A hyper-Kähler variety carries three compatible complex structures that rotate into each other. This extra symmetry constrains the geometry drastically. Among the consequences: the variety may admit a Lagrangian fibration — a map to a lower-dimensional base where each fiber is a Lagrangian submanifold, meaning it has half the dimension and the symplectic form vanishes on it.
The survey on hyper-Kähler varieties (arXiv: 2603.23033) reviews recent developments from three perspectives: Lagrangian fibrations, moduli spaces of stable sheaves, and derived categories.
These three viewpoints are not independent. Moduli spaces of sheaves on K3 surfaces produce hyper-Kähler varieties. Derived categories — the homological algebra of sheaves — detect when two hyper-Kähler varieties are equivalent in a strong sense (derived equivalent). Lagrangian fibrations provide the geometric structure theory: they decompose the variety into a family of abelian varieties parametrized by a base.
The interplay is productive. Stability conditions on derived categories determine which moduli spaces are well-behaved. Lagrangian fibrations appear naturally as limits of stability conditions (wall-crossing). The “atomic” sheaves — the indecomposable building blocks in the derived category — correspond to geometric features of the Lagrangian fibration.
The through-claim: the same object viewed from three directions is more visible than from one. Hyper-Kähler geometry progresses by rotating between perspectives — geometric (fibrations), algebraic (sheaves), and categorical (derived equivalences). None of the three is fundamental; the structure lives in their intersection.
2603.23033. Algebraic geometry / hyper-Kähler varieties / Lagrangian fibrations / derived categories / moduli spaces.