Hodge Structure on Symplectic Manifolds
Hodge Structure on Symplectic Manifolds
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DOI：
10.1006/aima.1996.0034
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发表时间：
1996-06
影响因子：
1.7
通讯作者：
D. Yan
中科院分区：
文献类型：
--
作者：
D. Yan
J. Brylinski在[B]引入了辛调和形式的概念。进一步，他推测在紧辛流形上，每一个de Rham上同调类都包含一个调和代表。Brylinski的猜想在辛意义上等价于Hodge分解的存在性问题。（在这种情况下，分解的唯一性显然是不正确的。）Olivier Mathieu在[M]中反驳了Brylinski的猜想。事实上，他证明了下面的定理，它适用于任何辛流形。
J. Brylinski introduced in [B] the notion of symplectic harmonic forms. Further he conjectured that on a compact symplectic manifold, every de Rham cohomology class contains a harmonic representative. Brylinski's conjecture is equivalent to the question of the existence of a Hodge decomposition in the symplectic sense.(The uniqueness of the decomposition in this case is evidently not true.) Olivier Mathieu disproved Brylinski's conjecture in [M]. In fact, he showed the following theorem, which applies to any symplectic manifold.