Minimizing normalized volumes of valuations

Abstract

For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, we introduce a normalized volume function $\widehat{\mathrm{vol}}$ that is defined on the space of real valuations centered at $o$ and consider the problem of minimizing $\widehat{\mathrm{vol}}$. We prove that the normalized volume has a uniform positive lower bound by proving an Izumi type estimate for any $\mathbb{Q}$-Gorenstein klt singularity. Furthermore, by proving a properness estimate, we show that the set of real valuations with uniformly bounded normalized volumes is compact, and hence reduce the existence of minimizers for the normalized volume functional $\widehat{\mathrm{vol}}$ to a conjectural lower semicontinuity property. We calculate candidate minimizers in several examples to show that this is an interesting and nontrivial problem. In particular, by using an inequality of de-Fernex-Ein-Mustaţă, we show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of $\widehat{\mathrm{vol}}$ for a smooth point. Finally the relation to Fujita's work on divisorial stability is also pointed out.

27 pages. Cut some calculations and add a second proof of Izumi type estimate by a referee's suggestion

References in corpus (6)

• Existence of valuations with smallest normalized volume
• Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, II
• Stability of Valuations and Kollár Components
• Optimal bounds for the volumes of Kähler-Einstein Fano manifolds
• Kähler-Einstein metrics and volume minimization
• The volume of a set of arcs on a variety

Cited by in corpus (7)

• A non-Archimedean approach to K-stability
• Calabi-Yau manifolds with isolated conical singularities
• On the stability of extensions of tangent sheaves on Kähler-Einstein Fano / Calabi-Yau pairs
• A Note on Equivariant K-stability
• Local models for conical Kähler-Einstein metrics
• Volume minimization and obstructions to solving some problems in Kähler geometry
• Stability of test ideals of divisors with small multiplicity