Group actions, non-Kähler complex manifolds and SKT structures

Abstract

Abstract:We give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-Kähler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the principal bundle admits SKT metrics. This generalizes recent results of Grantcharov et al. We study the Picard group and the algebraic dimension of the total space in some cases. We also use a slightly generalized version of the construction to obtain (non-Kähler) complex structures on tangential frame bundles of complex orbifolds.

A construction of SKT manifolds using toric geometry

“…If the bundle π : E → X is obtained by reduction of structure group from a holomorphic principal bundle over X, then the above construction of integrable complex structure on E is applicable, see [20,Section 5]. In the sequel, we will apply this construction when the torus bundle is obtained by reduction from a direct sum of holomorphic line bundles.…”

“…Compact connected Lie groups of even dimension admit invariant complex structures (cf. [19] or [20,Section 2]). Let G be such a group whose rank ≥ 2k.…”

“…Moreover, some of these examples may be used to produce an infinite family of examples by applying the J-construction in toric geometry. Finally, using some results of [20], we produce more SKT manifolds from each of the above examples by extension of structure group.…”

Picard groups of certain compact complex parallelizable manifolds and related spaces

“…See [LN], [ST]. Also Picard groups of certain compact complex manifolds which are holomorphically fibred by complex tori have been considered by Poddar and Thakur [PT,§6]. These manifolds are all non-Kähler as are the manifolds P Γ .…”

Hermitian non-Kähler structures on products of principal $S^{1}$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

“…concrete treatment in terms of differential forms, irreducible representations, and painted Dynkin diagrams for such constructions of complex structures. Also, by looking at the aspects of Hermitian geometry on manifolds provided by total spaces of principal torus bundles over flag manifolds [20,Proposition 4.26], [26], [57], as applications of our main results, we give a concrete description of Calabi-Yau connections with torsion (CYT structures) on certain Vaisman manifolds (generalized Hopf manifolds [72]). Recently, CYT structures on non-K ähler manifolds have attracted attention as models for string compactifications, see for instance [37] and references therein.…”