We study equivariant contact structures on complex projective varietiesarising as partial flag varieties $G/P$, where $G$ is a connected,simply-connected complex simple group of type $ADE$ and $P$ is a parabolicsubgroup. We prove a special case of the LeBrun-Salamon conjecture for partialflag varieties of these types. The result can be deduced from Boothby'sclassification of compact simply-connected complex contact manifolds withtransitive action by contact automorphisms, but our proof is completelyindependent and relies on properties of $G$-equivariant vector bundles on$G/P$. A byproduct of our argument is a canonical, global description of theunique $SO_{2n}(\mathbb C)$-invariant contact structure on the isotropicGrassmannian of $2$-planes in $\mathbb C^{2n}$.
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机译：我们研究复杂投影变种的等变接触结构，将其作为部分标志变种$ G / P $，其中$ G $是一个连接的，简单连接的$ ADE $类型的复杂简单群，而$ P $是一个抛物子群。我们证明了LeBrun-Salamon猜想对于这些类型的部分标志变体的特例。可以从Boothby对通过接触自同构具有传递作用的紧凑型简单连接复杂接触流形的分类推论得出，但是我们的证明是完全独立的，并且依赖于$ G $ -P等价向量束的性质。我们论证的副产品是在$ \ mathbb C ^ {2n} $中，$ 2 $平面的各向同性Grassmannian上的唯一$ SO_ {2n}（\ mathbb C）$-不变接触结构的规范全局描述。
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