The finite degree formula for normalized volumes

Abstract

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

arXiv comment20 pages. Comments are welcome! v2: references added