Principal twistor models and asymptotic hyperk\"ahler metrics

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.