Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form ∂∂-closed, i.e., to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed symplectic structure but non-admitting Kähler structures. We show some negative results for the existence of symplectic forms taming complex structures on compact quotients of Lie groups by discrete subgroups. In particular, we prove that if M is a nilmanifold (not a torus) endowed with an invariant complex structure J, then (M, J) does not admit any symplectic form taming J. Moreover, we show that if a nilmanifold M endowed with an invariant complex structure J admits an SKT metric, then M is at most 2-step. As a consequence we classify eight-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.
We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving tamed symplectic forms on a complex manifold, by considering evolution of symplectic forms via the flow induced by the Bismut Ricci form.
An SKT metric is a Hermitian metric on a complex manifold whose fundamental 2-form ω satisfies ∂∂ω = 0. Streets and Tian introduced in [27] a Ricci-type flow that preserves the SKT condition. This flow uses the Ricci form associated to the Bismut connection, the unique Hermitian connection with totally skew-symmetric torsion, instead of the Levi-Civita connection. A SKT metric is static if the (1,1)-part of the Ricci form of the Bismut connection satisfies (ρ B ) (1,1) = λω for some real constant λ. We study invariant static metrics on simply connected Lie groups, providing in particular a classification in dimension 4 and constructing new examples, both compact and non-compact, of static metrics in any dimension.
We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized Kähler structures. By considering the commutator Q of the two associated almost complex structures J ± , we prove that if either the manifold is 4dimensional or the distribution Im Q is involutive, then the manifold can be expressed locally as a disjoint union of twisted Poisson leaves.g is SKT with respect to J ± and both the torsion 3-forms of the Bismut connections ∇ B ± are exact. Note that a symplectic form Ω on a manifold M induces the generalized complex structurewhich is integrable since dΩ = 0. The following proposition allows us to see symplectic structures taming complex structures as particular cases of (almost) generalized Kähler structures.Proposition 2.1. Let (M, Ω) be a symplectic manifold. Giving a almost generalized Kähler structure on M such that one of the generalized complex structures is induced by Ω is equivalent to assigning an almost complex structure J such that Ω tames J, i.e. Ω(JX, X) > 0 for any vector field X = 0. Moreover, the almost generalized Kähler structure is integrable if and only if the almost complex structures J and −Ω −1 J * + Ω are integrable.
A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form ω is ∂∂-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a 2n-dimensional SKT Lie algebra g and using a Hermitian flat connection on g we construct a 4n-dimensional SKT Lie algebra. We apply this method to some 4-dimensional SKT Lie algebras. Moreover, we classify symplectic forms taming complex structures on 4-dimensional Lie algebras.
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