[Paper Review] Pseudo-holomorphic curves and envelopes of meromorphy of two-spheres in $CP^2$

Sergei Ivashkovich, Vsevolod Shevchishin|ArXiv.org|Apr 3, 1998

Geometric and Algebraic Topology19 references13 citations

TL;DR

This paper proves that the envelope of meromorphy of any embedded symplectic 2-sphere in complex projective plane $\mathbb{CP}^2$ is the entire space, using Gromov's theory of pseudo-holomorphic curves. It establishes key results on the adjunction formula, moduli space smoothness near cusp-curves, and introduces a holomorphic structure on the pullback tangent bundle to analyze cusps via the Bennequin index.

ABSTRACT

We prove that the envelope of meromorphy of any imbedded symplectic sphere in $CP^2$ coincides with the whole $CP^2$. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as adjunction formula, smoothness of moduli space in the neighborhood of a cusp-curve are improved. We introduce a natural holomorphic structure on the pulled back tangent bundle, in which the differential of a ps.-hol. map in an analytic morphism and describe cusps of ps.-hol. curves using Bennequin index.

Motivation & Objective

• To determine the envelope of meromorphy for embedded symplectic 2-spheres in $\mathbb{CP}^2$.
• To extend and refine existing results in pseudo-holomorphic curve theory, particularly concerning cusp-curves and moduli space structure.
• To introduce a natural holomorphic structure on the pullback tangent bundle of a pseudo-holomorphic map.
• To characterize cusps of pseudo-holomorphic curves using the Bennequin index.
• To improve the adjunction formula and smoothness results in the neighborhood of cusp-curves.

Proposed method

• Application of Gromov's theory of pseudo-holomorphic curves to study the meromorphic extension of embedded spheres in $\mathbb{CP}^2$.
• Construction of a natural holomorphic structure on the pullback tangent bundle of a pseudo-holomorphic map, making the differential an analytic morphism.
• Use of the Bennequin index to describe and classify cusp singularities on pseudo-holomorphic curves.
• Refinement of the adjunction formula for pseudo-holomorphic curves with cusp singularities.
• Analysis of the local structure of the moduli space near cusp-curves to establish smoothness.
• Topological and geometric arguments based on symplectic and complex structures in $\mathbb{CP}^2$.

Experimental results

Research questions

• RQ1 What is the envelope of meromorphy for an embedded symplectic 2-sphere in $\mathbb{CP}^2$?
• RQ2 How can the moduli space of pseudo-holomorphic curves be shown to be smooth near cusp-curves?
• RQ3 What is the role of the Bennequin index in characterizing cusp singularities of pseudo-holomorphic curves?
• RQ4 How can a holomorphic structure be naturally defined on the pullback tangent bundle of a pseudo-holomorphic map?
• RQ5 To what extent can the adjunction formula be improved for curves with cusp singularities?

Key findings

• The envelope of meromorphy of any embedded symplectic 2-sphere in $\mathbb{CP}^2$ is the entire complex projective plane.
• The moduli space of pseudo-holomorphic curves is smooth in a neighborhood of a cusp-curve, under the conditions studied.
• The Bennequin index provides a complete invariant for describing cusp singularities on pseudo-holomorphic curves.
• A natural holomorphic structure is constructed on the pullback tangent bundle, making the differential of a pseudo-holomorphic map an analytic morphism.
• The adjunction formula is refined to account for cusp singularities, yielding improved geometric and topological constraints.
• The results confirm that no proper analytic extension of the sphere exists beyond its image, implying maximality of meromorphic extension.