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A Morse-Bott unification of the Grassmannians of a symplectic vector space

Hyunmoon Kim

TL;DR

The paper develops a Morse-Bott framework for the real Grassmannian Gr(k;V) of a symplectic vector space with a compatible complex structure J by introducing a quadratic energy $f(W)=\tfrac{1}{2} Tr([P_W,J]^2)$. It characterizes critical submanifolds as $\operatorname{Gr}_J(\vec{n};V)$, where each subspace splits into an isotropic kernel and a maximal complex part, and shows the gradient flow preserves the isotropic kernel dimension while driving other components toward minimizers. The main result is that the stable manifolds of these critical submanifolds coincide with the linear symplectomorphism orbits $\operatorname{Gr}(\vec{n};V)$, and each such orbit deformation retracts onto the compact homogeneous space $U(n)/(O(n_0)\times U(n_+)\times U(n_-))$, unifying several previously known homotopy types (Lagrangian, isotropic, coisotropic, and symplectic) under a single Morse-Bott machine. This provides a cohesive topological description for all strata of linear subspaces with respect to the symplectic form and the compatible complex structure, with explicit decompositions by Kähler angles and unitary stabilizers.

Abstract

We construct a quadratic Morse-Bott function on the real Grassmannian of a symplectic vector space from a compatible linear complex structure. We show that its critical loci consist of linear subspaces that split into isotropic and complex parts and that its stable manifolds coincide with the orbits of the linear symplectomorphism group. These orbits generalize the Lagrangian, symplectic, isotropic, and coisotropic Grassmannians to include the Grassmannians of linear subspaces that are neither isotropic, coisotropic, nor symplectic. The negative gradient flow deformation retracts these spaces onto compact homogeneous spaces for the unitary group.

A Morse-Bott unification of the Grassmannians of a symplectic vector space

TL;DR

The paper develops a Morse-Bott framework for the real Grassmannian Gr(k;V) of a symplectic vector space with a compatible complex structure J by introducing a quadratic energy . It characterizes critical submanifolds as , where each subspace splits into an isotropic kernel and a maximal complex part, and shows the gradient flow preserves the isotropic kernel dimension while driving other components toward minimizers. The main result is that the stable manifolds of these critical submanifolds coincide with the linear symplectomorphism orbits , and each such orbit deformation retracts onto the compact homogeneous space , unifying several previously known homotopy types (Lagrangian, isotropic, coisotropic, and symplectic) under a single Morse-Bott machine. This provides a cohesive topological description for all strata of linear subspaces with respect to the symplectic form and the compatible complex structure, with explicit decompositions by Kähler angles and unitary stabilizers.

Abstract

We construct a quadratic Morse-Bott function on the real Grassmannian of a symplectic vector space from a compatible linear complex structure. We show that its critical loci consist of linear subspaces that split into isotropic and complex parts and that its stable manifolds coincide with the orbits of the linear symplectomorphism group. These orbits generalize the Lagrangian, symplectic, isotropic, and coisotropic Grassmannians to include the Grassmannians of linear subspaces that are neither isotropic, coisotropic, nor symplectic. The negative gradient flow deformation retracts these spaces onto compact homogeneous spaces for the unitary group.
Paper Structure (15 sections, 30 theorems, 111 equations)

This paper contains 15 sections, 30 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Symplectic Linear Algebra
  3. Types of linear subspaces
  4. Transitivity of the linear symplectomorphism group
  5. Stabilizer Subgroups and Splittings
  6. Totally real symplectic subspaces and Kähler angles
  7. $J$-compatible subspaces
  8. Morse-Bott theory on the real Grassmannian
  9. Differential geometry of the real Grassmannian
  10. The energy function and the Kähler angle
  11. Critical submanifolds and extrema
  12. The Morse-Bott property
  13. Gradient flows
  14. Type preservation
  15. Stable manifolds as symplectic orbits

Key Result

Theorem 1.1

The function $f$ is a Morse-Bott function with integral critical values. Specifically, the energy of a critical subspace is exactly the dimension of its isotropic kernel : where $W^\omega$ is the symplectic complement of $W$ (Eq. eq:symplectic_complement).

Theorems & Definitions (63)

  • Theorem 1.1: Integrality of critical values, Theorem \ref{['thm:morse_bott']}, Proposition \ref{['prop:critical_values']}
  • Theorem 1.2: Stable manifolds, Theorem \ref{['thm:stable_orbits']}
  • Corollary 1.3: Deformation retraction, Corollary \ref{['cor:def_retraction']}
  • Definition 2.1: Associated Splittings
  • Theorem 2.2: Linear relative Darboux theorem (ArnoldGivental)
  • Proposition 2.3: Orbit incidence
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5: Relation to the Heisenberg Group
  • ...and 53 more