The volume of singular Kähler–Einstein Fano varieties
By Yuchen Liu. Compositio Mathematica, 2016
69 citations recorded by the source corpus.
We show that the anti-canonical volume of an $n$ -dimensional Kähler–Einstein $\mathbb{Q}$ -Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/50a91080d97b0ae493207e94ffea4a5d0846806b
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- A Guided Tour to Normalized Volume (2018, methodology): “Theorem 2.6 ([Liu18a]).”
- Kähler–Einstein metrics and volume minimization (2016, methodology): “We prove this estimate using the method in [26,47].”
- Uniqueness of the minimizer of the normalized volume function (2020, background): “It is proved in [Liu18, Theorem 27] that (2.1) v̂ol(x,X,∆) = inf a• lct(X,∆; a•) n ·mult(a•), where the infimum runs through all graded ideal sequences a• of mx-primary ideals, and lct(X,∆; a•) n ·mult(a•) is set to be +∞ if lct(X,∆; a•) = +∞.”
- Uniqueness of K-polystable degenerations of Fano varieties (2018, background): “er k, set ak,• := a•(ordE k ) and ck:= lct(Z,Γ;ak,•). Note that ck≤ AZ,Γ(Ek) by (3). This implies f(0) ≤ c n+1 k ·mult(ak,•) ≤ f 1 k = AZ,Γ(Ek) ·mult(ak,•), where the first inequality follows from [Liu18, 7] and the assumption that (X,∆) is K-semistable. Therefore, 1 1+O(1/k2) 1 n+1 ≤ f(0) f(1/k”
- Stability of valuations and Kollár components (2016, methodology): “The proof of (i) is along the lines of [Liu18, proof of Lemma 33].”
- K-stability of Fano varieties: an algebro-geometric approach (2020, methodology): “First, in the course of generalizing [Fuj18] (see Example 3.9), in [Liu18, Theorem 1], Yuchen Liu found that there is an inequality connecting the local volume and global volume for a K-semistable Fano variety.”
- Existence of valuations with smallest normalized volume (2016, background): “of ideals a = fa mg m2N such that a ma n a m+n for all m;n2N. The following proposition relates minimizing the normalized volume function to minimizing the normalized multiplicity. Proposition4.3 ( [Liu16]). If x2Xis a closed point on a klt variety, then (1.1) inf v2Val X;x vol(cv) = inf am x-primar”
- Thresholds, valuations, and K-stability (2017, methodology): “The invariants S(ordE) and T (ordE) play an important role in the work of K. Fujita [Fuj19a], C. Li [Li17], and Y. Liu [Liu18], see Remark 3.10.”
- Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, methodology): “n-isolated singularity of Z, then again by the above slice argument we know a generic nearby singularity must locally be given by C× C2/Z2. Now we can apply the rigidity case of the Liu’s estimate in [41] Theorem 3, to conclude that Z = CP3/Z2, with [x : y : z : t] ≃ [x : y : −z : −t]. Moreover CP3/”
- Algebraicity of the metric tangent cones and equivariant K-stability (2018, methodology): “on that the minimizer is a higher rank quasi-monomial valuation. We will need a relation between the normalized volume and the normalized multiplicity of a graded sequence of ideals. Proposition 2.4 ([Liu18]). If x ∈(X,D) is an n-dimensional klt singularity, then we have vol(x,X,D) = inf b• mult(b•)”
- K-STABILITY OF FANO MANIFOLDS WITH NOT SMALL ALPHA INVARIANTS (2016, methodology): “Hence we have the equality ε(F ) = τ(F ) by [Liu16, Lemma 10].”