Let (J, g) be a Hermitian structure on a six-dimensional compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We show that, up to equivalence of the complex structure, the strong Kähler with torsion structures (J, g) on M are parametrized by the points in a subset of the Euclidean space, in particular, the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold and the region outside to strong Kähler with torsion structures with nonabelian J on the nilmanifold Γ\(H 3 × H 3 ), where H 3 is the Heisenberg group. A classification of sixdimensional nilmanifolds admitting balanced Hermitian structures (J, g) is given, and as an application we classify the nilmanifolds having invariant complex structures which do not admit Hermitian structure with restricted holonomy of the Bismut connection contained in SU(3). It is also shown that on the nilmanifold Γ\(H 3 × H 3 ) the balanced condition is not stable under small deformations. Finally, we prove that a compact quotient of H(2, 1) × R, where H(2, 1) is the five-dimensional generalized Heisenberg group, is the only six-dimensional nilmanifold having locally conformal Kähler metrics, and the complex structures underlying such metrics are all equivalent. Moreover, this nilmanifold is a Vaisman manifold for any invariant locally conformal Kähler metric.
We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behaviour of the associated Frölicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property are not closed under holomorphic deformation.1 Example 5.8. Let us consider the Lie algebra h 5 with the real basis {e 1 , . . . , e 6 } described in Theorem 2.1. Let us consider the complex structure J 0,0 given by J 0,0 e 1 = −e 2 , J 0,0 e 3 = −2e 2 − e 4 , J 0,0 e 5 = −e 6 , J 0,0 e 2 = e 1 , J 0,0 e 4 = −2e 1 + e 3 , J 0,0 e 6 = e 5 .
Abstract. We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if Γ\G is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology H * , * ∂ (Γ\G) is canonically isomorphic to thē ∂-cohomology H * , * ∂ (g C ) of the bigraded complex (Λ * , * (g C ) * ,∂) of complex valued left invariant differential forms on the nilpotent Lie group G.
We construct new explicit compact supersymmetric valid solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic equations of motion in dimension six. We present balanced Hermitian structures on compact nilmanifolds in dimension six satisfying the heterotic supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton which obey the three-form Bianchi identity with curvature term taken with respect to either the Levi-Civita, the (+)-connection or the Chern connection. Among them, all our solutions with respect to the (+)-connection on the compact nilmanifold M 3 satisfy the heterotic equations of motion.
The invariant balanced Hermitian geometry of nilmanifolds of dimension 6 is described. We prove that the (restricted) holonomy group of the associated Bismut connection reduces to a proper subgroup of SU.3/ if and only if the complex structure is abelian. As an application we show that if J is abelian, then any invariant balanced J -Hermitian structure provides solutions of the Strominger system.
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