The existence of the Kähler–Ricci soliton degeneration

Abstract

We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .

Convexity of multiplicities of filtrations on local rings

“…This definition was introduced by the first two authors, Xu, and Zhuang in [BLXZ23, § 3.1.2] to prove uniqueness results for certain optimal destabilizations of Fano varieties that arise from limits of Kähler–Ricci flow [BLXZ23, § 3]. Independently, Reboulet introduced an equivalent definition phrased in the language of norms on the section ring, rather than filtrations, and used it to define geodesics between metrics on a line bundle in non-Archimedean pluripotential theory [Reb22].…”

“…To prove Theorem 1.1, we define a measure on that encodes the multiplicities along the geodesic. The argument may be viewed as a local analogue of the construction in [BLXZ23, § 3.1] for the geodesic between two filtrations of the section ring of a polarized variety. The latter global construction is motivated by [BC11] and [BHJ17, Theorem 4.3], which constructs a measure on associated to a single filtration of the section ring of polarized variety.…”

“…We call (a •,t ) t∈ [0,1] the geodesic between a •,0 and a •,1 , since it is the local analogue of the geodesic between two filtrations of the section ring of a polarized variety [BLXZ23,Reb22]. This definition is also related to a construction in [XZ21].…”

“…In [BLXZ23], it was shown that several non-Archimedean functionals from the theory of K-stability are strictly convex along geodesics in the global setting. In a similar spirit, we prove a convexity result in the local setting for the multiplicity along geodesics.…”

“…Given two -filtrations and , we define a segment of -filtrations interpolating between and by setting We call the geodesic between and , since it is the local analogue of the geodesic between two filtrations of the section ring of a polarized variety [BLXZ23, Reb22]. This definition is also related to a construction in [XZ21].…”