Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Anti-self-dual blowups II

Vsevolod Shevchishin, Gleb Smirnov

TL;DR

The paper proves that on a closed oriented 4-manifold with $b_2^+\le 3$ containing disjoint $(-2)$-spheres, one can choose a metric so that each sphere’s Poincaré dual is represented by an ASD harmonic form. The authors develop a neck-stretching gluing technique, an orbifold-resolution construction, and a new existence theorem for closed SD forms with prescribed zeros on 4-orbifolds, aided by Eliashberg’s h-principle for overtwisted contact structures to control near-zero behavior. They show that, via long-necks and blow-ups, the relevant $(-2)$-classes can be simultaneously realized as ASD, including a mixed $(-1)$/$(-2)$ case, and they provide explicit local models for the orbifold-to-manifold transition. This work blends harmonic analysis, orbifold geometry, and near-symplectic/near-contact topology to realize ASD representatives for multiple cohomology classes and to extend previous results on $(-1)$-spheres.

Abstract

Let $X$ be a closed, oriented four-manifold with $b_2^+ \leq 3$, and suppose $X$ contains a collection of pairwise disjoint embedded $(-2)$-spheres. We prove that there is a Riemannian metric on $X$ such that the Poincare dual of each of these spheres is represented by an anti-self-dual harmonic form. This extends our earlier result for $(-1)$-spheres. The main new ingredient is an application of Eliashberg's $h$-principle for overtwisted contact structures, which we use to construct self-dual harmonic forms on four-orbifolds with prescribed local behaviour near the orbifold singular set.

Anti-self-dual blowups II

TL;DR

The paper proves that on a closed oriented 4-manifold with containing disjoint -spheres, one can choose a metric so that each sphere’s Poincaré dual is represented by an ASD harmonic form. The authors develop a neck-stretching gluing technique, an orbifold-resolution construction, and a new existence theorem for closed SD forms with prescribed zeros on 4-orbifolds, aided by Eliashberg’s h-principle for overtwisted contact structures to control near-zero behavior. They show that, via long-necks and blow-ups, the relevant -classes can be simultaneously realized as ASD, including a mixed / case, and they provide explicit local models for the orbifold-to-manifold transition. This work blends harmonic analysis, orbifold geometry, and near-symplectic/near-contact topology to realize ASD representatives for multiple cohomology classes and to extend previous results on -spheres.

Abstract

Let be a closed, oriented four-manifold with , and suppose contains a collection of pairwise disjoint embedded -spheres. We prove that there is a Riemannian metric on such that the Poincare dual of each of these spheres is represented by an anti-self-dual harmonic form. This extends our earlier result for -spheres. The main new ingredient is an application of Eliashberg's -principle for overtwisted contact structures, which we use to construct self-dual harmonic forms on four-orbifolds with prescribed local behaviour near the orbifold singular set.

Paper Structure

This paper contains 7 sections, 22 theorems, 187 equations.

Table of Contents

  1. Main result
  2. A neck-stretching argument
  3. Resolution formula
  4. Proof of Theorem \ref{['t:main']}
  5. Near-contact forms
  6. Near-symplectic forms
  7. Proof of Proposition \ref{['cone_zero']}

Key Result

Theorem 1

Let $X$ be a closed, oriented $4$-manifold with $b^{+}_{2} \leqslant 3$. Suppose $X$ contains a collection of pairwise disjoint embedded spheres $S_i$, each with self-intersection $(-2)$. Then there exists a Riemannian metric $g$ on $X$ such that every $g$-self-dual harmonic form $u$ satisfies: Equivalently, if $s_i \in H^2(X;\mathbb{Z})$ denotes the Poincaré dual of $S_i$, then each $s_i$ is rep

Theorems & Definitions (39)

  • Theorem 1
  • Remark
  • Proposition 1
  • Theorem
  • Remark
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 29 more