1 · The essentials
The question is whether one kind of agreement between two geometric structures can always be strengthened to another. One structure specifies a smooth rule on a four-dimensional space; the other is a form that must fit that rule. The manuscript shows that, under its stated conditions, the weaker agreement called “taming” guarantees that a form with the stronger agreement called “compatibility” exists. The rule stays fixed, but the form can change. The result applies to closed four-dimensional spaces, not to spaces of arbitrary dimension. It gives a positive answer to the question identified in the source as Donaldson’s tamed-to-compatible conjecture. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a fixed room layout and a piece of furniture that fits adequately but does not match it closely. The result says that an adequately fitting option guarantees a closely matching option, without rearranging the room; it does not say that the original furniture already matches.
Why it matters
The result turns the existence of a taming form into a guarantee that a compatible form exists, while keeping the almost complex structure unchanged. This resolves the conjecture as stated in the source. The manuscript also describes the collection of compatible forms through a strict positivity condition, adding a characterization beyond the existence statement.
2 · Going further
The precise problem concerns a smooth almost complex structure on a closed four-manifold . Suppose a symplectic form tames . The main result states that there is a symplectic form compatible with that same . The distinction between keeping fixed and keeping the form fixed is essential: the conclusion allows the form to change, and its cohomology class may change as well.
The source identifies this implication as Donaldson’s tamed-to-compatible conjecture and presents the manuscript as giving a positive solution. Its abstract also states a characterization of the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone. Thus the manuscript addresses both whether compatible forms exist and how their cone can be characterized.
The supplied abstract does not describe the construction of the compatible form or the main steps of the argument. In particular, it does not specify how the cone characterization enters the existence argument. Only one manuscript is supplied, so no relationship to companion manuscripts can be established from this source. The main result has a Lean formalization, but the supplied material does not specify whether that formalization also covers the cone characterization.
Useful notions
- Smooth manifolds and dimension
- Almost complex structures
- Symplectic forms
- Taming versus compatibility
- Cohomology classes and intersection positivity
What was known before / what is new
Before
The source identifies the question as Donaldson’s tamed-to-compatible conjecture. It supplies no account of earlier partial results, special cases, or methods, so no more detailed history can be given here.
New
The manuscript shows that a smooth almost complex structure on a closed four-manifold admits a compatible symplectic form whenever it admits a taming symplectic form. The almost complex structure remains fixed, while the cohomology class of the form is allowed to change. The manuscript also states a characterization of the compatible cone using strict intersection positivity.
Limits and assumptions
The existence statement assumes a smooth almost complex structure on a closed four-manifold and the existence of a symplectic form taming it. It does not assert that the compatible form can be chosen in the original form’s cohomology class. The main result has a Lean formalization; coverage of the additional cone characterization is not specified in the supplied material. The source gives no extension beyond the stated hypotheses and identifies no remaining open questions.
Mini glossary
- Almost complex structure —
- The geometric structure denoted by that remains fixed throughout the existence statement.
- Taming —
- The starting relationship between a symplectic form and the almost complex structure in the conjecture.
- Compatibility —
- The stronger relationship with the fixed almost complex structure that the resulting symplectic form is required to satisfy.
- Compatible cone —
- The collection associated with compatible symplectic forms that the manuscript characterizes through strict intersection positivity.
- Lean formalization —
- A verification of a mathematical result within the Lean proof assistant, available here for the main result.
Keywords : Almost complex structures · Symplectic compatibility · Donaldson’s conjecture
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Taming implies compatibility on four-manifolds
06/10/2026We prove that every smooth almost complex structure on a closed four-manifold which is tamed by a symplectic form is compatible with a symplectic form. This gives a positive solution to Donaldson's tamed-to-compatible conjecture. We also characterize the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone.