The generalized roof F(1,2,n): Hodge structures and derived categories

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.