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

TL;DR

This paper explores the geometric and categorical properties of generalized homogeneous roofs, focusing on the Hodge structures and derived categories of zero loci associated with hyperplane sections on these varieties, especially the flag variety F(1,2,n).

Contribution

It introduces a new framework connecting Hodge structures and derived categories for generalized homogeneous roofs, including explicit constructions for the flag variety F(1,2,n).

Findings

Analysis of zero loci properties at the Hodge level.

Construction of derived embeddings for zero loci.

Application of B-brane categories in the context of hyperplane sections.

Abstract

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.