Stable maps in higher dimensions

TL;DR

This paper extends the concept of stability for maps between polarized varieties to higher dimensions, establishing a moduli space and linking stability to canonical Kähler metrics.

Contribution

It introduces a new stability notion for higher-dimensional maps, constructs a moduli space, and proposes an analogue of the Yau-Tian-Donaldson conjecture.

Findings

Existence of a projective moduli space for stable maps in higher dimensions

Examples include Kodaira embeddings and fibrations

Formulation of an analogue of the Yau-Tian-Donaldson conjecture

Abstract

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective moduli space of canonically polarised stable maps, generalising the Kontsevich-Alexeev moduli space of stable maps in dimensions one and two. We also state an analogue of the Yau-Tian-Donaldson conjecture in this setting, relating stability of maps to the existence of certain canonical K\"ahler metrics.