Symplectically harmonic cohomology of nilmanifolds

Trans. Amer. Math. Soc.

2006

Self Cite

Abstract. For a symplectic manifold, the harmonic cohomology of symplectic divisors (introduced by Donaldson, 1996) and of the more general symplectic zero loci (introduced by Auroux, 1997) are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the s-Lefschetz property. In particular, we consider the symplectic blow-ups CP m of the complex projective space CP m along weakly Lefschetz symplectic submanifolds M ⊂ CP m . As an application we construct, for each even integer s ≥ 2, compact symplectic manifolds which are s-Lefschetz but not (s + 1)-Lefschetz.

“…has at each entry an -perturbation of the corresponding entry of B µ , by (14). Hence its determinant is λ · a + O( a+1 ), and it is non-zero for small > 0.…”

Section: We Have That [ω ] Is a Rational Class (And Hence A Multiple mentioning

confidence: 99%

Weakly Lefschetz symplectic manifolds

Fernández

Muñoz

Trans. Amer. Math. Soc.

2006

Self Cite

“…. , h 16 , and Family III to h By Nomizu theorem one can compute de Rham cohomology groups in terms of the complex structure equations (3.2), (3.3) and (3.4). From now on, the notation δ expression means that δ expression = 1 if expression = 0 is satisfied, and δ expression = 0 otherwise.…”

Section: Cohomological Decomposition Of 6-dimensional Complex Nilmanimentioning

confidence: 99%

“…For instance, the Kodaira-Thurston (nil)manifold [22] was the first example of a symplectic manifold with no Kähler metric, and more generally, Benson and Gordon proved [5] that a symplectic nilmanifold satisfies the Hard Lefschetz Condition (HLC) if and only if it is a torus. Regarding the HLC, given a symplectic manifold (M, ω), Mathieu [19] proved that any de Rham cohomology class of M has a symplectically harmonic representative (in the sense of Brylinski [6]) if and only if (M, ω) satisfies the HLC, and in [16] symplectic nilmanifolds were used to find the first examples of 6-dimensional compact manifolds that are symplectically flexible, giving in this way an affirmative answer to a question raised by Khesin and McDuff (see [24]). There are other interesting constructions in symplectic geometry where nilmanifolds are involved (see [23]).…”

Section: Introductionmentioning

confidence: 99%

Cohomological decomposition of complex nilmanifolds

Latorre

2015

TMNA

Abstract. We study pureness and fullness of invariant complex structures on nilmanifolds. We prove that in dimension six, apart from the complex torus, there exist only two non-isomorphic complex structures satisfying both properties, which live on the real nilmanifold underlying the Iwasawa manifold. We also show that the product of two almost complex manifolds which are pure and full is not necessarily full.

Symplectic harmonicity and generalized coeffective cohomologies

Villacampa

2019

Annali di Matematica

Relations between the symplectically harmonic cohomology and the coeffective cohomology of a symplectic manifold are obtained. This is achieved through a generalization of the latter, which in addition allows us to provide a coeffective version of the filtered cohomologies introduced by C.-J. Tsai, L.-S. Tseng and S.-T. Yau. We construct closed (simply connected) manifolds endowed with a family of symplectic forms ωt such that the dimensions of these symplectic cohomology groups vary with respect to t. A complete study of these cohomologies is given for 6-dimensional symplectic nilmanifolds, and concrete examples with special cohomological properties are obtained on an 8-dimensional solvmanifold and on 2-step nilmanifolds in higher dimensions.