Taming implies compatibility

Donaldson asked whether, on a closed four-manifold, an almost complex structure tamed by a symplectic form admits a compatible symplectic form. The answer is yes. If not, separation gives a positive current that annihilates closed invariant forms, and a closed lift adds an anti-invariant correction so that the sum annihilates all closed forms. Resolvent regularization shows that the correction is square integrable and that nearby complex lines of the current align; at points of positive density the current is planar, and its restriction there is closed. A final pairing removes the correction, and the current then annihilates the taming form, which is impossible. Proved in the programme from one named hypothesis, the short-time asymptotics of the heat kernel of the Hodge Laplacian. The course is an exposition of OpenAI's proof (2026).

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Taming, compatibility and a separating current
2. Quadratic mass growth
3. Regularization and angular control
4. Densities and planar tangent measures
5. Closedness of the density part and the proof

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Order, positivity and distributional limits (Distributions, kernels and analytic singularities)
• Connections and parallel transport (Bundles, curvature and holonomy)
• Geodesics, normal coordinates and curvature (Bundles, curvature and holonomy)
• Jacobi fields, conjugate points and the Morse index theorem (Bundles, curvature and holonomy)
• Riemannian connections and convex neighbourhoods (Bundles, curvature and holonomy)
• Symbols, operators and Sobolev scales
• Detecting regularity without choosing coordinates
• Symbols, finite defects, and the index on a closed manifold
• Hahn–Banach, Baire and the basic theorems on Banach spaces · Foundations of von Neumann algebras
• Hilbert spaces and compact operators · Foundations of von Neumann algebras
• Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian · Foundations of von Neumann algebras
• Besicovitch's covering theorem and the differentiation of Radon measures · Open Mathematics Courses (Measurable fields and direct integrals — lifting and differentiation)
• Elliptic operators on closed manifolds (course Elliptic boundary reduction); normal coordinates, parallel transport and Jacobi fields (course Bundles, curvature and holonomy); positive distributions and the differentiation of Radon measures (courses Distributions, kernels and analytic singularities, and Measurable fields and direct integrals).