The Kawatani--Okawa conjecture in dimensions three and four

We prove that a smooth projective threefold or fourfold whose bounded derived category admits no nontrivial semiorthogonal decomposition has nef canonical divisor, confirming the Kawatani--Okawa conjecture in these dimensions. The proof combines the classification of extremal contractions with constructions of admissible subcategories. In dimension four, we refine the descriptions of isolated surface fibers and construct exceptional objects supported on certain fibers.