Asymptotic stability for Kähler–Ricci solitons

Abstract

Let X be a Fano manifold. We say that a hermitian metric φ on −K X with positive curvature ω φ is a Kähler-Ricci soliton if it satisfies the equation Ric(ω φ ) − ω φ = L V K S ω φ for some holomorphic vector field V K S . The candidate for a vector field V K S is uniquely determined by the holomorphic structure of X up to conjugacy, hence depends only on the holomorphic structure of X . We introduce a sequence {V k } of holomorphic vector fields which approximates V K S and fits to the quantized settings. Moreover, we also discuss about the existence and convergence of the quantized Kähler-Ricci solitons attached to the sequence {V k }.

Balanced metrics for Kähler-Ricci solitons and quantized Futaki invariants

“…This question has already been addressed in previous works of Berman and Witt Nyström in [4,Th. 1.7] and Takahashi in [31,Th. 1.2], where Theorem 1.1 is established in the sense of currents under the hypothesis that a modified Ding functional with respect to ξ ∈ Lie Aut(X) is coercive modulo Aut 0 (X).…”

Ricci Iteration for Coupled Kähler–Einstein Metrics

“…On the other hand, the notion of our balanced metrics (3.3) has already been studied in several papers [BBGZ13,BN14,Tak15] in the N = 1 case, where the existence and weak convergence results of the balanced metrics were given under some suitable assumptions.…”

Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka

“…Kähler-Ricci g-solitons) implies that we can find a sequence of anticanonically balanced metrics that converges to the Kähler-Einstein metric (resp. the Kähler-Ricci g-soliton), in the sense of currents; see also [59] for the case of solitons. The convergence can in fact be improved to the smooth convergence by [60,Theorem 1.3 with N = 1] and [39,40].…”