TITLE: complement of Lagrangian Grassmannian
QUESTION [6 upvotes]: The Lagrangian Grassmannian $LG(2,4)$ is a 3 dimensional sub-manifold of the original 4 dimensional Grassmannian manifold $Gr(2,4)$. Is there a nice description of the complement $Gr(2,4) \setminus LG(2,4)$ ?

REPLY [7 votes]: The Grassmannian $Gr(2,4)$ is a smooth quadric in $\mathbb{P}^5$, and $LGr(2,4)$ is its smooth hyperplane section. Therefore the complement is a smooth affine quadric. Over $\mathbb{C}$ it can be described in the affine space $\mathbb{A}^5$ as the hypersurface
$$
x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 1.
$$
EDIT. There is also a similar construction for $Gr(3,6) \setminus LGr(3,6)$. If $U \subset V := \mathbb{C}^6$ is a 3-subspace, which is not isotropic, then the restriction of the symplectic form to $U$ has rank 2, hence has a 1-dimensional kernel space. This gives a regular map
$$
Gr(3,6) \setminus LGr(3,6) \to \mathbb{P}(V).
$$
Its fiber over point $v \in V$ is easily seen to be $Gr(2,\bar{V}) \setminus LGr(2,\bar{V})$, where $\bar{V} := v^\perp/v$ with its induced symplectic form. Thus, the map is a fibration with fibers being 4-dimensional affine quadrics. Note, however, that the ambient affine space fibration for it is non-trivial!