Symplectically harmonic cohomology of nilmanifolds

Commentarii Mathematici Helvetici, 2001

In the present paper we study the variation of the dimensions h k of spaces of symplectically harmonic cohomology classes (in the sense of Brylinski) on closed symplectic manifolds. We give a description of such variation for all 6-dimensional nilmanifolds equipped with symplectic forms. In particular, it turns out that certain 6-dimensional nilmanifolds possess families of homogeneous symplectic forms $ \omega_t $ for which numbers $ h_k({\rm M},\omega_t) $ vary with respect to t. This gives an affirmative answer to a question raised by Boris Khesin and Dusa McDuff. Our result is in contrast with the case of 4-dimensional nilmanifolds which do not admit such variations by a remark of Dong Yan.

Rocky Mountain Journal of Mathematics, 2010

We prove that an (ϕ, J)-holomorphic maps from a compact cosymplectic manifold to a Kähler manifold is not only a harmonic map but also an energy minimizer on its homotopy class. We also prove a converse result.

Research Square (Research Square), 2024

Brylinski introduced the symplectic analogue of harmonic forms on symplectic manifolds, thereby defining the symplectically harmonic cohomology space of a symplectic manifold. In this paper, we derive a formula for determining the symplectically harmonic Betti numbers of an arbitrary symplectic manifold of finite type, based on the dimensions of the kernels of the Lefschetz homomorphisms. Utilizing this formula, we compute the symplectically harmonic Betti numbers of a specific class of 4-dimensional symplectic manifolds, namely, McMullen-Taubes symplectic 4-manifolds, which includes the Kodaira-Thurston symplectic nilmanifold as a special case.

2000

We consider invariant symplectic connections ∇ on homogeneous symplectic manifolds (M, ω) with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible with the symplectic structure. If M is compact with finite fundamental group then (M, ω) is symplectomorphic to P n (C) with a multiple of its Kähler form and ∇ is affinely equivalent to the Levi-Civita connection. Research of the first three authors supported by an ARC of the communauté française de Belgique.

2003

We consider the question of existence of symplectic and Kähler structures on compact homogeneous spaces of solvable triangular Lie groups. The aim of the article is to clarify the situation with examples in this area. We prove that it is impossible to complete the construction of examples in the well-known article by Benson and Gordon on the structure of compact solvmanifolds with Kähler structure. We do this by proving the absence of lattices (and thereby a compact form) in the Lie groups of the above-mentioned article. We construct a new (similar) example for which, unlike the above examples, a compact form exists. We consider one class of solvable Lie groups, namely the class of almost abelian groups, and obtain for this class a characterization of those Lie groups for which the cohomologies of their compact solvmanifolds are isomorphic to the cohomologies of the corresponding Lie algebras. Until recently, such isomorphism has been known only for one specific class of Lie groups, namely the class of triangular groups. We give examples of new (almost abelian) Lie groups with such isomorphism.

arXiv (Cornell University), 2023

For a compact subset K of a closed symplectic manifold (M, ω), we prove that K is heavy if and only if its relative symplectic cohomology over the Novikov field is non-zero. As an application we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results are also included. Conjecture 1.6 (Conjecture 1.52 ). A compact subset of a closed symplectic manifold is heavy if and only if it is SH-heavy. Here is a summary of our main results. Theorem 1.7. Let (M, ω) be a closed symplectic manifold. For any compact subset K of M , the following statements are true. (A) (Corollary 3.4, Theorem 3.5) K is SH-visible if and only if K is heavy. (B) (Proposition 3.9 The proof of our most fundamental result, that SH-visible implies heavy, relies on a chain level product structure, which was recently constructed in . We only use a small part of the structure constructed in that paper. Here is an interesting corollary of Theorem 1.7 part (A) and the Mayer-Vietoris property of relative symplectic cohomology: Corollary 1.8. Let K 1 and K 2 be Poisson commuting compact subsets of M , see Definition 2.17. If they are both not heavy, then neither is their union K 1 ∪ K 2 .

arXiv (Cornell University), 2008

For any compact almost complex manifold (M, J), the last two authors [8] defined two subgroups H + J (M ), H - J (M ) of the degree 2 real de Rham cohomology group H 2 (M, R). These are the sets of cohomology classes which can be represented by J-invariant, respectively, Janti-invariant real 2-forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of H 2 (M, R). This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini [6]. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.

2000

The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds.

Journal of Symplectic Geometry, 2012

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.

Mathematische Annalen, 2011

We consider actions of reductive complex Lie groups G = K C on Kähler manifolds X such that the K -action is Hamiltonian and prove then that the closures of the G-orbits are complex-analytic in X . This is used to characterize reductive homogeneous Kähler manifolds in terms of their isotropy subgroups. Moreover we show that such manifolds admit K -moment maps if and only if their isotropy groups are algebraic.