Asymptotic stability for Kähler–Ricci solitons

Abstract

Let X be a Fano manifold. We say that a hermitian metric $\phi$ on $-K_X$ with positive curvature $\omega _{\phi }$ is a Kähler–Ricci soliton if it satisfies the equation $\mathrm{Ric}(\omega _{\phi }) - \omega _{\phi } = L_{V_{KS}} \omega _{\phi }$ for some holomorphic vector field $V_{KS}$. The candidate for a vector field $V_{KS}$ 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 $V_{KS}$ 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 $\{V_k\}$.