Hodge theory on transversely symplectic foliations
Hodge theory on transversely symplectic foliations
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DOI：
10.1093/qmath/hax051
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发表时间：
2016-09
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影响因子：
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通讯作者：
Yi Lin
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文献类型：
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作者：
Yi Lin
在本文中，我们发展了关于横辛叶理的辛Hodge理论。特别地，我们建立了具有（横向）$s$ -Lefschetz性质的任何这类叶的辛$d\delta$ -引理。由于横向辛叶形包括许多几何结构，如接触流形、共辛流形、辛轨道和辛拟褶皱等，我们的工作提供了这些几何结构中辛霍奇理论的统一处理。作为一个应用，我们证明了在紧致$K$ -接触流形上，$s$ -Lefschetz性质暗示了杯积消失的一般结果，并且具有（横向）$s$ -Lefschetz性质的$2n+1$维紧致$K$ -接触流形的杯长最多为$2n-s$。对于任何偶数$s\geq 2$，我们也应用我们的主要结果来产生$K$ -接触流形的例子，这些流形是$s$ -Lefschetz而不是$(s+1)$ -Lefschetz。
In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic $d\delta$-lemma for any such foliations with the (transverse) $s$-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds, and symplectic quasi-folds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries. As an application, we show that on compact $K$-contact manifolds, the $s$-Lefschetz property implies a general result on the vanishing of cup products, and that the cup length of a $2n+1$ dimensional compact $K$-contact manifold with the (transverse) $s$-Lefschetz property is at most $2n-s$. For any even integer $s\geq 2$, we also apply our main result to produce examples of $K$-contact manifolds that are $s$-Lefschetz but not $(s+1)$-Lefschetz.