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A guide to wall crossing for moduli of varieties

Kristin DeVleming

Abstract

There have been major developments in the theory of moduli of varieties in the past decade, essentially settling the construction of moduli spaces of log canonically polarized slc pairs and moduli spaces of K-polystable log Fano pairs. Given the construction of these moduli spaces of pairs $(X, D)$, it is natural to ask how the moduli spaces vary as the coefficients of $D$ are perturbed. This phenomenon is known as wall crossing, the theory of which has been developed in several important cases in the past five years. This semi-expository article is an introduction to moduli of varieties and wall crossing, capturing a portion of the theory developed in the past several years. It also introduces tools and techniques used in explicit computations and examples, applying them in new examples.

A guide to wall crossing for moduli of varieties

Abstract

There have been major developments in the theory of moduli of varieties in the past decade, essentially settling the construction of moduli spaces of log canonically polarized slc pairs and moduli spaces of K-polystable log Fano pairs. Given the construction of these moduli spaces of pairs , it is natural to ask how the moduli spaces vary as the coefficients of are perturbed. This phenomenon is known as wall crossing, the theory of which has been developed in several important cases in the past five years. This semi-expository article is an introduction to moduli of varieties and wall crossing, capturing a portion of the theory developed in the past several years. It also introduces tools and techniques used in explicit computations and examples, applying them in new examples.
Paper Structure (23 sections, 44 theorems, 73 equations, 4 figures, 1 table)

This paper contains 23 sections, 44 theorems, 73 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singularities of the minimal model program, stable pairs, and K-stability
  4. Conventions
  5. Singularities of pairs
  6. K-stability via the Futaki invariant
  7. K-stability via the $\delta$ invariant
  8. Families of stable pairs and moduli functors
  9. Wall crossing for moduli of pairs
  10. K-moduli of log Fano pairs
  11. Wall crossing for canonically polarized pairs
  12. Moduli of boundary polarized Calabi Yau pairs
  13. Wall crossing for quartic plane curves
  14. Toolbox
  15. K-moduli wall crossing for quartic curves
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\mathcal{M}^\mathrm{K}_{n,v,r,c}$ (respectively, $M^\mathrm{K}_{n,v,r,c}$) be the proper K-moduli stack (respectively, projective K-moduli space) of K-semistable (respectively, K-polystable) log Fano pairs $(X,cD)$ such that $X$ is a smoothable klt Fano variety of $\dim X = n$, $(-K_X)^n = v$, such that $c$-K-(poly/semi)stability conditions do not change for $c\in (c_i,c_{i+1})$. For each $1

Figures (4)

  • Figure 1: Replacement of the rational octic curve with cusp $y^3 = x^{22}$.
  • Figure 2: Replacement of the double conic.
  • Figure 3: The common polystable degeneration of tacnodal curves.
  • Figure 4: Wall crossings for moduli of quartic curves.

Theorems & Definitions (84)

  • Theorem 1.1: ADLpub
  • Theorem 1.2: ABIP
  • Theorem 1.3: BABWILDBL
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • ...and 74 more