Anti-self-dual blowups II

2025/12/03 by Vsevolod Shevchishin, Shevchishin, Vsevolod, Gleb Smirnov +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2512.03352

openalex publication_date 2025/12/03 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/28

Abstract

Let X be a closed, oriented four-manifold with b₂⁺ ≤ 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.

Citations

• Metric stretching and the period map for smooth 4-manifolds
• Anti-self-dual blowups
• Zero-sets of near-symplectic forms
• On the Cappell-Lee-Miller glueing theorem
• Yamabe Constants and the Perturbed Seiberg-Witten Equations

Related

• Anti-self-dual blowups
• Anti-Self-Dual 4-Manifolds, Quasi-Fuchsian Groups, and Almost-Kaehler Geometry
• Blow-up formulas for (-2)-spheres
• Local Properties of Self-Dual Harmonic 2-forms on a 4-Manifold
• Anti-self-dual bihermitian structures on Inoue surfaces