The finite degree formula for normalized volumes

Zhiyu Liu

algebraic geometry

The finite degree formula for normalized volumes

arXiv:2607.27032

summary

The paper proves that for a finite surjective morphism between klt singularities preserving the log canonical divisor, the normalized volume at the source equals the degree of the map times the normalized volume at the target, confirming a recent conjecture.

Abstract

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

20 pages. Comments are welcome! v2: references added

Topics & keywords

#normalized volume #klt singularities #finite morphism #birational geometry #singularity invariantsnormalized volumekltfinite surjective morphismdegree formulalog canonical divisorconjecture