Principal twistor models and asymptotic hyperk\"ahler metrics

TL;DR

This paper constructs a universal twistor model for hyperk"ahler metrics on crepant resolutions of conical symplectic varieties, enabling the classification of asymptotic hyperk"ahler structures.

Contribution

It introduces a principal twistor model that uniquely recovers algebraic hyperk"ahler metrics asymptotic to a cone, advancing the understanding of their moduli space.

Findings

The principal twistor model is universal for hyperk"ahler cone metrics.

Any algebraic hyperk"ahler metric asymptotic to a cone is recovered by slicing the model.

The moduli space of such hyperk"ahler structures embeds into a finite-dimensional real vector space.

Abstract

Let $X$ be a conical symplectic variety admitting a crepant resolution $Y$. Based on the theory of universal Poisson deformations, we construct a complex manifold called the principal twistor model associated with $Y$. We prove a universality theorem for this model: if the regular locus of $X$ admits a hyperk\"ahler cone metric, then the twistor space of any algebraic hyperk\"ahler metric on $Y$ asymptotic to this cone metric is uniquely recovered by slicing the principal twistor model. As an application, we use this universality to study the moduli space of hyperk\"ahler structures with asymptotic behavior, and show that it admits an inclusion into a finite-dimensional real vector space.