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Positivity of Boundary Intersections of Complex Curves and Adjunction Formula

S. Ivashkovych

Abstract

One of the goals of this paper is to prove that the index of intersection of two complex curves in a two-dimensional complex manifold tangent to each other at a common boundary point is positive. This is achieved via the construction of a totally real surface such that the curves in question are attached to it by some parts of their boundaries and then defining a certain ``boundary intersection index'' of two complex ``half-disks'' with their edges on a totally real surface. We prove that this index is always positive. This second result holds true, more generally, for complex curves in a two dimensional almost complex manifold with boundaries on totally real submanifold, but to our best knowledge is new even for integrable structures, unless the totally real surface in question is supposed to be real analytic. We also formulate and prove the Adjunction Formula for complex curves with boundaries on totally real submanifolds in an almost complex manifold of dimension four.

Positivity of Boundary Intersections of Complex Curves and Adjunction Formula

Abstract

One of the goals of this paper is to prove that the index of intersection of two complex curves in a two-dimensional complex manifold tangent to each other at a common boundary point is positive. This is achieved via the construction of a totally real surface such that the curves in question are attached to it by some parts of their boundaries and then defining a certain ``boundary intersection index'' of two complex ``half-disks'' with their edges on a totally real surface. We prove that this index is always positive. This second result holds true, more generally, for complex curves in a two dimensional almost complex manifold with boundaries on totally real submanifold, but to our best knowledge is new even for integrable structures, unless the totally real surface in question is supposed to be real analytic. We also formulate and prove the Adjunction Formula for complex curves with boundaries on totally real submanifolds in an almost complex manifold of dimension four.

Paper Structure

This paper contains 29 sections, 27 theorems, 274 equations, 3 figures.

Table of Contents

  1. 1. Introduction and Statement of Main Results
  2. 2. Extension by Reflection and the Order of Vanishing
  3. 3. Comparison of Tangent Analytic Half-Disks
  4. 4. Perturbations of Half-Disks and Proof of the Main Theorem
  5. 5. Application I: Boundary Intersection of Two Analytic Disks in ${\mathbb C}^2$
  6. 6. Application II: Adjunction Formula
  7. 7. Smoothing of Boundary Cusps

Key Result

Theorem 1

Let $J$ and $W$ be as above and let $u_k:(\Delta^+,\partial_0\Delta^+,0)\to (X, W, p_0)$ be two $J$-holomorphic maps continuous up to $\partial_0\Delta^+$ such that the order of vanishing of $u_k(\zeta)-u_k(0)$ is one for both of them, and such that one is not a reparametrization of another on the b

Figures (3)

  • Figure 1: Surgery.
  • Figure 2: Gluing $\nu^+$ and $\nu^-$. Case 1.
  • Figure 3: Gluing $\nu^+$ and $\nu^-$. Case 2.

Theorems & Definitions (53)

  • Definition 1.1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 43 more