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 $\widehat{\rm vol}(v)=A_{\mathcal{C}}^n(v)\cdot {\rm vol}(v)$ 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 Kähler cone. These results strengthen the minimization result of Martelli-Sparks-Yau [Martelli et al 08].
- Publication:
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arXiv e-prints
- Pub Date:
- February 2016
- DOI:
- 10.48550/arXiv.1602.05094
- arXiv:
- arXiv:1602.05094
- Bibcode:
- 2016arXiv160205094L
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Differential Geometry
- E-Print:
- 43 pages. A section on minimizers from smooth Sasaki-Einstein metrics is added