Families of strong KT structures in six dimensions

Abstract

Abstract. 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 (2000). 53C55; 32G05, 17B30, 81T30.

Product of generalized p-Kähler manifolds

“…Notice that it is known that the same metric cannot be SKT and balanced (see f.i. [11], Proposition 1.4), but here we get more: I 3,1 cannot have balanced metrics.…”

“…In [9] pluriclosed (i.e. 1−pluriclosed) metrics are defined (see also [24]); a 1PL metric (manifold) is often called a strong Kähler metric (manifold) with torsion (SKT) (see among others [11]). Finally, 1WK forms are used, f.i., in [7], Theorem 1.2.…”

“…In [11] the authors study SKT metrics on (real) six-dimensional nilmanifolds, and prove that the condition ∂∂ω = 0 only depends on the underlying complex structure. In particular, they give in Theorem 1.2 a necessary and sufficient condition to the existence of a SKT metric in terms of the differentials of a basis of invariant (1, 0)−forms {ϕ 1 , ϕ 2 , ϕ 3 }.…”

Non-Kähler solvmanifolds with generalized Kähler structure

“…Therefore, the Hermitian structures (J ± , g) are strong Kähler with torsion (SKT). Such structures have been studied by many authors in [9,11,15,23,30].…”

“…Compact examples in six dimensions are given in [11] where the Hermitian manifolds are provided by nilmanifolds, i.e. compact quotients of nilpotent Lie groups endowed with a Hermitian structure (J, g) in which J and g arise from left-invariant tensors.…”