Minimizing normalized volumes of valuations

By Chi Li. 2015

106 citations recorded by the source corpus.

For any $${\mathbb {Q}}$$Q-Gorenstein klt singularity (X, o), we introduce a normalized volume function $${\widehat{\mathrm{vol}}}$$vol^ that is defined on the space of real valuations centered at o and consider the problem of minimizing $${\widehat{\mathrm{vol}}}$$vol^. We prove that the normalized volume has a uniform positive lower bound by proving an Izumi type estimate for any $${\mathbb {Q}}$$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 $${\widehat{\mathrm{vol}}}$$vol^ to a conjectural lower semicontinuity property. For a smooth point, we use an inequality of de-Fernex-Ein-Mustaţă to show that the divisorial valuation associated to the exceptional divisor of the standard blow up is a minimizer of $${\widehat{\mathrm{vol}}}$$vol^. Finally the relation to Fujita’s work on divisorial stability is pointed out.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f4fbdca4f93abc25a611aa67bb97a1bde31d6798

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

K-semistability is equivariant volume minimization (2015, methodology): “In [Li15a], v∗ was shown to minimize v̂ol among valuations of the special form vx determined by weight x ∈ Rn+1>0 .”
A non-Archimedean approach to K-stability (2018, background): “Another way of studying K-stability of Fano varieties was introduced by Chi Li [Li15, Li17] and involves considering the cone over the variety.”
K-stability of Fano varieties: an algebro-geometric approach (2020, methodology): “A striking development is the local K-stability theory, built on Li’s definition of the normalized volume function in [Li18] defined on valuations over a klt singularities, and the study of its minimizer.”
Uniqueness of the minimizer of the normalized volume function (2020, methodology): “The study of v̂olX,∆ is closely related to K-stability of log Fano pairs, guided by the so-called Stable Degeneration Conjecture as formulated in [Li18, Conjecture 7.1] and [LX18, Conjecture 1.2].”
Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, background): “nduced by the order of vanishing νKE, which appeared in the above recalled deﬁnition of weighted tangent cone W, since a GH limit Z is indeed K-polystable by Berman’s result [6]. It is conjectured in [38], and partially veriﬁed [39], that such valuation minimizes the normalized volume. A crucial obs”
ACC for Minimal Log Discrepancies of Exceptional Singularities (2019, unknown): “[Li18] introduced the normalized volumes to study K-stability and the existence of Kähler-Einstein metrics on Fano varieties.”
Thresholds, valuations, and K-stability (2017, methodology): “orem D. Let Lbe a big line bundle. Then we have vol(L) n+ 1 n n (L) nvol(cv): for any valuation von Xcentered at a closed point. Here vol(cv) is the normalized volume of v, introduced by C. Li [Li18]. When X is a Q-Fano variety and L= K X, Theorem D generalizes the volume bounds found in [Fuj18] ”
Uniqueness of K-polystable degenerations of Fano varieties (2018, methodology): “The reverse implication was shown in [Fuj16b, 1.6] and [Li18, 3.7].”
Stability of valuations and Kollár components (2016, methodology): “As a candidate the first named author [Li18] defined the normalized volume function v̂ol(X,D),o on the space of valuations centered at o.”
Existence of valuations with smallest normalized volume (2016, background): “ng Eto x. We denote such a valuation by ord E 2Val X;x. Li introduced the normalized volume function volc X;x: Val X;x! R >0 [f+1g that sends a valuation vto its normalized volume, denoted volc(v) [Li15]. To dene the normalized volume, we recall the following. Given a valuation v2Val X;x, we have”
Algebraicity of the metric tangent cones and equivariant K-stability (2018, methodology): “mediate semistable cone W in Donaldson-Sun’s work (see [DS17]) only depends on the algebraic structure. Our strategy is to systematically use minimizers of the normalized volume functional (deﬁned in [Li18]) to characterize valuations associated to metric tangent cones. Aiming at a vast generalizati”
The volume of singular Kähler–Einstein Fano varieties (2016, methodology): “…AY (u0) +∞, a singular version of [JM13, Corollary 5.11] for klt singularities (such a statement is true thanks to the Izumi inequality for klt singularities [Li15a, Proposition 3.1]) implies that u0 has a unique extension, say û0, as a valuation on Spec ÔY,o that centered at the closed point.”
Openness of K-semistability for Fano varieties (2019, methodology): “[Li17] C. Li, K-semistability is equivariant volume minimization, Duke Math.”
Kähler–Einstein metrics and volume minimization (2016, methodology): “For all other cases of (n, k), by the calculations in [42] it was conjectured that v̂ol is minimized at the valuation associated to the weight ( 1, · · · , 1, n−2 n−1 ) .”