Kähler-Einstein metrics and volume minimization

Abstract

We prove that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a Kähler-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 $\widehat{\mathrm{vol}}(v)=A_{\mathcal{C}}^n(v)\cdot \mathrm{vol}(v)$ is globally minimized at the canonical valuation ${\mathrm{ord}}_V$ 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 $\widehat{\mathrm{vol}}$ over the corresponding Kähler cone. These results strengthen the minimization result of Martelli-Sparks-Yau [Martelli et al 08].

43 pages. A section on minimizers from smooth Sasaki-Einstein metrics is added

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Cited by in corpus (14)

• Existence of valuations with smallest normalized volume
• A non-Archimedean approach to K-stability
• Stability of Valuations and Kollár Components
• Openness of uniform K-stability in families of $\mathbb{Q}$-Fano varieties
• A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties
• Minimizing normalized volumes of valuations
• Calabi-Yau manifolds with isolated conical singularities
• Stability of Valuations: Higher Rational Rank
• On the stability of extensions of tangent sheaves on Kähler-Einstein Fano / Calabi-Yau pairs
• Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds
• A Note on Equivariant K-stability
• Interaction Between Singularity Theory and the Minimal Model Program
• On the semi-continuity problem of normalized volumes of singularities
• Volume minimization and obstructions to solving some problems in Kähler geometry