Uniqueness of tangent cone of Kähler-Einstein metrics on singular varieties with crepant singularities. (English) Zbl 1539.53084
This paper is about proving the uniqueness of the local tangent cone for Kähler-Einstein metrics on polarized varieties with crepant singularities. The bulk of the paper focuses on the Calabi-Yau case, the canonically polarized case is sketched at the end.
The basic idea is very similar to S. Donaldson and S. Sun’s fundamental work [Acta Math. 213, No. 1, 63–106 (2014; Zbl 1318.53037); J. Differ. Geom. 107, No. 2, 327–371 (2017; Zbl 1388.53074)] on the local tangent cones for Gromov-Hausdorff limits of polarized Kähler-Einstein manifolds. One uses the Hörmander technique to produce many holomorphic sections, uses the \(3\)-circle type inequality to define a degree filtration on the local ring, and then the rigidity of the holomorphic spectrum to show the uniqueness of the tangent cone. The main difference with [{S. Donaldson} and {S. Sun}, loc. cit.] is that here the sequence of smooth metric approximations coming from the crepant resolution is not uniformly polarized by a line bundle. Instead the authors use the more technical variants of the Hörmander technique from [G. Liu and G. Székelyhidi, Geom. Funct. Anal. 32, No. 2, 236–279 (2022; Zbl 1493.53097)] to produce holomorphic sections directly on the singular variety itself, rather than on its crepant resolution.
In the case of non-zero Einstein constant, an extra difficulty is that the crepant resolution may not have a Kähler-Einstein metric, but only a Kähler metric with Ricci lower bound. Thus one cannot appeal to the Cheeger-Naber regularity theory for Einstein manifolds, and instead the authors appeal to the work from [G. Tian and B. Wang, J. Am. Math. Soc. 28, No. 4, 1169–1209 (2015; Zbl 1320.53052)] on almost Kähler-Einstein manifolds, which relies on some results about the Kähler-Ricci flow.
The basic idea is very similar to S. Donaldson and S. Sun’s fundamental work [Acta Math. 213, No. 1, 63–106 (2014; Zbl 1318.53037); J. Differ. Geom. 107, No. 2, 327–371 (2017; Zbl 1388.53074)] on the local tangent cones for Gromov-Hausdorff limits of polarized Kähler-Einstein manifolds. One uses the Hörmander technique to produce many holomorphic sections, uses the \(3\)-circle type inequality to define a degree filtration on the local ring, and then the rigidity of the holomorphic spectrum to show the uniqueness of the tangent cone. The main difference with [{S. Donaldson} and {S. Sun}, loc. cit.] is that here the sequence of smooth metric approximations coming from the crepant resolution is not uniformly polarized by a line bundle. Instead the authors use the more technical variants of the Hörmander technique from [G. Liu and G. Székelyhidi, Geom. Funct. Anal. 32, No. 2, 236–279 (2022; Zbl 1493.53097)] to produce holomorphic sections directly on the singular variety itself, rather than on its crepant resolution.
In the case of non-zero Einstein constant, an extra difficulty is that the crepant resolution may not have a Kähler-Einstein metric, but only a Kähler metric with Ricci lower bound. Thus one cannot appeal to the Cheeger-Naber regularity theory for Einstein manifolds, and instead the authors appeal to the work from [G. Tian and B. Wang, J. Am. Math. Soc. 28, No. 4, 1169–1209 (2015; Zbl 1320.53052)] on almost Kähler-Einstein manifolds, which relies on some results about the Kähler-Ricci flow.
Reviewer: Yang Li (Cambridge, MA)
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53E30 | Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows) |
| 32Q20 | Kähler-Einstein manifolds |
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
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