Constructing Mironov cycles in complex Grassmannians

Abstract: 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 Pl\"{u}cker 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 ${\rm Gr}(k, n+1)$.