Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties

Han, Jiyuan¹ ; Li, Chi²

¹ Westlake University, Hangzhou, China
² Department of Mathematics, Rutgers University, Piscataway, NJ, United States

Geometry & topology, Tome 28 (2024) no. 2, pp. 539-592.

Résumé

We prove that for any Fano manifold $X$, the special $ℝ$–test configuration that minimizes the $H^{NA}$–functional is unique and has a K–semistable $\mathbb{Q}$–Fano central fiber $(W,\xi )$. Moreover there is a unique K–polystable degeneration of $(W,\xi )$. As an application, we confirm the conjecture of Chen, Sun and Wang about the algebraic uniqueness for Kähler–Ricci flow limits on Fano manifolds, which implies that the Gromov–Hausdorff limit of the flow does not depend on the choice of initial Kähler metrics. The results are achieved by studying algebraic optimal degeneration problems via new functionals for real valuations over $\mathbb{Q}$–Fano varieties, which are analogous to the minimization problem for normalized volumes.

Détail

DOI : 10.2140/gt.2024.28.539

Keywords: Fano varieties, optimal degeneration, Kähler–Ricci flow, Hamilton–Tian conjecture, K–stability, test configuration

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