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On moduli of Fano varieties: an introduction to K-stability and K-moduli

Kristin DeVleming

Abstract

This survey article is an accompaniment to the 2025 Summer Research Institute in Algebraic Geometry Bootcamp on K-stability and K-moduli. It is aimed at graduate students and intended to provide the necessary background to begin research on explicit K-moduli problems.

On moduli of Fano varieties: an introduction to K-stability and K-moduli

Abstract

This survey article is an accompaniment to the 2025 Summer Research Institute in Algebraic Geometry Bootcamp on K-stability and K-moduli. It is aimed at graduate students and intended to provide the necessary background to begin research on explicit K-moduli problems.
Paper Structure (17 sections, 21 theorems, 84 equations)

This paper contains 17 sections, 21 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Singularities of the Minimal Model Program
  3. Exercises
  4. Introduction to K-stability
  5. K-stability via test configurations
  6. K-stability via the $\alpha$-invariant
  7. K-stability via the $\beta$ and $\delta$ invariants
  8. Exercises
  9. Abban-Zhuang theory of admissible flags
  10. Exercises
  11. Results on moduli of K-semistable Fano varieties
  12. Exercises
  13. K-stability of singular Fano varieties and K-moduli of cubic surfaces
  14. Local-to-global principles and normalized volume
  15. K-moduli of cubic surfaces
  16. ...and 2 more sections

Key Result

Theorem 2.3

KM98 Let $\pi: (Y,\Delta_Y) \to (X,\Delta)$ be a log resolution such that $\Delta_Y$ is smooth, where $\Delta_Y$ is the strict transform of $\Delta$ on $Y$. If $a_{X,\Delta}(E_i) \ge -1$ for every exceptional divisor $E_i$ of $\pi$, then $\mathrm{ discrep } (X,\Delta) = \min \{ \min_i \{a_{X, \Delta

Theorems & Definitions (76)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Definition 3.1: Tian Tian, Donaldson Donaldson
  • Remark 3.2
  • Remark 3.3
  • ...and 66 more