论文标题
在复杂的拉格曼尼亚人中建造米龙诺夫周期
Constructing Mironov cycles in complex Grassmannians
论文作者
论文摘要
A. Mironov在$ \ Mathbb {C}^n $和$ \ Mathbb {C} \ Mathbb {p}^n $中提出了Lagrangian Submanifolds的构造。在那儿，他主要是因为这些拉格朗日的submanifolds（可以在一般的自我交叉口，因此我们称它们称为拉格朗日周期）以下事实，这是最小或汉密尔顿的最小拉格朗日submanifolds的新例子。 However the Mironov construction of lagrangian cycles itself can be directly extended to much wider class of compact algrebraic varieties: namely it works in the case when algebraic variety $X$ of complex dimension $n$ admits $T^k$ - action and an anti - holomorphic involution such that the real part $X_{\mathbb{R}} \subset X$ has real dimension $ n $，并且是横向曲线动作的。对于这种情况，人们有Lagrangian Submanifolds和周期的家庭。 在目前的小文本中，我们展示了Mironov Cycles的构造如何适合复杂的司羊曼尼亚人，从而简单地示例了$ {\ rm gr}（\ rm gr}（k，n+1）$，配备了标准的kahler形式的$ {\ rm gr}（k，n+1）$。可以肯定的是，文本还不完整，但是在新的现实中，我们想修复它，希望继续进行调查，并在未来的Mironov Cycles以$ {\ rm gr}（k，n+1）$的完整列表中介绍。
A. Mironov proposed a construction of lagrangian submanifolds in $\mathbb{C}^n$ and $\mathbb{C} \mathbb{P}^n$; there he was mostly motivated by the fact that these lagrangian submanifolds (which can have in general self intersections, therefore below we call them lagrangian cycles) present new example of minimal or Hamiltonian minimal lagrangian submanifolds. However the Mironov construction of lagrangian cycles itself can be directly extended to much wider class of compact algrebraic varieties: namely it works in the case when algebraic variety $X$ of complex dimension $n$ admits $T^k$ - action and an anti - holomorphic involution such that the real part $X_{\mathbb{R}} \subset X$ has real dimension $n$ and is transversal to the torus action. For this case one has families of lagrangian submanifolds and cycles. In the present small text we show how the construction of Mironov cycles works for the complex Grassmannians, resulting in simple examples of smooth lagrangian submanifolds in ${\rm Gr}(k, n+1)$, equipped with a standard Kahler form under the Plücker embedding. For sure the text is not complete but in the new reality we would like to fix it, hoping to continue the investigations and to present in a future complete list of Mironov cycles in ${\rm Gr}(k, n+1)$.
