  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 ${\rm Gr}(k,
n+1)$.
