Positivity of Boundary Intersections of Complex Curves and Adjunction Formula

S. Ivashkovych

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arXiv · 2504.00805

Positivity of Boundary Intersections of Complex Curves and Adjunction Formula

S. Ivashkovych

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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.

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S. Ivashkovych. 2025-04-01. Positivity of Boundary Intersections of Complex Curves and Adjunction Formula. https://arxiv.org/abs/2504.00805

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