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| In [[mathematics]], a '''Lefschetz manifold''' is a particular kind of [[symplectic manifold]]<math>(M^{2n}, \omega)</math>, sharing a certain cohomological property with [[Kaehler manifold]]s, that of satisfying the conclusion of the [[Hard Lefschetz theorem]]. More precisely, the '''strong Lefschetz property''' asks that for <math>k = 1 \ldots n</math>, the cup product
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| :<math>\cup [\omega^k]: H^{n-k}(M, \mathbb R)\to H^{n+k}(M, \mathbb R) </math>
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| be an isomorphism.
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| The topology of these symplectic manifolds is severely constrained, for example their odd [[Betti number]]s 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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| ==Lefschetz maps==
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| Let <math>M</math> be a (<math>2n</math>)-dimensional smooth manifold. Each element
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| :<math>[\omega] \in H_{DR}^2 (M)</math>
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| of the second [[de Rham cohomology]] space of <math>M</math> induces a map
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| :<math>L_{[\omega]}: H_{DR} (M) \to H_{DR} (M), [\alpha] \mapsto [\omega \wedge \alpha]</math>
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| called the '''Lefschetz map''' of <math>[\omega]</math>. Letting <math>L_{[\omega]}^i</math> be the <math>i</math>th iteration of <math>L_{[\omega]}</math>, we have for each <math>0 \leq i \leq n</math> a map
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| :<math>L_{[\omega]}^i : H_{DR}^{n-i}(M) \to H_{DR}^{n+i}(M).</math>
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| If <math>M</math> is [[compact space|compact]] and [[oriented]], then [[Poincaré duality]] tells us that <math>H_{DR}^{n-i}(M)</math> and <math>H_{DR}^{n+i}(M)</math> are vector spaces of the same dimension, so in these cases it is natural to ask whether or not the various iterations of Lefschetz maps are isomorphisms.
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| The [[Hard Lefschetz theorem]] states that this is the case for the symplectic form of a compact Kähler manifold.
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| ==Definitions==
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| If
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| :<math>L_{[\omega]}^{n-1}: H_{DR}^1(M) \to H_{DR}^{2n-1}</math>
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| and
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| :<math>L_{[\omega]}^{n}: H_{DR}^0(M) \to H_{DR}^{2n}</math>
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| are isomorphisms, then <math>[\omega]</math> is a '''Lefschetz element''', or '''Lefschetz class'''. If
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| :<math>L_{[\omega]}^i : H_{DR}^{n-i}(M) \to H_{DR}^{n+i}(M)</math> | |
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| is an isomorphism for all <math>0 \leq i \leq n</math>, then <math>[\omega]</math> is a '''strong Lefschetz element''', or a '''strong Lefschetz class'''.
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| Let <math>(M,\omega)</math> be a <math>2n</math>-dimensional [[symplectic manifold]].Then it is orientable, but maybe not compact. One says that <math>(M,\omega)</math> is a '''Lefschetz manifold''' if <math>[\omega]</math> is a Lefschetz element, and <math>(M,\omega)</math> is a '''strong Lefschetz manifold''' if <math>[\omega]</math> is a strong Lefschetz element.
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| ==Where to find Lefschetz manifolds==
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| The real manifold underlying any [[Kähler manifold]] is a symplectic manifold. The [[strong Lefschetz theorem]] tells us that it is also a strong Lefschetz manifold, and hence a Lefschetz manifold. Therefore we have the following chain of inclusions.
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| <center>{Kähler manifolds} <math>\subset</math> {strong Lefschetz manifolds} <math>\subset</math>{Lefschetz manifolds} <math>\subset</math> {symplectic manifolds}</center>
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| In,<ref>C. Benson and C. Gordon, Kahler and symplectic structures on nilmanifolds, ''Topology'' 27 (1988), 513-518.</ref> Chal Benson and Carolyn S. Gordon proved that if a [[compact manifold|compact]] [[nilmanifold]] is a Lefschetz manifold, then it is diffeomorphic to a [[torus]]. The fact that there are nilmanifolds that are not diffeomorphic to a torus shows that there is some space between Kähler manifolds and symplectic manifolds, but the class of nilmanifolds fails to show any differences between Kähler manifolds, Lefschetz manifolds, and strong Lefschetz manifolds.
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| Gordan and Benson conjectured that if a compact [[complete solvmanifold]] admits a Kähler structure, then it is diffeomorphic to a [[torus]]. This has been proved. Furthermore, many examples have been found of solvmanifolds that are strong Lefschetz but not Kähler, and solvmanifolds that are Lefschetz but not strong Lefschetz. Such examples can be found in.<ref>Takumi Yamada, Examples of Compact Lefschetz Solvmanifolds, ''Tokyo J. Math'' Vol. 25, No. 2, (2002), 261-283.</ref>
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| ==Notes==
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| <references/>
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| [[Category:Symplectic geometry]]
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