[Paper Review] Enumerative vs. Symplectic Invariants and Obstruction Bundles

Aleksey Zinger|ArXiv.org|Jan 26, 2002

Algebraic Geometry and Number Theory4 references4 citations

TL;DR

This paper establishes a rigorous analytic foundation for comparing enumerative and symplectic invariants in symplectic geometry by developing a gluing theory for pseudoholomorphic maps using obstruction bundles. It proves key theorems on the local structure of moduli spaces and the convergence of sequences under Gromov topology, enabling precise computation of invariants for genus-one and genus-two curves in projective spaces.

ABSTRACT

We give detailed descriptions of gluing pseudoholomorphic maps in symplectic geometry, especially in the presence of an obstruction bundle. The main motivation is to try to compare the symplectic and enumerative invariants of algebraic manifolds. These descriptions can also be used to enumerate rational curves with high-order degeneracies of local nature in projective spaces.

Motivation & Objective

• To resolve analytic challenges in computing the difference between enumerative and symplectic invariants for genus-one and genus-two Riemann surfaces in complex projective spaces.
• To provide a rigorous justification for the obstruction-bundle approach used in prior works (e.g., [I], [Z2]) on symplectic invariants of $π^n$.
• To establish the local structure of moduli spaces of holomorphic rational maps under regularity conditions, enabling enumeration of curves with higher-order degeneracies.
• To prove the continuity, injectivity, and surjectivity of the gluing map in the context of bubble tree compactifications.
• To extend the applicability of symplectic invariants to complex homogeneous manifolds by analyzing the behavior of nearly holomorphic bubble maps.

Proposed method

• Constructs nearly holomorphic bubble maps via a scale-dependent gluing procedure on bubble trees of spheres attached to a base Riemann surface.
• Applies an implicit function theorem to the $̄partial$-operator on weighted Sobolev spaces to define local coordinates on the moduli space of bubble maps.
• Uses fiber-uniform inverse estimates for the linearized $̄partial$-operator $D_\nu$ to control the size of variations in the gluing construction.
• Employs $L^p$-norm estimates and Sobolev inequalities for metrics $g_\upsilon$ on varying Riemann surfaces to control the convergence of sequences of maps.
• Introduces a balanced map space and analyzes the $C^0$-convergence of variations $\xi_k$ to show that asymptotic matching conditions (e.g., $\xi_h^*(\infty) = \xi_{\iota_h}^*(x_h^*)$) hold in the limit.
• Applies elliptic estimates and Holder’s inequality to bound the difference in asymptotic values of Jacobi fields across neck regions, proving the gluing map is well-defined and continuous.

Experimental results

Research questions

• RQ1 How can the obstruction-bundle method be rigorously adapted to the symplectic setting to compare enumerative and symplectic invariants?
• RQ2 What is the local structure of the moduli space of pseudoholomorphic maps near bubble trees, and how does it depend on regularity conditions?
• RQ3 Under what conditions does the gluing map between moduli spaces of bubble maps preserve orientation and ensure injectivity and surjectivity?
• RQ4 How do sequences of nearly holomorphic maps converge in the Gromov topology, and what conditions ensure the limit is a holomorphic bubble map?
• RQ5 Can the analytic framework be extended to count curves with higher-order degeneracies, such as cuspidal or tacnodal rational curves?

Key findings

• Theorem 3.29 establishes that the obstruction-bundle approach of [T] can be rigorously applied in the symplectic setting, providing a foundation for computing differences between symplectic and enumerative invariants.
• Theorem 3.33 describes the local structure of moduli spaces of holomorphic rational maps as smooth, oriented manifolds under regularity conditions, enabling explicit enumeration of curves.
• The gluing map is shown to be continuous, injective, and surjective under appropriate regularity and semiregularity conditions, ensuring a well-defined parametrization of the moduli space.
• The asymptotic matching condition $\xi_h^*(\infty) = \xi_{\iota_h}^*(x_h^*)$ holds in the limit, which is essential for constructing consistent gluing maps across neck regions.
• The $L^p$-norm of the differential of the map is controlled via Sobolev and elliptic estimates, ensuring uniform convergence in the Gromov topology.
• The proof of Proposition 5.13 establishes a uniform $W^{1,p}$-bound on Jacobi fields via contradiction, showing that $\|D_\nu \xi\|_p \to 0$ implies $\|\xi\|_{p,1} \to 0$, which confirms the Fredholm property of the linearized operator.