Kähler–Einstein metrics and volume minimization
By Chi Li, Yuchen Liu. Advances in Mathematics, 2016
66 citations recorded by the source corpus.
We prove that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a K\"{a}hler-Einstein $\mathbb{Q}$-Fano variety $V$, then for any ample Cartier divisor $H=-r^{-1} K_V$ with $r\in \mathbb{Q}_{>0}$, the normalized volume $\widehat{\rm vol}(v)=A_{\mathcal{C}}^n(v)\cdot {\rm vol}(v)$ is globally minimized at the canonical valuation ${\rm ord}_V$ among all real valuations which are centered at the vertex of the affine cone $\mathcal{C}:=C(V,H)$. This is also generalized to the logarithmic and the orbifold setting. As a consequence, we complete the confirmation of a conjecture in [arXiv:1511.08164] on an equivalent characterization of K-semistability for any smooth Fano manifold. We also prove that the valuation associated to the Reeb vector field of a smooth Sasaki-Einstein metric minimizes $\widehat{\rm vol}$ over the corresponding K\"ahler cone. These results strengthen the minimization result of Martelli-Sparks-Yau [Martelli et al 08].
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/4ca8a4a1a4efe3b7ddd4c022cae51cc452602714
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Stability of Valuations: Higher Rational Rank (2017, background): “ons in [Li15a] and the arguments in [LX16], we have the following almost complete picture: 1. n+ 1 = 2, then X∼= {z2 1 + z22z3 + zk 3 = 0} = C2/D k+1where D is the (k+ 1)-th binary dihedral group. By [LL16], the valuation v0 induced by the weight 1,1− 1 k, 2 k is a global minimizer of vol.c 2. n+ 1 ”
- Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, result): “hing 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 observation for us is that the ”
- Existence of valuations with smallest normalized volume (2016, background): “ty and C(V; K V) := Spec( m0H0(V; mK V)) the ane cone over V with cone point 0 2C(V; K V). The blowup of C(V; K V) at 0 has a unique exceptional divisor, which we denote by V~. Theorem 1.3 ([Li15a] [LL16]). Let V be a smooth Fano variety. The following are equivalent: (1)The Fano variety V is K-semi”
- A minimizing valuation is quasi-monomial (2019, background): “aluation obtained by blowing up the vertex. Then v∗ is a minimizer of volc (X,∆),x if and only if (V,∆V ) is K-semistable. Proof. This was proved in [LX16, Theorem 4.5], after the works in [Li17] and [LL19]. 2.5.2. Invariance of local volumes. The following theorem is a local version of [HMX13, Theo”