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

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.

• The paper introduces a quadratic Morse-Bott function that quantifies non-J-invariance through a generalized Kähler angle, classifying subspaces by isotropic kernel dimensions.
• It shows that gradient flows retract symplectic orbits onto J-compatible submanifolds, thereby unifying Lagrangian, isotropic, coisotropic, and symplectic Grassmannians.
• The analysis establishes uniform topological classifications and homotopy equivalences, yielding practical insights for symplectic topology and representation theory.

Morse-Bott Unification of Symplectic Grassmannians

Introduction and Motivation

This paper introduces a quadratic Morse-Bott function on the real Grassmannian $Gr(k; V)$ of a symplectic vector space $(V, \omega)$ equipped with a compatible linear complex structure $J$. The construction quantifies the degree to which a $k$-dimensional subspace fails to be $J$-invariant, encodes this via a generalized Kähler angle decomposition, and interprets the function dynamically as the kinetic energy of a fundamental vector field generated by the $U(1)$ action of $J$ on $Gr(k; V)$. Through Morse-theoretic analysis, the authors unify the Lagrangian, isotropic, coisotropic, and symplectic Grassmannians within a single framework, further generalizing to non-standard Grassmannians of subspaces that are neither isotropic, coisotropic, nor symplectic.

Classification of Linear Subspaces

Within the $2n$-dimensional symplectic vector space $(V,\omega)$, any linear subspace $W$ is categorized by three invariants $\vec{n} = (n_0, n_+, n_-)$:

• $n_0$ is the dimension of the isotropic kernel $W_0 = W \cap W^\omega$.
• $2n_+$ is the dimension of the reduced symplectic part $W/W_0$.
• $2n_-$ is the analogous dimension for $W^\omega/W_0$. This tripartite structure partitions $n = n_0 + n_+ + n_-$ and facilitates the analysis of symplectic group orbits in the Grassmannian.

The $Sp(V)$ group acts transitively on subspaces of fixed type, establishing $Gr(\vec{n}; V)$ as connected, and structuring $Gr(k; V)$ as a disjoint union of such orbits indexed by $\vec{n}$.

Symplectic Group Structure and Splittings

Associated splittings $(W_+, W_-, W^0)$ refine the understanding of a subspace $W \subset V$. Stabilizer subgroups $Sp_W(V)$ admit a Levi decomposition into a Levi factor and a unipotent radical $H(W)$, which is shown to be a simply connected nilpotent group, generalizing the Heisenberg group for isotropic and mixed-type subspaces. The variety of splittings associated to $W$ forms a principal homogeneous space for $H(W)$.

Kähler Angle Decomposition

Through the introduction of a compatible complex structure $J$, the authors analyze the Kähler angle for two-dimensional planes, extending this notion to higher dimensions as an additive function measuring deviation from being complex or isotropic. The decomposition $W = W_0 \oplus W_+^J \oplus W_+^\theta$ separates the subspace into isotropic, maximal complex, and totally real symplectic components, each with distinct geometric and topological interpretations. Notably, the dimension of totally real components in $W$ and $W^\omega$ are always equal.

$J$-Compatible Subspaces and Unitary Actions

$J$-compatible subspaces admit orthogonal decompositions aligned with $J$, forming a manifold $Gr_J(\vec{n}; V)$ preserved under the action of $U(V, J) \cong U(n)$. Stabilizers of these subspaces are products of orthogonal and unitary groups, $O(n_0) \times U(n_+) \times U(n_-)$, making $Gr_J(\vec{n}; V)$ a homogeneous space.

Quadratic Energy Function and Morse-Bott Analysis

The central construct, the Morse-Bott function $f: Gr(k; V) \rightarrow \mathbb{R}$, is defined by

$f(W) := \frac{1}{2} \text{Tr}([P_W, J]^2)$

which quantifies the "non-$J$-invariance" of a subspace. The function is minimized precisely on complex subspaces and dynamically represents the squared norm of the vector field induced by the $U(1)$ action of $J$.

A critical result is the integrality of critical values: for $W$ in the critical locus, $f(W) = \dim_\mathbb{R}(W \cap W^\omega)$. Thus, the function organizes subspaces by the dimension of their isotropic kernel.

The Morse-Bott structure is rigorously established. Critical submanifolds are precisely the Grassmannians $Gr_J(\vec{n}; V)$ of $J$-compatible subspaces, and the Hessian is shown to be nondegenerate in normal directions. The energy function strictly bounds $f(W)$ above by $\min(k, 2n-k)$ when nontrivial complex parts are present, realizing minima and maxima at complex and maximally isotropic/coisotropic loci, respectively.

Gradient Flow and Stable Manifolds

The negative gradient flow of $f$ deformation retracts each symplectic orbit $Gr(\vec{n}; V)$ onto its corresponding critical submanifold $Gr_J(\vec{n}; V)$. The gradient flow decouples along the components of the Kähler decomposition, preserving the type vector $\vec{n}$ and signaling strong topological rigidity within the strata.

Strongly, it is proved that the stable manifold of $Gr_J(\vec{n}; V)$ coincides with the symplectic orbit $Gr(\vec{n}; V)$. This unifies the Lagrangian, isotropic, coisotropic, and symplectic cases, and fully resolves the topology for mixed-type Grassmannians, not previously accessible via standard symmetric space methods due to non-reductivity of stabilizers.

Homotopy and Deformation Retraction

It is established that each orbit $Gr(\vec{n}; V)$ deformation retracts onto the $J$-compatible submanifold, making it homotopy equivalent to the homogeneous space $U(n) / (O(n_0) \times U(n_+) \times U(n_-))$. This result consolidates scattered observations in the literature, providing a uniform topological description for all cases, including those outside classical types.

Implications and Future Directions

The paper rigorously unifies distinct Grassmannian types under Morse-Bott theory, leveraging symplectic group actions and complex structures to elucidate both geometric and topological properties. Practically, this framework facilitates explicit calculations of homotopy types and stable manifolds, with significant ramifications for symplectic topology, representation theory, and the study of stratified homogeneous spaces.

Theoretically, the approach suggests further investigation into Morse functions invariant under other non-reductive group actions, as well as potential applications in moduli problems and symplectic field theory where stratifications by isotropic and complex data play a key role. Moreover, exploring generalizations to infinite-dimensional or non-linear settings may yield deeper insights into topological invariants of symplectic and complex manifolds.

Conclusion

This work provides a comprehensive Morse-theoretic framework for decomposing and retracting the real Grassmannian of a symplectic vector space, revealing rich interconnections between symplectic, complex, and isotropic structures. The interplay between symplectic group orbits and compatible complex structures, quantified via the Morse-Bott function, leads to definitive topological classifications and unifications for all subspace types, with broad mathematical implications in geometry and topology (2601.16441).