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 <PRE_TAG>zero loci</POST_TAG> 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 P^{n-1} and G(2, n), we construct a derived embedding of the relevant <PRE_TAG>zero loci</POST_TAG> by methods based on the study of B-brane categories in the context of a gauged linear sigma model.