Uniqueness of asymptotically conical shrinking gradient K\"ahler-Ricci solitons

TL;DR

This paper proves the uniqueness of asymptotically conical shrinking gradient K"ahler-Ricci solitons on noncompact complex manifolds without fixing asymptotic data, and applies the method to related geometric problems.

Contribution

It establishes the uniqueness of such solitons without fixing asymptotic conditions and introduces a method applicable to Calabi-Yau metrics and algebraic fibrations.

Findings

Uniqueness of shrinking gradient K"ahler-Ricci solitons up to biholomorphism.

Method for proving soliton vector field uniqueness applicable to Calabi-Yau metrics.

Biholomorphic polarized Fano fibrations are isomorphic as algebraic varieties.

Abstract

We show that, up to biholomorphism, a given noncompact complex manifold only admits one shrinking gradient K\"ahler-Ricci soliton with Ricci curvature tending to zero at infinity. Our result does not require fixing the asymptotic data of the metric, nor fixing the soliton vector field. The method used to prove the uniqueness of the soliton vector field can be applied more widely, for example to show that conical Calabi-Yau metrics on a given complex manifold are unique up to biholomorphism. We also use it to prove that if two polarized Fano fibrations, as introduced by Sun-Zhang, are biholomorphic, then they are isomorphic as algebraic varieties.