>on</strong>al nilpotent Lie groups G bydiscrete subgroups Γ.By the classificati<strong>on</strong> obtained in [17] it turns out that <strong>on</strong>ly 4 classes of 6-dimensi<strong>on</strong>al nilpotent Lie algebras admit a str<strong>on</strong>g KT structure and that theexistence of a str<strong>on</strong>g KT metric compatible with a left-invariant complex structureJ <strong>on</strong> the nilmanifold Γ\G depends <strong>on</strong>ly <strong>on</strong> the complex structure J.More preciselyTheorem 5.1. [17] Let M = Γ\G be a 6-dimensi<strong>on</strong>al nilmanifold, J be a leftinvariantcomplex structure and g any J-Hermitian metric. Then the Hermitianstructure (J,g) is str<strong>on</strong>g KT if and <strong>on</strong>ly if there exists a basis (α 1 ,α 2 ,α 3 ) ofleft-invariant (1,0)-forms such that⎧⎪⎨⎪⎩with A,B,C,D,E ∈ C such thatdα 1 = dα 2 = 0,dα 3 = Aα 1 ∧ α 2 + Bα 2 ∧ α 2 + Cα 1 ∧ α 1 +Dα 1 ∧ α 2 + Eα 1 ∧ α 2|A| 2 + |D| 2 + |E| 2 + 2Re (BC) = 0.