The Ricci iteration and its applications

TL;DR

This paper introduces Ricci iteration as a discrete dynamical system on Kahler metrics, exploring its convergence and applications in Kahler geometry, including solving open questions and constructing multiplier ideal sheaves.

Contribution

It proposes a new Ricci iteration framework, analyzes its convergence in various Chern class cases, and applies it to solve problems in Kahler geometry.

Findings

Conjecture on convergence towards canonical Kahler metrics.

Application to Nadel's question.

Construction of multiplier ideal sheaves.

Abstract

In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kahler metrics and study the case where the first Chern class is negative, zero or positive. This construction has several applications in Kahler geometry, among them an answer to a question of Nadel and a construction of multiplier ideal sheaves.