Tamed Symplectic forms and Strong Kahler with torsion metrics

Abstract

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.

Product of generalized p-Kähler manifolds

“…We refer to [10], p. 216: in this paper the authors classify eight-dimensional nilpotent Lie algebras admitting an SKT (= 1PL) structure. More precisely, in Theorem 4.2 they prove that, given a nilmanifold (different from a torus) M = G/Γ with dim R M = 8, with an invariant complex structure J, there exists an SKT metric on M compatible with J if and only if the Lie algebra of G belongs to one of the two families described in (4.1) and (4.3) ibidem.…”

“…(Recall that if the structure equations are rational, then there exists a Γ such that the quotient M is compact, see [10] p. 205). Let…”

“…Let us choose a very simple nilmanifold (M, J) belonging to the first family of [10]: that is, pick in (4.1):…”

Cohomological decomposition of compact complex manifolds and holomorphic deformations

“…Recall that Enrietti, Fino, and Vezzoni proved in [14] that an invariant complex structure J on a nilmanifold M is tamed by a symplectic form if and only if (M, J) is a complex torus. Any Hermitian-symplectic structure is in particular SKT, i.e.…”

Extending invariant complex structures

“…As we shall see the existence of a complex or a totally real ideal on g imposes extra conditions on the algebraic structure of g. Note that this viewpoint which makes use of algebraic tools was useful in several works (see [8,2,10,22,23,25,26] for instance).…”