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-How to read this file. This is a study guide to one research paper: the proof of the strong hyperkähler SYZ conjecture, which says every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic (IHS) Kähler manifold X is semiample. It is written for a patient high-school reader. Each block starts with "Exercise:", then a short question, then a self-contained answer. A "Proof:" sketch inside an answer can be skipped on a first read and returned to later. Named results carry a short nickname in parentheses, e.g. "Theorem (Main)"; find the full statement by searching for the same nickname. Most notation is defined at first use; the symbols you will meet first are ℂ, ℝ, ℤ (complex numbers, reals, integers); X (the manifold, complex dimension 2n); q (the Beauville–Bogomolov–Fujiki form, called BBF for short); c_1(L) (the first Chern class of a line bundle L); ω (a Kähler form); σ (a nondegenerate holomorphic 2-form); and ω_I, ω_J, ω_K (three Kähler forms coming from three complex structures I, J, K on the same real manifold, related by quaternions). Citations are given inline, mostly arXiv IDs. The companion paper Projective-space bases of Lagrangian fibrations (OpenAI, September 2026) supplies the base-is-projective-space conclusion for the projective case; it is at https://github.com/openai/math/blob/main/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf.
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 Exercise: The main theorem. What does the strong hyperkähler SYZ conjecture assert, and what is proven here?
 For a compact IHS Kähler manifold X of complex dimension 2n with n ≥ 1, the strong hyperkähler SYZ conjecture says: if a holomorphic line bundle L has first Chern class c_1(L) that is nonzero, nef (lies in the closure of the Kähler cone of X), and isotropic (q(c_1(L)) = 0), then L is semiample, meaning some positive tensor power L^{⊗m} is globally generated. The paper proves it in full (Theorem (Main)). More precisely, there exist a positive integer m, a normal projective variety B of dimension n, an ample line bundle A on B, and a surjective holomorphic map f: X → B with connected fibers such that L^m ≅ f^*A; every irreducible component of every fiber is n-dimensional, and the holomorphic 2-form σ vanishes on its smooth locus. This turns a class on the boundary of the Kähler cone into a genuine fibration half-dimensional in the fibers. Citation: Verbitsky 2010, Conjecture 1.7 (Hyperkähler SYZ conjecture and semipositive line bundles), https://arxiv.org/abs/0811.0639v2.