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Asymptotically flat divisors and strongly $F$-regular type varieties

Donghyeon Kim

Abstract

In this paper, we define the notion of asymptotically flat divisor on a normal variety over $\C$, and prove that if $X$ is a strongly $F$-regular type variety and $K_X$ is asymptotically flat, then $X$ is of klt type.

Asymptotically flat divisors and strongly $F$-regular type varieties

Abstract

In this paper, we define the notion of asymptotically flat divisor on a normal variety over , and prove that if is a strongly -regular type variety and is asymptotically flat, then is of klt type.

Paper Structure

This paper contains 11 sections, 18 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strongly $F$-regular couple
  4. Valuation
  5. Log discrepancy
  6. Log canonical threshold and Jonsson-Mustață conjecture
  7. Models
  8. Toroidal blowup
  9. Natural valuation and natural pullback
  10. Log discrepancies in sense of de Fernex-Hacon
  11. Proof of the main theorem

Key Result

Theorem 1.1

Let $(X,\Delta)$ be an affine pair over $\mathbb{C}$. Then $(X,\Delta)$ is of strongly $F$-regular type (for the definition, see Definition str) if and only if $(X,\Delta)$ is klt.

Theorems & Definitions (50)

  • Theorem 1.1: cf. HW02 and Tak04
  • Definition 1.2
  • Conjecture 1.3: cf. HW02 and SS10
  • Proposition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2: cf. SS10
  • proof
  • Theorem 2.3: cf. HW02
  • Definition 2.4
  • ...and 40 more