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Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories

Alessio Bottini, Emanuele Macrì, Paolo Stellari

Abstract

We review recent developments in the theory of compact hyper-Kähler varieties, from the viewpoint of Lagrangian fibrations, moduli spaces of stable sheaves, and derived categories. These notes originated from the lecture by the second named author at the \emph{2025 Summer Institute in Algebraic Geometry}, Colorado State University, Fort Collins (USA), July 14 -- August 1, 2025.

Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories

Abstract

We review recent developments in the theory of compact hyper-Kähler varieties, from the viewpoint of Lagrangian fibrations, moduli spaces of stable sheaves, and derived categories. These notes originated from the lecture by the second named author at the \emph{2025 Summer Institute in Algebraic Geometry}, Colorado State University, Fort Collins (USA), July 14 -- August 1, 2025.
Paper Structure (22 sections, 22 theorems, 90 equations)

This paper contains 22 sections, 22 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Hyper-Kähler varieties
  3. Definition
  4. Primitive symplectic varieties and the Torelli Theorem
  5. Irreducible symplectic varieties and the Decomposition Theorem
  6. Examples
  7. Lagrangian fibrations
  8. Definition and the main conjecture
  9. The Saccà Theorem
  10. Examples
  11. Recent major advances
  12. Sheaves on hyper-Kähler manifolds
  13. Lagrangian subvarieties
  14. Hyperholomorphic bundles
  15. Atomic objects
  16. ...and 7 more sections

Key Result

Lemma 2.3

Let $M$ be a projective hyper-Kähler variety. Then $(M,\sigma)$ is symplectic.

Theorems & Definitions (70)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Conjecture 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 60 more