Automatic transversality and orbifolds of punctured holomorphic curves in dimension four

Abstract

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 [HLS97] and Ivashkovich-Shevchishin [IS99] 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. Mathematics Subject Classification (2000). Primary 32Q65; Secondary 57R17.

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

“…By the choices made in the splitting construction, this plane is moreover asymptotic to a geodesic on L for the flat metric. The totality of the one-dimensional family of such planes is shown to form a smoothly embedded solid torus having boundary equal to L. Here we need the automatic transversality result [55] by C. Wendl together with the asymptotic intersection results shown in [28] by R. Hind and S. Lisi. We also make heavy use of positivity of intersection for pseudoholomorphic curves; see the work [36] by D. McDuff.…”

“…[5], [55 [10, (3)]. Assume that the almost complex structure is cylindrical with respect to the contact form α 0 on the cylindrical ends of (W, ω).…”

On nonseparating contact hypersurfaces in symplectic 4–manifolds

“…Note that in the above proof, Fredholm regularity does not require any genericity assumptions, rather it comes for free due to "automatic" transversality (cf [34]). As a consequence, u 0 can be deformed with sufficiently small perturbations of J and so that Theorem 4.7 still applies.…”

“…In the linearization this amounts to replacing c z D by c z D ; this idea is explained in detail in Wendl [41; 39]. The problem with the additional constraint then has index 1 and is again regular by an argument using the formal adjoint of D N u , as in [34].…”

“…In proving Theorem 4.8, we will make use of a few concepts from the intersection theory of punctured holomorphic curves; this theory is developed in detail in the papers [31; 32], and the last section of [34] also contains a summary. Assume…”