Hyperkähler SYZ and projective-space bases

1 · The essentials

These two manuscripts describe how certain highly structured geometric spaces can be organized into families of smaller pieces. The first asks what the space labeling those pieces can look like. Under the stated conditions, it is shown in the manuscript that this labeling space must be a standard space called projective space. The second concerns a kind of geometric data attached to the original space, called a line bundle. It is shown in that manuscript that line bundles satisfying three specific conditions have enough global sections, after taking a positive power, to avoid vanishing everywhere at any point. Both claims concern a restricted class of complex geometric spaces, not arbitrary shapes. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a building divided into rooms, with a directory that labels each room. The first manuscript says that, for buildings obeying very special geometric rules, the directory must have one standard geometric form; the analogy does not capture the actual mathematical restrictions.

Why it matters

The main manuscript describes its result as resolving the projective-space base conjecture for the specified fibrations in every dimension and deformation type. The companion manuscript claims the strong hyperkähler SYZ conjecture under the precise line-bundle hypotheses stated below. Together, they address the form of the base of a fibration and the behavior of particular line bundles on the same class of ambient spaces.

2 · Going further

Let $X$ be a compact irreducible holomorphic symplectic Kähler manifold. The main manuscript considers a projective Lagrangian fibration $f : X \to B$ whose base $B$ is normal and projective. Its claim is that $B$ is projective space. The abstract describes this as resolving the projective-space base conjecture for such fibrations in every dimension and deformation type.

The companion manuscript considers a holomorphic line bundle $L$ on such an $X$ , with nonzero, nef, isotropic first Chern class $c_{1} (L)$ . It claims that $L$ is semiample: some positive tensor power $L^{\otimes m}$ is generated by its global sections. Here nefness is a numerical nonnegativity condition, while isotropy means that the class has square zero for the relevant quadratic form.

The manuscripts therefore address two related questions: what the base of an existing projective Lagrangian fibration must be, and whether line bundles with the specified numerical properties are semiample. The supplied abstracts state the conclusions but give no argument, construction, or intermediate steps. Consequently, they do not support a description of the proof strategy or of a logical dependency between the two manuscripts.

Useful notions

Complex manifolds and Kähler geometry
Holomorphic symplectic forms
Line bundles and first Chern classes
Fibrations and their bases
Global sections and tensor powers

What was known before / what is new

Before

The source identifies the projective-space base statement as a conjecture and the line-bundle statement as the strong hyperkähler SYZ conjecture. It supplies no earlier partial results, historical chronology, or previously known special cases.

New

The main manuscript claims that every projective Lagrangian fibration in its stated setting has projective space as its normal projective base, without restricting dimension or deformation type. The companion manuscript claims semiampleness for every holomorphic line bundle whose first Chern class is nonzero, nef, and isotropic on a compact irreducible holomorphic symplectic Kähler manifold.

Limits and assumptions

Both statements require a compact irreducible holomorphic symplectic Kähler ambient manifold. The base result additionally requires a projective Lagrangian fibration and a normal projective base; the line-bundle result requires a nonzero, nef, isotropic first Chern class. The supplied abstracts do not establish conclusions outside those hypotheses and do not identify remaining open cases. They also provide no details with which to assess the arguments. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Base of a fibration: The target space of a fibration, whose points label its fibers.
Projective space: The geometric space whose points represent one-dimensional linear subspaces of a vector space.
Lagrangian fibration: A fibration whose smooth fibers have half the ambient complex dimension and on which the holomorphic symplectic form restricts to zero.
Semiample: A line bundle is semiample if some positive tensor power is generated by global sections, so those sections have no common zero.
Isotropic class: A class is isotropic when its value under the relevant quadratic form is zero.

Keywords : Hyperkähler SYZ · Lagrangian fibrations · Projective-space bases

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 the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

★

Projective-space bases of Lagrangian fibrations

23/09/2026

To be verified

We prove that the normal projective base of a projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space. This resolves the projective-space base conjecture for such fibrations in every dimension and deformation type.

PDF GitHub folder

The strong hyperkähler SYZ conjecture

23/09/2026

To be verified

We prove the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample.

PDF GitHub folder