Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds

TL;DR

This paper explores the use of dynamic multiplier ideal sheaves and complex Monge-Ampere equations to construct rational curves in Fano manifolds, highlighting a novel approach under development.

Contribution

It introduces a new method employing dynamic multiplier ideal sheaves and singularity-magnifying equations for constructing rational curves in Fano manifolds.

Findings

Initial framework for using multiplier ideal sheaves in Fano manifolds

Connection between complex Monge-Ampere equations and rational curve construction

Ongoing development with details still being refined

Abstract

This note is written for the Festschrift in honor of Professor Christer Kiselman. Multiplier ideal sheaves identify the location and the extent of the failure of crucial estimates. In this note we will discuss and explain the historic evolution of the notion of multiplier ideal sheaves, especially the interpretation from the viewpoint of destabilizing subsheaves in the context of terminating or bounding an infinite process. We will also discuss the approach of constructing rational curves in Fano manifolds by using dynamic multiplier ideal sheaves and singularity-magnifying complex Monge-Ampere equations. This approach is still under development with details in the process of being worked out. We will indicate where details still need to be worked out.