Survey of finiteness results for hyperk\"ahler manifolds

TL;DR

This survey reviews key finiteness theorems in hyperk"ahler geometry, highlighting classical and recent results, and includes an improved finiteness theorem by the author.

Contribution

It consolidates and discusses classical and recent finiteness results in hyperk"ahler geometry, including a strengthened theorem by the author.

Findings

Comprehensive overview of finiteness theorems in hyperk"ahler geometry

Extension and strengthening of existing finiteness results

Connections between classical and recent research in the field

Abstract

This paper is a survey of finiteness results in hyperk\"ahler geometry. We review some classical theorems by Sullivan, Koll\'ar-Matsusaka, Huybrechts, as well as theorems in the recent literature by Charles, Sawon, and joint results of the author with Verbitsky. We also strengthen a finiteness theorem of the author. These are extended notes of the author's talk during the closing conference of the Simons Semester in the Banach Center in B\c{e}dlewo, Poland.