[Paper Review] Symplectic neighborhood of crossing divisors

Roberta Guadagni|arXiv (Cornell University)|Nov 8, 2016

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper establishes a symplectic neighborhood theorem for finite collections of symplectic submanifolds that intersect orthogonally, generalizing Weinstein's tubular neighborhood theorem. It constructs a symplectic plumbing of their normal bundles, showing that a neighborhood of the union is symplectomorphic to a neighborhood of the zero section in the plumbing, with equivariant versions when a compact Lie group acts symplectically.

ABSTRACT

This paper presents a proof of the existence of standard symplectic coordinates near a set of smooth, orthogonally intersecting symplectic submanifolds. It is a generalization of the standard symplectic neighborhood theorem. Moreover, in the presence of a compact Lie group $G$ acting symplectically, the coordinates can be chosen to be $G$-equivariant.

Motivation & Objective

• To generalize the symplectic tubular neighborhood theorem to finite families of symplectic submanifolds intersecting orthogonally.
• To construct a symplectic model for the neighborhood of such a union using a plumbing of normal bundles.
• To ensure the model is compatible across all intersections, preserving symplectic structure.
• To extend the result to equivariant settings under compact Lie group actions.
• To provide a foundation for understanding degenerations in Kähler geometry and generalized symplectic sums.

Proposed method

• Define a rigid plumbing construction of normal bundles $NX_i$ with additional data to ensure symplectic compatibility.
• Use connections $\alpha_i$ on each $NX_i$ to define a closed, non-degenerate 2-form $\omega_{\alpha_i}$ on the normal bundle.
• Establish compatibility conditions between embeddings $\phi_i$ and $\phi_j$ via a common embedding $\phi_{ij}$ factoring through both.
• Prove that such a plumbing admits a global symplectic form $\omega_{\mathcal{I}}$ when connections are compatible and the intersection is orthogonal.
• In the equivariant case, construct $G$-invariant connections and $G$-equivariant symplectic embeddings using equivariant splitting maps.
• Apply induction and local model construction to show that the symplectic form on $M$ near $\bigcup X_i$ is determined by its restriction to $\bigcup T X_i$.

Experimental results

Research questions

• RQ1 Can a standard symplectic neighborhood exist for a finite union of symplectic submanifolds intersecting orthogonally?
• RQ2 How can symplectic tubular neighborhood embeddings be made compatible across multiple intersecting submanifolds?
• RQ3 Is it possible to construct a global symplectic model for such a union via a plumbing of normal bundles?
• RQ4 Can the construction be made equivariant under a compact Lie group action?
• RQ5 What is the relationship between positive intersection and the existence of a standard symplectic model?

Key findings

• A symplectic neighborhood of a finite union of orthogonally intersecting symplectic submanifolds is symplectomorphic to a neighborhood of the zero section in a symplectic plumbing of their normal bundles.
• The symplectic form on the neighborhood is fully determined by the restriction of $\omega$ to the tangent spaces of the submanifolds, as shown in Lemma 1.
• Compatible symplectic embeddings $\phi_i$ exist such that their common intersection is modeled by a symplectic embedding $\phi_{ij}$ factoring through both $\phi_i$ and $\phi_j$.
• In the presence of a compact Lie group $G$ acting symplectically, the symplectic neighborhood and the plumbing can be made $G$-equivariant by choosing $G$-equivariant connections.
• The result provides a local model for generalized symplectic sums and degenerating families of Kähler manifolds relevant to mirror symmetry.
• For positively intersecting submanifolds, a symplectic isotopy exists that deforms them into orthogonal intersections, allowing the use of the main theorem after deformation.