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Boundedness of log Fano pairs with certain K-stability

Konstantin Loginov, Chuyu Zhou

TL;DR

The paper investigates boundedness phenomena for log Fano pairs under K-stability, proving that K-semistable log Fano pairs of Maeda type in dimension $d$ form a log bounded family and establishing boundedness for the parameterized set $\mathcal{E}(d,k,v,I)$. It leverages complements, alpha- and delta-invariants, and BAB-type results to control both the base varieties and the boundary divisors, then derives uniform delta-invariant lower bounds via a gap argument. Additionally, the authors compute K-semistable domains for explicit projective-space configurations and show these domains are polytopes, with vertices corresponding to known K-semistable log pairs. Together, these results extend prior 2- and 3-dimensional boundedness phenomena to Maeda-type settings and enrich the understanding of moduli for K-stable Fano varieties under controlled boundary data.

Abstract

We prove several boundedness results for log Fano pairs with certain K-stability. In particular, we prove that K-semistable log Fano pairs of Maeda type form a log bounded family. We also compute K-semistable domains for some examples.

Boundedness of log Fano pairs with certain K-stability

TL;DR

The paper investigates boundedness phenomena for log Fano pairs under K-stability, proving that K-semistable log Fano pairs of Maeda type in dimension form a log bounded family and establishing boundedness for the parameterized set . It leverages complements, alpha- and delta-invariants, and BAB-type results to control both the base varieties and the boundary divisors, then derives uniform delta-invariant lower bounds via a gap argument. Additionally, the authors compute K-semistable domains for explicit projective-space configurations and show these domains are polytopes, with vertices corresponding to known K-semistable log pairs. Together, these results extend prior 2- and 3-dimensional boundedness phenomena to Maeda-type settings and enrich the understanding of moduli for K-stable Fano varieties under controlled boundary data.

Abstract

We prove several boundedness results for log Fano pairs with certain K-stability. In particular, we prove that K-semistable log Fano pairs of Maeda type form a log bounded family. We also compute K-semistable domains for some examples.
Paper Structure (8 sections, 16 theorems, 57 equations)

This paper contains 8 sections, 16 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. K-stability
  4. Complements
  5. Boundedness of Fano varieties
  6. Boundedness I
  7. Boundedness II
  8. On K-semistable domains

Key Result

Theorem 1.1

Let $(X, \sum_ic_iD_i)$ be a log Fano pair of Maeda type such that $\dim X=2$. Then $(X, \sum_ic_iD_i)$ is K-semistable if and only if it is isomorphic to

Theorems & Definitions (36)

  • Theorem 1.1: Fujita20
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 26 more