Families of strong KT structures in six dimensions

Anna Fino, Maurizio Parton, Simon Salamon
2004 Commentarii Mathematici Helvetici  
This paper classifies Hermitian structures on 6-dimensional nilmanifolds M = Γ\G for which the fundamental 2-form is ∂∂-closed, a condition that is shown to depend only on the underlying complex structure J of M . The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limaçon-shaped curve in the complex plane. Related theory is used to provide examples of various types of Ricci-flat structures. Mathematics Subject Classification
more » ... lassification (2000). 53C55; 32G05, 17B30, 81T30. A. Fino, M. Parton and S. Salamon CMH investigate the situation with regard to other groups. In this paper, we study KT geometry on 6-dimensional nilmanifolds in which J and g arise from corresponding left-invariant tensors. If G is a simply-connected nilpotent Lie group, and if the structure equations of its Lie algebra are rational, then there exists a discrete subgroup Γ of G for which M = Γ\G is compact [22, 25] . Any left-invariant complex structure on G will pass to a complex structure J on M but, unless G is abelian, the ∂∂-lemma is not valid for J and in particular there is no compatible Kähler metric [3, 7, 17, 24] . As we explain in §6, there may or may not be invariant pseudo-Kähler metrics on (M, J). Eighteen of the thirty-four classes of real 6-dimensional nilpotent Lie algebras g admit a complex structure. Exactly four of these classes, all of them 2-step with b 1 4 and including the case in which g underlies the complex Heisenberg algebra, give rise to strong KT metrics. Given that compact nilmanifolds with a strong KT structure exist, it is perhaps surprising that there are so few classes. The classification over R is accomplished in §3, after an analysis of the relevant structure equations over C in § §1,2. A matrix formalism for describing (1,1)-forms is introduced in an attempt to make the calculations of this paper rather more enlightening. A striking feature of our classification is that the existence of a strong KT structure depends only on the complex structure of g, and this poses the question of understanding the solutions as a subset of an appropriate moduli space of complex structures. With this aim, we proceed to a detailed study of the strong KT equations when G is the complex Heisenberg group and M = Γ\G is the Iwasawa manifold. It is easy to check that none of the standard complex structures [1] on G are strong KT, so we were intrigued to discover which ones are. According to the third author's joint paper with Ketsetzis [21], essential features of an invariant complex structure J on M depend on XX, where X is a 2 × 2 matrix representing the induced action of J on M/T 2 ∼ = T 4 . In §4, we prove that the strong KT condition constrains the eigenvalues of XX to be complex conjugates lying on a curve in the complex plane. We interpret this result in terms of the action of the automorphism group of g in §5, and this leads to an explicit description of the solution space. An analogous study can probably be carried out when G = H 3 × H 3 is the product of real Heisenberg groups, using methods from [14] . A Hermitian manifold is called conformally balanced if the Lee 1-form θ (the 'trace' of dΩ) is exact. The study of such structures in relation to the Bismut connection ∇ is motivated by work of [26] , though there are less subtleties in our invariant context. It was observed in [2] that the vanishing of θ is complementary to the SKT condition (see §1), and we begin the final section by discussing some known facts concerning the situation in which the holonomy of ∇ reduces to SU (n). We list some 6-dimensional Lie algebras giving rise to nilmanifolds admitting such a reduction, and give an example of a pseudo-Riemannian metric with zero Ricci tensor.
doi:10.1007/s00014-004-0803-3 fatcat:uejc6hghabexjiezo7cziofboy