Polarized Relations on Horizontal $\mathrm{SL}(2)$'s

Abstract

We introduce a relation on real conjugacy classes of $\mathrm{SL}(2)$-orbits in a Mumford-Tate domain $D$. The relation answers the question when is one $\mathbb{R}$-split polarized mixed Hodge structure more singular/degenerate than another?. The relation is compatible with natural partial orders on the sets of nilpotent orbits in the corresponding Lie algebra and boundary orbits in the compact dual.

A generalization of the $\mathrm{SL}(2)$-orbit theorem to such domains leads to an algorithm for computing this relation. The relation is then worked out in several examples and special cases, including period domains, Hermitian symmetric domains, and complete flag domains.

Although the above relation is not in general a partial order, it leads (via cubical sets) to a poset of equivalence classes of multivariable nilpotent orbits on $D$. The elements of this poset encode the possible degeneracy relations amongst the polarized mixed Hodge structures that arise in a several-variable degeneration of Hodge structure. We conclude with an example illustrating a link to mirror symmetry for Calabi-Yau VHS.