Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positivity of Intersections and Tameness of Almost Complex 4-manifolds

Spencer Cattalani

Abstract

We prove that pseudoholomorphic curves intersect complex 2-cycles positively in almost complex 4-manifolds. This makes possible a general and conceptually simple proof that an almost complex 4-manifold with many curves admits a taming symplectic structure, as envisioned by Gromov. Furthermore, we prove that the positivity of intersections between pseudoholomorphic curves is stable, in a geometric sense.

Positivity of Intersections and Tameness of Almost Complex 4-manifolds

Abstract

We prove that pseudoholomorphic curves intersect complex 2-cycles positively in almost complex 4-manifolds. This makes possible a general and conceptually simple proof that an almost complex 4-manifold with many curves admits a taming symplectic structure, as envisioned by Gromov. Furthermore, we prove that the positivity of intersections between pseudoholomorphic curves is stable, in a geometric sense.
Paper Structure (7 sections, 14 theorems, 56 equations)

This paper contains 7 sections, 14 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Discussion of results
  3. Preliminaries: currents
  4. Preliminaries: curves
  5. Construction of Poincaré dual
  6. Proofs of the main theorems
  7. Proof of Proposition \ref{['CurveMS_prp']}

Key Result

Theorem 1

Let $(X,J)$ be a closed almost complex 4-manifold such that every pair of points can be joined by a closed connected pseudoholomorphic curve $C$. Suppose $(X,J)$ contains Then $(X,J)$ admits a taming symplectic form $\omega$.

Theorems & Definitions (37)

  • Theorem 1
  • Proposition 1.1: Sullivan
  • Theorem 2
  • Corollary 1.2
  • Theorem 3
  • Example 2.1
  • Example 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 27 more