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Seshadri constants and K-stability of Fano manifolds. (English) Zbl 1511.14068

The authors use the techniques they previously developed in [H. Abban and Z. Zhuang, Forum Math. Pi 10, Paper No. e15, 43 p. (2022; Zbl 1499.14066)] to prove K-stability of most families of Picard rank \(1\) Fano varieties in dimension three and of many Fano hypersurfaces in higher dimensions.
In [H. Abban and Z. Zhuang, Forum Math. Pi 10, Paper No. e15, 43 p. (2022; Zbl 1499.14066)] the authors introduced a way to compute local \(\delta\)-invariants. One disadvantage of their method was that it was very computation-intensive. In this paper the authors introduce a way to bound \(\delta\)-invariants using Seshadri constants. This allows them to consider multiple families at once in an efficient manner. Following [H. Abban and Z. Zhuang, Forum Math. Pi 10, Paper No. e15, 43 p. (2022; Zbl 1499.14066)] the authors seek to reduce computations of \(\delta\)-invariants to a surface-case, where the problem is easier. In this paper, they consider complete intersection surface \(S\) passing through a point \(x\) to bound \(\delta_x\) using Seshadri-constants on \(S\). The result is a unified proof of K-stability for \(12\) families of Fano manifolds on Picard rank \(1\) and the proof of K-stability of Fano hypersurfaces of index at least \(3\) of sufficiently high dimension (depending on index).

MSC:

14J45 Fano varieties
14J70 Hypersurfaces and algebraic geometry

Citations:

Zbl 1499.14066
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References:

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