[Paper Review] Almost holomorphic embeddings in Grassmannians with applications to singular simplectic submanifolds

Vicente Muñoz, Francisco Presas+1 more|ArXiv.org|Feb 25, 2000

Geometric Analysis and Curvature Flows3 citations

TL;DR

This paper introduces asymptotically holomorphic embeddings of closed symplectic manifolds with integer symplectic class into Grassmannians using Donaldson's approximately holomorphic techniques. It constructs singular determinantal submanifolds as intersections with generalized Schubert cycles, yielding symplectic submanifolds not symplectomorphic to those in Auroux's framework, especially in dimensions >2, thus extending the scope of symplectic submanifold constructions beyond existing methods.

ABSTRACT

We use Donaldson's approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the grassmannians Gr(r,N). We assure that these embeddings are asymptotically holomorphic in a precise sense. We study first the particular case of embeddings in the projective space $CP^N$ obtaining control on N. The main reason of our study is the construction of singular determinantal submanifolds as the intersection of the embedding with certain ``generalized Schur cycles'' defined on a product of grassmannians. It is shown that the symplectic type of these submanifolds is quite more general that the ones obtained by Auroux as zero sets of approximately holomorphic sections of ``very ample'' vector bundles.

Motivation & Objective

• To extend Donaldson's approximately holomorphic techniques to symplectic embeddings into Grassmannians rather than just projective spaces.
• To construct singular determinantal submanifolds as intersections of asymptotically holomorphic embeddings with generalized Schubert cycles in products of Grassmannians.
• To demonstrate that these new submanifolds are not symplectomorphic to those obtained via Auroux’s method of zero sets of sections of twisted bundles.
• To show that the symplectic invariants of these determinantal submanifolds differ significantly from those in the Auroux construction, especially in dimensions greater than 2.
• To establish a broader class of symplectic submanifolds than previously known, using asymptotic holomorphicity in the non-integrable setting.

Proposed method

• Define γ-asymptotically holomorphic embeddings using a compatible almost complex structure and scaled metrics g_k = k g on the tangent bundle.
• Generalize the Kodaira embedding theorem to the symplectic category by constructing asymptotically holomorphic embeddings φ_k: M → ℂP^{2n+1} with φ_k^*[ω_FS] = kω.
• Extend the construction to embeddings into Grassmannians Gr(r,N), ensuring φ_k^*𝒰 = E ⊗ L^⊗k for a hermitian vector bundle E and line bundle L with c_1(L) = [ω/2π].
• Use generalized Schubert cycles (‘generalized Schur cycles’) on products of Grassmannians to define determinantal submanifolds as intersections of the embedded image with these cycles.
• Apply asymptotic analysis to control curvature and norm behavior of derivatives, ensuring the embeddings satisfy O(k^{-1/2}) decay in the almost complex structure distortion.
• Compute symplectic invariants such as volume, first Chern class, and second Chern class (n_1, n_11, n_2) to compare with Auroux’s examples and prove non-symplectomorphism.

Experimental results

Research questions

• RQ1 Can asymptotically holomorphic embeddings of symplectic manifolds into Grassmannians be constructed such that their images intersect generalized Schubert cycles in singular determinantal submanifolds with controlled symplectic type?
• RQ2 Are the symplectic invariants of these determinantal submanifolds distinct from those of submanifolds obtained as zero sets of sections of twisted bundles, as in Auroux’s construction?
• RQ3 For which complex dimensions n > 2 do the symplectic invariants of the determinantal submanifolds fail to match those of Auroux’s examples, even asymptotically?
• RQ4 Can the Segre classes of singular symplectic submanifolds be defined in a way compatible with algebraic geometry, enabling topological classification?
• RQ5 Is the class of symplectic submanifolds obtainable via asymptotically holomorphic determinantal constructions strictly larger than the class obtained via Donaldson–Auroux methods?

Key findings

• The paper constructs asymptotically holomorphic embeddings φ_k: M → Gr(r,N) such that φ_k^*𝒰 = E ⊗ L^⊗k, with N > n + r − 1 and r(N − r) > 2n, ensuring the existence of such embeddings for large k.
• For complex dimension n > 2, the symplectic invariants of the determinantal submanifolds D_1 — specifically, the ratio n_2(D_1)/n_11(D_1) = (n² + n − 4)/(2(n² − 5)) + O(k⁻¹) — differ from those of Auroux’s submanifolds, which have ratio (n−1)/(2(n−2)) + O(k⁻¹), proving non-symplectomorphism.
• In the case r = 1, r_e = 2, r_f = n, the volume of D_1 is ((n−1)2^{n−2} + O(k⁻¹))vol_ω_k(M), and n_11(D_1) = (4(n−1)(n²−5)2^{n−2} + O(k⁻¹))vol_ω_k(M), showing distinct scaling behavior.
• For n > 2, no choice of k_1, k_2 can make the determinantal submanifold D_1^k_1 isotopic to an Auroux-type zero set Z_k_2, as their symplectic invariants do not match asymptotically.
• The determinantal submanifolds are not even homeomorphic to Auroux’s examples in dimension 4, since the ratio n_2/n_11 is a topological invariant and differs between the two constructions.
• The paper establishes that the class of symplectic submanifolds obtainable via asymptotically holomorphic determinantal constructions is strictly larger than the class obtained via sections of twisted bundles, especially in dimensions greater than 2.