Two-dimensional equivariant symplectic submanifolds in toric manifolds

To find all two-dimensional equivariant symplectic submanifolds in symplectic toric manifolds, we combine the convex geometry of Delzant polytopes with local equivariant symplectic models and obtain a criterion for determining when a two-dimensional submanifold is an equivariant symplectic submanifold in a toric manifold.

💡 Research Summary

The paper addresses the problem of classifying all two‑dimensional equivariant symplectic submanifolds inside compact symplectic toric manifolds. A toric manifold (M^{2n}, ω, T^{n}, μ) is completely encoded by its Delzant polytope Δ^{n} ⊂ ℝ^{n}. The authors ask: given a smooth curve inside Δ^{n}, when does it lift to a 2‑dimensional S^{1}‑equivariant symplectic submanifold of M?

The work proceeds in several stages. First, basic definitions of toric manifolds, Delzant polytopes, facial submanifolds, and the characteristic map l are recalled. Using the Buchstaber–Panov construction (M^{2n} = (T^{n}×Δ^{n})/∼) the authors set up the combinatorial–geometric framework needed for later arguments.

The first major result (Theorem 1) shows that any equivariant symplectic submanifold N^{2k} ⊂ M^{2n} is itself a toric manifold. The proof is straightforward: the embedding i : N → M together with a Lie‑group homomorphism ρ : T^{k} → T^{n} induces a moment map μ_N = ρ^{*}∘μ∘i, which satisfies the Delzant conditions, hence N inherits a toric structure.

Theorem 2 establishes that the restriction of the ambient moment map to N^{2k} yields a k‑dimensional “section” \tildeΔ^{k} of the original polytope Δ^{n}. By examining facial submanifolds L_{2l} = μ_N^{-1}(Ďelta^{l}) and the free action of subtori T^{l}, the authors compare the rank of dμ on N with the dimension of the isotropy subgroups, concluding that the image μ(N) is exactly a smooth k‑dimensional slice of Δ^{n}.

Specializing to the case k = 1 (i.e., a 2‑dimensional S^{1}‑equivariant submanifold), Theorem 3 proves that for any interior point z of the slice \tildeΔ^{1}, the tangent space T_z\tildeΔ^{1} cannot be orthogonal to the line ρ^{*}(ℝ) ⊂ ℝ^{n}. If orthogonality held, the infinitesimal generator of the S^{1}‑action would lie in the kernel of dμ|_{N}, contradicting the non‑degeneracy of ω on N.

To obtain concrete lifting criteria, the authors invoke the local model near a vertex of Δ^{n} (Theorem 4, essentially the standard Delzant/ Guillemin–Sternberg normal form). Near a vertex p, the toric manifold is equivariantly symplectomorphic to a neighborhood in ℂ^{n} with the standard Hamiltonian T^{n}‑action and moment map μ(z)=½∑|z_i|^{2}α_i, where α_i are the isotropy weights at p.

Finally, Theorem 5 translates the geometric data of a curve segment ℓ ⊂ Δ^{n} (with one endpoint v₁ on a face F_{v₁} and the other endpoint v₂ on the boundary) into explicit algebraic conditions for the existence of a smooth S^{1}‑equivariant surface S ⊂ μ^{-1}(ℓ). Writing the curve in coordinates (x₁, g₂(x₁),…, g_n(x₁)) relative to the basis of isotropy weights, the lift is constructed by rotating the complex coordinates with weights (k₁,…,k_n). The necessary and sufficient conditions are:

1. If x₁≠0, the functions y = p g_i(x₁) must be smooth.
2. If x₁=0, then k₁ must vanish; for indices i belonging to the set Q (spanning the lowest‑dimensional face containing v₁) we also require k_i=0 and smoothness of the corresponding g_i; for indices outside Q we need the ratios k_i/k₁ to be integers, smoothness of y = p g_i(x₁)/ (k_i/k₁), and vanishing of all odd‑order derivatives at x₁=0.

These conditions guarantee that the parametrization F(x₁,t) = (x₁ cos k₁t, x₁ sin k₁t, √g₂(x₁) cos k₂t, √g₂(x₁) sin k₂t,… ) defines a smooth embedded surface whose projection under μ is exactly ℓ.

The paper thus provides a complete, verifiable criterion: a smooth curve in a Delzant polytope lifts to a 2‑dimensional equivariant symplectic submanifold if and only if the curve satisfies the integer‑weight and smoothness constraints described above. This bridges the combinatorial data of the polytope with the differential‑geometric structure of the toric manifold, offering a concrete tool for constructing low‑dimensional equivariant submanifolds and suggesting pathways for extending the method to higher‑dimensional equivariant symplectic embeddings.