Hyper-K\"ahler varieties: Lagrangian fibrations, atomic sheaves, and categories
Alessio Bottini, Emanuele Macr\`i, Paolo Stellari
TL;DR
This paper reviews recent advances in the theory of compact hyper-K"ahler varieties, focusing on Lagrangian fibrations, moduli spaces of stable sheaves, and derived categories, highlighting new perspectives and developments.
Contribution
It synthesizes recent progress in understanding hyper-K"ahler varieties through Lagrangian fibrations, sheaf moduli spaces, and categorical approaches, offering a comprehensive overview.
Findings
Enhanced understanding of Lagrangian fibrations in hyper-K"ahler varieties
Connections between moduli spaces of stable sheaves and hyper-K"ahler geometry
Insights into derived categories related to hyper-K"ahler varieties
Abstract
We review recent developments in the theory of compact hyper-K\"ahler 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 2025 Summer Institute in Algebraic Geometry, Colorado State University, Fort Collins (USA), July 14 - August 1, 2025.
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