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 103Moreover, this example shows also that the property “vanishing Ricci tensorfor the Bismut c<strong>on</strong>necti<strong>on</strong>” is not stable under small deformati<strong>on</strong>s of the complexstructure.The balanced c<strong>on</strong>diti<strong>on</strong> is complementary to the str<strong>on</strong>g KT <strong>on</strong>e. Indeed, by[4], a (n<strong>on</strong>-Kähler) Hermitian metric g <strong>on</strong> a complex manifold (M,J) of complexdimensi<strong>on</strong> n > 2, can be str<strong>on</strong>g KT, <strong>on</strong>ly if the Lee form θ does not vanish.In [34], it was shown that a c<strong>on</strong>formally balanced str<strong>on</strong>g KT structure (J,g) <strong>on</strong>a compact manifold M of real dimensi<strong>on</strong> 2n with Hol(∇ B ) ⊆ SU(n) is necessarilyKähler and therefore it gives rise to a Calabi-Yau structure.It is an interesting problem to find examples of complex manifolds whichadmit str<strong>on</strong>g CYT structures. Sufficient c<strong>on</strong>diti<strong>on</strong>s for principal toric bundlesover compact Kähler manifolds to admit Calabi-Yau with torsi<strong>on</strong> structures andas well str<strong>on</strong>g KT metrics have been found in [21].We will provide a compact example of str<strong>on</strong>g CYT manifold, c<strong>on</strong>structed asa T 2 -bundle over the Hopf surface. This complex manifold is locally c<strong>on</strong>formallybalanced, i.e. its Lee form θ is closed and n<strong>on</strong>-exact.Example 2.4. We recall that a Hopf surface is diffeomorphic to a fiber bundleX = S 1 × Zm S 3 /K over S 1 with fiber S 3 /K, where K is a finite subgroup ofU(2) acting freely <strong>on</strong> X [29]. Then, the Hopf surface can be also viewed as acompact quotient X = L/Θ, where the Lie algebra of L is l = su(2) ⊕ IR and Θis a uniform discrete subgroup of L.C<strong>on</strong>sider the 6-dimensi<strong>on</strong>al Lie algebra g with structure equati<strong>on</strong>s:⎧de 1 = e 2 ∧ e 3 ,de 2 = e 3 ∧ e 1 ,⎪⎨ de 3 = e 1 ∧ e 2 ,(1)de 4 = 0,de⎪⎩5 = e 6 ∧ e 4 ,de 6 = e 4 ∧ e 5 .The Lie algebra g is the direct sum su(2) ⊕ h, where h is a 3-dimensi<strong>on</strong>aln<strong>on</strong>-completely solvable Lie algebra. Moreover, l = span < e 1 ,e 2 ,e 3 ,e 4 > is a Liesubalgebra of g, where we denote by {e 1 ,...,e 6 } the dual basis of {e 1 ,...,e 6 }.Define the almost complex structures J <strong>on</strong> g, by settingη 1 = e 1 + ie 4 , η 2 = e 2 + ie 3 , η 3 = e 5 + ie 6 .Then by definiti<strong>on</strong> (η 1 ,η 2 ,η 3 ) are the (1,0)-forms associated with J. The almostcomplex structure J is integrable. Indeed:dη 1 = i 2 η2 ∧ η 2 ,dη 2 = i 2 (η1 ∧ η 2 + η 1 ∧ η 2 ),dη 3 = 1 2 (η1 ∧ η 3 − η 1 ∧ η 3 ).