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Symmetric and Kähler--Einstein Fano polygons

DongSeon Hwang, Yeonsu Kim

Abstract

We investigate \emph{singular} symmetric and Kähler--Einstein Fano polytopes. More precisely, we show that every symmetric Fano polytope is Kähler--Einstein generalizing the work by Batyrev and Selivanova, and study the automorphism groups of symmetric and Kähler--Einstein Fano polygons in detail. In particular, every finte subgroup of $GL_2(\mathbb{Z})$ is an automorphism group of a Kähler--Einstein Fano polygon.

Symmetric and Kähler--Einstein Fano polygons

Abstract

We investigate \emph{singular} symmetric and Kähler--Einstein Fano polytopes. More precisely, we show that every symmetric Fano polytope is Kähler--Einstein generalizing the work by Batyrev and Selivanova, and study the automorphism groups of symmetric and Kähler--Einstein Fano polygons in detail. In particular, every finte subgroup of is an automorphism group of a Kähler--Einstein Fano polygon.

Paper Structure

This paper contains 9 sections, 25 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fano polygons
  4. Kähler-Einstein Fano polygons
  5. Symmetric and Kähler-Einstein Fano polygons
  6. Automorphisms
  7. Automorphisms of Lattice symmetric Fano polygons
  8. Automorphisms of Kähler-Einstein Fano polygons
  9. Demazure root systems

Key Result

Theorem 1.2

$($SZ, BB$)$ Let $X$ be a toric Fano variety. Then $X$ admits a Kähler--Einstein metric if and only if the barycenter of its moment polytope is the origin.

Figures (1)

  • Figure 1: $P_{m,n}$

Theorems & Definitions (49)

  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.5
  • Lemma 2.6
  • ...and 39 more