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

##### Doc. Math. 24, 1295-1360 (2019)
DOI: 10.25537/dm.2019v24.1295-1360

### Summary

We introduce a relation on real conjugacy classes of $\operatorname{SL}(2)$-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. \par A generalization of the $\operatorname{SL}(2)$-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. \par 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.

### Mathematics Subject Classification

14D07, 14M17, 17B45, 32M10, 32G20

### Keywords/Phrases

degeneration of Hodge structure, Mumford-Tate domain, boundary component, nilpotent orbit, $\operatorname{SL}(2)$-orbit, mirror symmetry

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### Affiliation

Kerr, Matt
Department of Mathematics, Washington University, St. Louis, Campus Box 1146, One Brookings Drive, St. Louis, MO 63130-4899, USA
Pearlstein, Gregory J.
Mathematics Department, Mail stop 3368, Texas A \& M University, College Station, TX 77843, USA
Robles, Colleen
Mathematics Department, Duke University, Box 90320, Durham, NC, USA