Kähler-Einstein metrics and volume minimization

2016/02/16 by Chi Li, Yuchen Liu, Li, Chi +1 · 3 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1602.05094

openalex publication_date 2016/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if a ℚ-Fano variety V specially degenerates to a Kähler-Einstein ℚ-Fano variety V, then for any ample Cartier divisor H=-r^-1 K_V with r∈ ℚ_>0, the normalized volume \widehat\rm vol(v)=A_Cⁿ(v)⋅ \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 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 Kähler cone. These results strengthen the minimization result of Martelli-Sparks-Yau [Martelli et al 08].

Citations

• A refinement of Izumi's Theorem
• Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds
• Testing log K-stability by blowing up formalism
• Connecting toric manifolds by conical Kahler-Einstein metrics
• A generalization of Ross-Thomas' slope theory
• Bergman kernels and the pseudoeffectivity of relative canonical bundles
• Calabi-Yau manifolds with isolated conical singularities
• Multiplicities Associated to Graded Families of Ideals
• Toric Kähler-Einstein metrics and convex compact polytopes
• Conical Kahler-Einstein metric revisited

Cited by

• Optimal bounds for local volumes of threefold singularities
• On the volume of K-semistable Fano manifolds
• Interaction Between Singularity Theory and the Minimal Model Program

Related

• Optimal bounds for the volumes of Kähler-Einstein Fano manifolds
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties
• Volume minimization and obstructions to solving some problems in Kähler geometry
• Kähler-Einstein metrics on Q-smoothable Fano varieties, their moduli and some applications