On the existence of minimizer on a log Fano cone singularity

Abstract

We prove that if $x\in (X,\Delta,\mathbb{T})$ is a log Fano cone singularity over an uncountable algebraically closed field, and $\nu_0$ is a $\mathbb{T}$-invariant valuation with center $x$ and $A_{X,\Delta}(\nu_0)<\infty$, then the value $$ \delta(X,\Delta;\nu_0):=\inf_{\nu\in \mathrm{Val}^{\mathbb{T},*}_{X,\ni x}}\frac{A_{X,\Delta}(\nu)}{S(\nu_0;\nu)}$$ admits a minimum. The proof uses the generic limit argument. Note that there is a counterexample if we discard the log Fano cone structure.