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In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold {\displaystyle } , sharing a certain cohomological property with Kähler manifolds, that of satisfying the conclusion of the Hard Lefschetz theorem. More precisely, the strong Lefschetz property asks that for k ∈ { 1 , … , n } {\displaystyle k\in \{1,\ldots ,n\}} , the cup product
be an isomorphism.
The topology of these symplectic manifolds is severely constrained, for example their odd Betti numbers are even. This remark leads to numerous examples of symplectic manifolds which are not Kähler, the first historical example is due to William Thurston.
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