  We consider generalized homogeneous roofs, i.e. quotients of simply
connected, semisimple Lie groups by a parabolic subgroup, which admit two
projective bundle structures. Given a general hyperplane section on such a
variety, we consider the zero loci of its pushforwards along the projective
bundle structures and we discuss their properties at the level of Hodge
structures. In the case of the flag variety $F(1,2,n)$ with its projections to
$\mathbb{P}^{n-1}$ and $G(2, n)$, we construct a derived embedding of the
relevant zero loci by methods based on the study of $B$-brane categories in the
context of a gauged linear sigma model.
