Symplectic Hypersurface Structures

• Symplectic hypersurface is a codimension-one (real codimension-two in complex settings) submanifold that inherits a symplectic structure, allowing both smooth and controlled singular configurations.
• Donaldson’s construction enables forming smooth polarizations by realizing the hypersurface as the Poincaré dual of a high multiple of the symplectic form, aiding in manifold decomposition.
• These hypersurfaces are instrumental in solving embedding problems, computing symplectic invariants, and shedding light on mirror symmetry and contact topology.

A symplectic hypersurface is, in its most basic sense, a codimension-one (real co-dimension two in the complex setting) submanifold or subvariety of a symplectic manifold that is itself symplectic in the induced structure. In contemporary research, the notion encompasses both smooth and singular cases, admitting controlled degeneracies in the symplectic form or in the geometry of the hypersurface. Symplectic hypersurfaces are central in the decomposition and topological analysis of symplectic manifolds, play a key role in embedding problems, invariants, and mirror symmetry, and manifest in both algebraic and differential geometric contexts.

1. Fundamental Constructions and Local Models

The classical approach to symplectic hypersurfaces follows Donaldson’s technique, wherein, given a rational (or even just integral after appropriate approximation) symplectic manifold $(M^{2n}, \omega)$, one constructs a codimension-2 symplectic submanifold $E$ (the hypersurface) Poincaré dual to $k[\omega]$ for some large integer $k$. Such $E$ is called a Donaldson hypersurface or smooth polarization. In irrational or more restrictive settings, the construction is generalized to singular polarizations (Opshtein, 2010).

The key structural result is that in a compact symplectic $4$-manifold, the cohomology class $[\omega]$ can be decomposed as a positive linear combination of Poincaré duals to (possibly singular) symplectic hypersurfaces:

$$[\omega] = \sum_{i=1}^n a_i [\Sigma_i], \quad a_i > 0,$$

where each $\Sigma_i$ is a (possibly singular) symplectic surface, with all intersections being positive and transverse. In higher dimensions, analogous decompositions hold with higher-codimension singular loci required to be "generic" and the intersections symplectic.

Locally, near a component $\Sigma_i$, one uses a symplectic disc bundle model:

$$\omega_i = \pi_i^* \tau_i + d(r_i^2 \alpha_i) + \sum_{j,k} \pi_i^* \tau_{ijk},$$

with $\pi_i : L_i \to \Sigma_i$ the projection, $\alpha_i$ a connection one-form (with $\alpha_i|_{\text{fiber}} = a_i d\theta_i$), and the last term accounting for transverse intersections. A compatible Liouville one-form takes the form

$$\lambda_i = (1 - r_i^2) \alpha_i + (1 - \gamma_i) \pi_i^* \lambda'_i + \sum \pi_i^* \lambda_{ijk},$$

ensuring compatibility and facilitating the embedding of standard symplectic ellipsoids via the Liouville flow.

2. Singularities and Local Invariants

Symplectic hypersurfaces often admit controlled singularities, crucial for both flexibility and classification. Several frameworks analyze these features:

• Log Symplectic Hypersurfaces: On a complex $2n$-fold, $(X, \omega)$ is log symplectic if $\omega$ is a logarithmic 2-form with simple poles along an anticanonical divisor $D$—the degeneracy hypersurface. Locally,

$$\omega = \frac{dx_1}{x_1} \wedge dy_1 + \sum_{i=2}^n dx_i \wedge dy_i,$$

so $\omega$ degenerates transversely along $D$. Singularities of $D$ are heavily constrained. At so-called elliptic points (where $D$ is normal, a symplectic leaf through the point has codimension four, and the modular vector field is transverse), the singularity is locally equivalent to one of the simple elliptic surface singularities ($\tilde{E}_6,\tilde{E}_7,\tilde{E}_8$), with prescribed quasi-homogeneous equations. This rigidifies possibilities for the global topology and arithmetic of $D$ (Pym, 2015).

• b\textsuperscript{m}-Symplectic and Folded Hypersurfaces: For a $b^m$-symplectic manifold, the symplectic form degenerates as

$\omega = \frac{dx}{x^m} \wedge \alpha + \beta,$

near a critical hypersurface $Z = \{x = 0\}$, with $\alpha$ a closed one-form on $Z$, $\beta$ a smooth two-form, and $m \in \mathbb{N}$. For $m$ odd, the singularity is "folded," meaning $\omega^n$ vanishes transversally: folded symplectic forms have a null line field along the fold and admit local models such as

$$\omega = x_1\,dx_1 \wedge dy_1 + dx_2 \wedge dy_2 + \ldots + dx_n \wedge dy_n.$$

These structures generalize the Darboux-Givental theorem to singular settings, and a complete set of local invariants—such as the Martinet hypersurface ($\Sigma_2$), the restriction of $\omega$ to $T\Sigma_2$, the kernel of $\omega^{n-1}$, and the canonical orientation—classifies local singular symplectic forms (Domitrz, 2016).

3. Global Decomposition and Applications

The decomposition of symplectic manifolds into "standard" model pieces—such as ellipsoids or ball-like regions—is a unifying theme. In dimension four, singular polarizations enable the packing of $M$ by symplectic ellipsoids $E(A_i, a_i)$, constructed from local data on each $\Sigma_i$ and covering the manifold up to an arbitrarily small residue of volume (Opshtein, 2010):

$$\Phi_i: E(A_i - \delta, a_i) \longrightarrow M \quad (\delta > 0).$$

This decomposition translates challenging global symplectic embedding problems and isotopy issues into tractable local or combinatorial ones. It further demonstrates that singular objects with well-controlled, positive intersection theory play as strong a role in governing geometry as their smooth counterparts.

This methodology has important implications: for instance, subcritical polarizations (where the complement of the hypersurface has the homology type of a subcritical Stein manifold) are only possible for degree one polarizations, confirming a conjecture of Biran and Cieliebak (Geiges et al., 2021). In contact topology, folded symplectic forms are intricately tied to convex hypersurface theory, and their existence is closely related to the topology of contact hypersurface thickenings (Breen, 2023).

4. Foliations and Holomorphic Symplectic Hypersurfaces

In complex symplectic geometry, particularly in projective or holomorphic settings, hypersurfaces in irreducible holomorphic symplectic manifolds possess a canonical characteristic foliation: the kernel of the restricted symplectic form on the hypersurface. For a smooth hypersurface $Y$ in a projective irreducible holomorphic symplectic manifold $X$ of dimension $2n$ equipped with a Lagrangian fibration $\pi : X \to B$, vertical hypersurfaces $Y = \pi^{-1}(D)$ (with $D \subset B$) inherit a characteristic foliation whose very general leaves are Zariski dense in the fibers of $\pi$ (Abugaliev, 2019).

When the closure of a general leaf is a surface, the existence of a Lagrangian fibration on $X$ is forced (Amerik et al., 2016). Such results encode deep rigidity within the intersection of complex symplectic and algebraic geometry, constraining both the types of hypersurfaces and the possible ambient structures.

5. Singularities, Symplectic Cohomology, and Mirror Symmetry

Isolated hypersurface singularities, especially those of invertible (weighted homogeneous) type, are now understood via the bridge between symplectic cohomology of the associated Milnor fiber and the Hochschild cohomology of the dg category of matrix factorizations. For an invertible matrix singularity given by $w_A(x)$, one constructs a finite symmetry group $\Gamma_A$ and analyzes the category of $\Gamma_A$-equivariant matrix factorizations. A pivotal result is the isomorphism:

$$\mathrm{HH}(\mathrm{Matf}(R, w_A, \Gamma_A)) \cong \mathrm{SH}(F_{w_{A^T}}),$$

with $F_{w_{A^T}}$ the Milnor fiber of the dual singularity (Peters, 6 May 2024). The deep algebraic structure—encompassing the Gerstenhaber algebra structure, bigrading by the adjoint of the Euler vector field, and explicit generators via Koszul-type matrix factorizations—yields practical computations of invariants in both symplectic and contact topology (e.g., for cDV singularities, negative-graded SH detects contact invariants of the link).

This correspondence has major repercussions for questions in homological mirror symmetry, topological invariants of links of singularities, and symplectic invariants up to symplectomorphism.

6. Symplectic Embedding Rigidity and Related Structures

Symplectic hypersurfaces serve as barriers and guide the paper of symplectic capacities. J-holomorphic techniques, such as fillings by Levi-flat hypersurfaces (locally foliated by complex curves), are leveraged to prove quantitative rigidity phenomena—most notably Gromov’s non-squeezing theorem (Sukhov et al., 2011). The existence of such fillings imposes non-trivial invariants like the “holomorphic radius,” and constrains symplectic embeddings.

Relatedly, the Chow groups of hypersurfaces in isotropic or bisymplectic Grassmannians inject into cohomology in a range of degrees, providing vanishing results for homologically trivial algebraic cycles (Laterveer, 2020). These results are obtained via motivic decompositions (applying techniques like Cayley’s trick and the blow-up formula) and connect to broader questions in motives and Hodge theory.

7. Future Directions and Open Problems

Emerging directions focus on further exploring singular symplectic structures (log symplectic, $b^m$-symplectic, folded, etc.), the structure of symplectic and contact invariants associated to singularities, and the interplay between symplectic decompositions and mirror symmetry. The precise impact of singularities on symplectic and contact invariants (such as the Gerstenhaber bigrading or SH in negative degrees for links of higher-dimensional singularities) presents an active area of investigation.

It remains to be seen to what extent the patchwork decomposition philosophy, central to embedding and isotopy problems, can be extended in higher dimensions or to more complicated singularities, and how the algebra-geometric invariants (via derived categories and matrix factorizations) can be fully integrated with symplectic invariants in the classification of hypersurfaces and their Milnor fibers.

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This synthesis assembles the principal structural theorems, local and global models, invariants, applications, and new algebraic-symplectic dualities that define the modern paper of symplectic hypersurfaces.