[Paper Review] Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four

Chris Wendl|ArXiv.org|Feb 26, 2008

Geometric and Algebraic Topology19 references4 citations

TL;DR

This paper establishes a numerical transversality criterion for punctured J-holomorphic curves in 4-dimensional symplectic cobordisms without requiring generic almost complex structures, generalizing prior results to include non-injective and non-immersed curves. The key contribution is proving that certain moduli spaces of such curves are globally smooth orbifolds, generically consisting of embedded curves with isolated singularities from unbranched multiple covers, using intersection theory and a novel normal Cauchy-Riemann operator analysis in dimension four.

ABSTRACT

We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that form isolated orbifold singularities.

Motivation & Objective

• To generalize automatic transversality results in 4-dimensional symplectic topology beyond somewhere injective or immersed curves.
• To establish a numerical criterion for transversality in 4D symplectic cobordisms without genericity assumptions on the almost complex structure.
• To prove that certain moduli spaces of punctured J-holomorphic curves are globally smooth orbifolds, even when curves are multiply covered.
• To apply intersection theory of punctured curves to show that multiple covers form isolated orbifold singularities in the moduli space.

Proposed method

• Derives a transversality criterion for J-holomorphic curves in 4D symplectic cobordisms using the generalized normal bundle and Cauchy-Riemann operators on line bundles.
• Introduces a functional analytic setup for the normal operator of a pseudoholomorphic curve, splitting the linearization into components using Teichmüller slices.
• Applies a counting method for boundary zeros of admissible sections on bordered Riemann surfaces, defining an algebraic count via winding numbers.
• Uses the doubling construction to relate the algebraic zero count to the relative first Chern class and boundary Maslov index.
• Combines the transversality criterion with intersection theory of punctured curves to analyze the global structure of moduli spaces.
• Establishes that the moduli space is a smooth orbifold by showing that multiple covers are unbranched and isolated, with embedded curves forming the generic part.

Experimental results

Research questions

• RQ1 Can transversality for J-holomorphic curves in 4D symplectic cobordisms be guaranteed without genericity assumptions on the almost complex structure?
• RQ2 Do moduli spaces of punctured J-holomorphic curves in 4D remain smooth even when curves are not somewhere injective or immersed?
• RQ3 Can the structure of the moduli space be described as a global smooth orbifold when multiple covers are present?
• RQ4 What role does intersection theory play in characterizing the singularities of such moduli spaces?
• RQ5 How can the algebraic count of zeros of sections of the normal bundle be related to topological invariants like the first Chern class and Maslov index?

Key findings

• The moduli space of unparametrized J-holomorphic curves in 4D is smooth if the index satisfies $\operatorname{ind}(u) > 2g - 2$, generalizing results of Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin.
• The transversality criterion applies to arbitrary J-holomorphic curves with totally real boundary and cylindrical ends, not requiring injectivity or immersion.
• Moduli spaces of punctured J-holomorphic curves in 4D are globally smooth orbifolds, consisting generically of embedded curves and isolated singularities from unbranched multiple covers.
• The algebraic zero count $Z(\sigma)$ of a section $\sigma$ of the normal bundle satisfies $Z(\sigma) = c_1^\Phi(E) + \frac{1}{2}\mu^\Phi(E,\ell) + \operatorname{wind}_{\partial_1 S}^\Phi(\sigma)$, linking topology to analysis.
• The doubling construction yields $Z(\sigma^D) = 2Z(\sigma)$, enabling topological invariants to be computed via closed surface extensions.
• The boundary Maslov index and relative Chern class fully determine the zero count, ensuring consistency under homotopy and enabling global moduli space analysis.