This paper is concerned with the behaviour of the Kähler-Ricci flow on Fano manifolds. This is a parabolic flow designed to detect the existence of Kähler-Einstein metrics and Kähler-Ricci solitons on Fano manifolds, and starting from an initial Kähler metric \(\omega \in c_1(X)\) on \(X\), produces a family of Kähler metrics \[ \frac{d\omega(t)}{dt} = -\operatorname{Ric} \omega(t) + \omega(t), \] with \(\operatorname{Ric} \omega(t)\) the Ricci curvature and \(\omega(0)=\omega\). By the resolution of the Hamilton-Tian conjecture by X. Chen and B. Wang [J. Differ. Geom. 116, No. 1, 1–123 (2020; Zbl 1479.53103)], it is known that the Gromov-Hausdorff limit \((X_{\infty},\omega_{\infty})\) of \((X,\omega(t))\) is a \(\mathbb Q\)-Fano variety with \(\omega_{\infty}\) a Kähler-Ricci soliton.
This paper considers the uniqueness of \((X_{\infty},\omega_{\infty})\), namely how \((X_{\infty},\omega_{\infty})\) depends on the initial Kähler metric \(\omega \in c_1(X)\). The authors prove uniqueness under the following assumption: assuming that \(\operatorname{Aut}(X_{\infty})\) is reductive, any other Gromov-Hausdorff limit must equal \((X_{\infty},\omega_{\infty})\). In essence, the assumption that \(\operatorname{Aut}(X_{\infty})\) is reductive allows the authors to adapt techniques used to prove the analogous results for uniqueness of Kähler-Einstein degenerations (where reductivity is automatic). The proofs are interesting and quite technical.
The reader may be interested in recent work of J. Han and C. Li which uses much more algebro-geometric techniques to prove uniqueness in general [“Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties”, Preprint, arXiv:2009.01010].