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 109As a c<strong>on</strong>sequence of the last Theorem, the nilpotent Lie group G has to be2-step and the existence of a str<strong>on</strong>g KT metric depends <strong>on</strong>ly <strong>on</strong> the complexstructure. By applying Nomizu’s result [37] about the de Rham cohomology of anilmanifold, <strong>on</strong>e obtains that the first Betti number b 1 (M) of M is at least 4.The previous result has been used in [17] to classify explicitly the real 6-dimensi<strong>on</strong>al nilpotent Lie algebras admitting a str<strong>on</strong>g KT structure. Indeed, <strong>on</strong>ehas that a 6-dimensi<strong>on</strong>al nilmanifold M = Γ\G, endowed with a left-invariantcomplex structure J, has a J-Hermitian str<strong>on</strong>g KT metric if and <strong>on</strong>ly if the Liealgebra g of G is isomorphic to <strong>on</strong>e of the following(0,0,0,0,13 + 42,14 + 23),,(0,0,0,0,12,14 + 23),(0,0,0,0,12,34),(0,0,0,0,0,12),where, for instance for (0,0,0,0,0,12) we mean the Lie algebra with structureequati<strong>on</strong>s{de i = 0, i = 1,...,5,de 6 = e 1 ∧ e 2 .A detailed study of the str<strong>on</strong>g KT structures, up to equivalence of the complexstructure <strong>on</strong> 6-dimensi<strong>on</strong>al nilpotent Lie algebra, was also carried out in [46].By [45] the str<strong>on</strong>g KT nilmanifolds can be also obtained applying repeatedlythe “twist c<strong>on</strong>structi<strong>on</strong> ”to a torus.Since the c<strong>on</strong>diti<strong>on</strong> str<strong>on</strong>g KT <strong>on</strong> the 6-dimensi<strong>on</strong>al nilmanifolds depends <strong>on</strong>ly<strong>on</strong> the complex structure, in [17] we studied the str<strong>on</strong>g KT equati<strong>on</strong>s when G isthe complex Heisenberg group HC 3 and the compact quotient M = Γ\H3 C is theIwasawa manifold. N<strong>on</strong>e of the standard complex structures (see [1]) <strong>on</strong> HC 3 arestr<strong>on</strong>g KT, so it was interesting to discover which <strong>on</strong>es are.By the results of [38], the features of an invariant complex structure J <strong>on</strong> Mdepend <strong>on</strong> a matrix XX, where X is a 2 × 2 matrix representing the inducedacti<strong>on</strong> of J <strong>on</strong> M/T 2 ∼ = T 4 , by viewing the Iwasawa manifold M as the totalspace of a T 2 -bundle over T 4 . In [17] it is showed that the str<strong>on</strong>g KT c<strong>on</strong>diti<strong>on</strong>c<strong>on</strong>strains the eigenvalues of XX to be complex c<strong>on</strong>jugates lying <strong>on</strong> the curve ofequati<strong>on</strong>(1 + |z| 2 ) |1+z| 2 = 8|z| 2in the complex plane.Moreover, by [19] the Iwasawa manifold Γ\H C 3 is also an example for whichthe c<strong>on</strong>diti<strong>on</strong> for a Hermitian metric to be str<strong>on</strong>g KT is not stable under smalldeformati<strong>on</strong>s of the complex structure underlying the str<strong>on</strong>g KT structure.