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arXiv · 2607.27032

The finite degree formula for normalized volumes

Zhiyu Liu
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Abstract

Let $f\colon \big(x\in (X, \Delta_X)\big)\to \big(y\in (Y, \Delta_Y)\big)$ be a finite surjective morphism between klt singularities such that $K_X+\Delta_X=f^*(K_Y+\Delta_Y)$. We show that the normalized volumes satisfy \[\widehat{\mathrm{vol}}(x, X, \Delta_X)=\mathrm{deg}(f)\cdot \widehat{\mathrm{vol}}(y, Y, \Delta_Y).\] This proves a conjecture in [LLX20, Zhu25, XZ26a].

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Zhiyu Liu. 2026-07-29. The finite degree formula for normalized volumes. https://arxiv.org/abs/2607.27032

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