Minimizing normalized volumes of valuations

Abstract

For any Q-Gorenstein klt singularity (X, o), we introduce a normalized volume function vol that is defined on the space of real valuations centered at o and consider the problem of minimizing vol. We prove that the normalized volume has a uniform positive lower bound by proving an Izumi type estimate for any 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 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-Mustaţȃ, we show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of vol for a smooth point. Finally the relation to Fujita's work on divisorial stability is also pointed out.

Interaction Between Singularity Theory and the Minimal Model Program

“…See [Laz04,9.3.14] for the definition of the log canonical threshold of a klt pair (X, ∆) with respect to an ideal a. Then in [Blu16], using an argument combining estimates on asymptotic invariants and the generic limiting construction, it is show that there always exists a valuation v such that vol(x, X) = vol(v), i.e., the infimum is indeed a minimum, confirming a conjecture in [Li15a]. Therefore the main questions left are two-fold.…”

“…X,x . In [Li15a], it was shown that vol(x, X) > 0. In [Liu16], a different characterisation is given:…”

“…Conjecture 4.4 (Stable Degeneration Conjecture, [Li15a,LX17a]). Given any arbitrary klt singularity x ∈ (X = Spec(R), ∆).…”

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

“…Numerical consequence: Let ν Y denote the volume density of the Calabi-Yau cone metric, i.e., the ratio of the volume of the link by that of S 2n−1 (1). The number 0 < ν Y < 1 is an algebraic invariant of the singularity (Y, o), see [Li18]. The Bishop-Gromov volume monotonicity applied to g RF implies that the volume density of g RF at any point in X is greater (strictly in the non-flat case) than that of the tangent cone at infinity.…”

Volume minimization and obstructions to solving some problems in Kähler geometry

“…Odaka observed the decrease of the Donaldson-Futaki invariant along the minimal model program using the concavity of the volume functional. Li [38], [39] considered normalized volume functional on the space of valuations on Fano manifolds and characterized K-semistabilty in terms of volume minimization. Note that when a Sasakian manifold is the circle bundle of an ample line bundle L over M , then the Reeb vector field defines a valuation of the ring ⊕ ∞ k=0 H 0 (M, L k ).…”