[Paper Review] Polarized relations on horizontal SL(2)s

Matt Kerr, Gregory Pearlstein+1 more|arXiv (Cornell University)|May 8, 2017

Advanced Algebra and GeometryMathematics38 references21 citations

TL;DR

This paper introduces a combinatorially computable relation on real conjugacy classes of SL(2)-orbits in Mumford-Tate domains to classify degeneracy relations among polarized mixed Hodge structures in multivariable degenerations. Using the multivariable SL(2)-orbit theorem and representation theory, it constructs a poset of equivalence classes of nilpotent orbits that encodes admissible degeneracy patterns, with applications to period domains, Hermitian symmetric domains, and mirror symmetry for Calabi-Yau VHS.

ABSTRACT

We introduce a relation on real conjugacy classes of SL(2)-orbits in a Mumford-Tate domain D which 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 SL(2)-orbit theorem to such domains leads to an algorithm for computing this relation, which is worked out in several examples and special cases including period domains, Hermitian symmetric domains, and complete flag domains, and used to define a poset of equivalence classes of multivariable nilpotent orbits on D.

Motivation & Objective

• To understand the constraints on multivariable degenerations of Hodge structures, particularly how polarized mixed Hodge structures can degenerate into one another.
• To classify nilpotent orbits and their degeneracy relations in Mumford-Tate domains, especially in the absence of a finite orbit classification for r > 1.
• To generalize the SL(2)-orbit theorem to Mumford-Tate domains to compute degeneracy relations algorithmically.
• To construct a poset of equivalence classes of multivariable nilpotent orbits that encode admissible degeneracy patterns in Hodge-theoretic degenerations.
• To clarify the role of the Mumford-Tate group in restricting possible limiting mixed Hodge structures and their stratifications.

Proposed method

• Introduces a relation ≤ on real conjugacy classes of SL(2)-orbits in a Mumford-Tate domain D, defining when one polarized mixed Hodge structure is more degenerate than another.
• Uses the multivariable SL(2)-orbit theorem adapted to Mumford-Tate domains to compute the relation algorithmically via the action of G(R)+ on nilpotent orbits.
• Applies representation theory, particularly standard triples (N+, Y, N) and Jacobson–Morosov filtrations, to analyze monodromy weight filtrations and grading elements.
• Defines a secondary poset of equivalence classes of multivariable nilpotent orbits under the action of G(R)+ on each face of a nilpotent cone, forming a combinatorial structure encoding degeneracy relations.
• Utilizes distinguished grading elements and DKS-triples to characterize horizontal SL(2) actions and ensure compatibility with Hodge-theoretic constraints.
• Analyzes special cases including period domains, Hermitian symmetric domains, and complete flag domains to demonstrate when the relation yields linear or partial orders.

Experimental results

Research questions

• RQ1 When is one R-split polarized mixed Hodge structure more singular or degenerate than another in a multivariable degeneration?
• RQ2 How can the degeneracy relations among nilpotent orbits in a polarized mixed Hodge structure be systematically classified and computed?
• RQ3 What is the structure of the poset of equivalence classes of multivariable nilpotent orbits under the action of G(R)+, and how does it reflect the stratification of the boundary of D?
• RQ4 In which cases (e.g., Hermitian symmetric or flag domains) does the degeneracy relation yield a partial or linear order?
• RQ5 How do the constraints imposed by the Mumford-Tate group affect the possible limiting mixed Hodge structures in degenerations?

Key findings

• The proposed relation ≤ on real conjugacy classes of SL(2)-orbits is compatible with the partial orders on nilpotent cones and boundary orbits in the compact dual.
• The relation is not a partial order in general, but leads to a well-defined poset of equivalence classes of multivariable nilpotent orbits via cubical sets, encoding all admissible degeneracy patterns.
• In Hermitian symmetric domains, the relation ≤ yields a linear order; in complete flag domains, ≤ gives a partial order (though the relation ≼ does not).
• The multivariable SL(2)-orbit theorem provides an algorithmic method to compute the degeneracy relation, enabling explicit classification in specific cases.
• The framework reveals a link to mirror symmetry for Calabi-Yau variations of Hodge structure, as illustrated in the final example.
• Representation-theoretic tools such as standard triples, Jacobson–Morosov filtrations, and distinguished grading elements are essential for characterizing horizontal SL(2) actions and Hodge-theoretic constraints.