[Paper Review] Schubert varieties as variations of Hodge structure

Colleen Robles|arXiv (Cornell University)|Aug 27, 2012

Algebraic Geometry and Number Theory17 references4 citations

TL;DR

This paper characterizes Schubert varieties in the compact dual of a Hodge domain as variations of Hodge structure (VHS), showing that those satisfying the infinitesimal period relation are precisely the Schubert VHS. It proves that the isotropy orbits of these Schubert VHS span the space of all infinitesimal VHS, and that the cohomology classes dual to Schubert VHS form a basis for the invariant characteristic cohomology of the IPR, resolving a question posed by Green–Griffiths–Kerr.

ABSTRACT

We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual the Schubert VHS form a basis of the invariant characteristic cohomology associated to the infinitesimal period relation (a.k.a. Griffiths transversality).

Motivation & Objective

• To determine which Schubert varieties in the compact dual of a Hodge domain arise as variations of Hodge structure (VHS).
• To show that the isotropy orbits of infinitesimal Schubert VHS span the entire space of infinitesimal VHS.
• To identify a basis for the invariant characteristic cohomology of the infinitesimal period relation (IPR), using cohomology classes dual to Schubert VHS.
• To resolve a question posed by Green–Griffiths–Kerr regarding explicit representatives for invariant characteristic cohomology.
• To establish that maximal Schubert VHS are maximal VHS, and to characterize their geometric structure in types A and C.

Proposed method

• Uses the Green–Griffiths–Kerr structure theorem to generalize the study of VHS from period domains to Hodge domains.
• Applies grading elements and parabolic subalgebras to analyze the Hodge-theoretic structure of the compact dual $\check{D} = G_{\mathbb{C}}/P$.
• Employs the Bruhat order on the Weyl group to characterize Schubert varieties via root systems and positive roots.
• Reduces the study of the IPR to the bracket-generating case via a reduction theorem (Proposition 3.10), simplifying the analysis.
• Uses the correspondence between root systems $\Delta(w)$ and nilpotent subalgebras $\mathfrak{n}_w$ to identify VHS as abelian subalgebras of $\mathfrak{g}_{-1}$.
• Applies representation-theoretic techniques to show that the cohomology classes dual to Schubert VHS span the invariant characteristic cohomology of the IPR.

Experimental results

Research questions

• RQ1 Which Schubert varieties in the compact dual of a Hodge domain are variations of Hodge structure?
• RQ2 Do the isotropy orbits of infinitesimal Schubert VHS span the entire space of infinitesimal VHS?
• RQ3 Can the invariant characteristic cohomology of the infinitesimal period relation be explicitly described using Schubert VHS?
• RQ4 Are maximal Schubert VHS also maximal VHS in the general sense?
• RQ5 Under what conditions are maximal Schubert VHS homogeneous Hermitian symmetric spaces?

Key findings

• The Schubert varieties that are variations of Hodge structure are precisely those whose associated root systems satisfy the infinitesimal period relation.
• The isotropy orbits of infinitesimal Schubert VHS span the entire space of infinitesimal VHS, meaning every infinitesimal VHS arises from such orbits.
• The cohomology classes dual to Schubert VHS form a basis for the invariant characteristic cohomology of the IPR, providing explicit representatives as sought by Green–Griffiths–Kerr.
• Maximal Schubert VHS are maximal variations of Hodge structure, resolving a potential discrepancy between Schubert and general VHS maximality.
• In types A and C, maximal Schubert VHS are homogeneously embedded Hermitian symmetric spaces, though this fails in other types.
• Counter-examples to Mayer’s rigidity theorem for maximal VHS are constructed using Schubert VHS, showing the theorem does not hold in general.