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arXiv · 2608.23019

On the existence of minimizer on a log Fano cone singularity

Donghyeon Kim
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Abstract

We prove that if $x\in (X,Δ,\mathbb{T})$ is a log Fano cone singularity over an uncountable algebraically closed field, and $ν_0$ is a $\mathbb{T}$-invariant valuation with center $x$ and $A_{X,Δ}(ν_0)<\infty$, then the value $$ δ(X,Δ;ν_0):=\inf_{ν\in \mathrm{Val}^{\mathbb{T},*}_{X,\ni x}}\frac{A_{X,Δ}(ν)}{S(ν_0;ν)}$$ admits a minimum. The proof uses the generic limit argument. Note that there is a counterexample if we discard the log Fano cone structure.

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Donghyeon Kim. 2026-08-24. On the existence of minimizer on a log Fano cone singularity. https://arxiv.org/abs/2608.23019

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