摘要翻译：
真正的传说子变体是微分几何和经典力学的经典对象，自古以来就有研究。然而，复杂的传奇亚变种更加僵硬，具有更多的特殊性质。最显著的例子是射影空间的Legendrian子类，在作者的研究之前，这些子类的光滑例子很少。本文的第一系列结果与任意射影接触流形中任意Legendrian子簇的自同构群有关。这个群的连通分量（在适当的次要假设下）完全由接触流形上的区分线丛的截面在勒让德里变体上消失而确定。而且，它的作用保留了接触结构。第二系列结果致力于寻找射影空间光滑Legendrian子簇的新例子。本文的贡献主要体现在三个方面：首先，我们给出了光滑曲面的一个例子。其次，我们找到了一个允许Legendrian嵌入的光滑拟齐次8重Fano。最后，我们认识到这两个都是非常一般构造的特例：光滑Legendrian变体的一般超平面截面，经过适当的投影后，是维数较小的光滑Legendrian变体。通过将这一结果应用于已知的例子和可分解的Legendrian变体，我们在每个维上构造了无限多个新的例子，这些例子具有不同的Picard秩、规范度、Kodaira维数和其他不变量。
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英文标题：
《Algebraic Legendrian Varieties》
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作者：
Jaros{\l}aw Buczy\'nski
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最新提交年份：
2010
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space and prior to the author's research only few smooth examples of these were known.   The first series of results of this thesis is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable minor assumptions) is completely determined by the sections of the distinguished line bundle on the contact manifold vanishing on the Legendrian variety. Moreover its action preserves the contact structure.   The second series of results is devoted to finding new examples of smooth Legendrian subvarieties of projective space. The contribution of this thesis is in three steps: First we find an example of a smooth toric surface. Next we find a smooth quasihomogeneous Fano 8-fold that admits a Legendrian embedding. Finally, we realise that both of these are special cases of a very general construction: a general hyperplane section of a smooth Legendrian variety, after a suitable projection, is a smooth Legendrian variety of smaller dimension. By applying this result to known examples and decomposable Legendrian varieties, we construct infinitely many new examples in every dimension, with various Picard rank, canonical degree, Kodaira dimension and other invariants.
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PDF链接：
https://arxiv.org/pdf/0805.3848