Optimal bounds for the volumes of Kähler-Einstein Fano manifolds

By Kento Fujita. 2015

120 citations recorded by the source corpus.

abstract:We show that any $n$-dimensional Ding semistable Fano manifold $X$ satisfies that the anti-canonical volume is less than or equal to the value $(n+1)^n$. Moreover, the equality holds if and only if $X$ is isomorphic to the $n$-dimensional projective space. Together with a result of Berman, we get the optimal upper bound for the anti-canonical volumes of $n$-dimensional K\"ahler-Einstein Fano manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c524be6ea09bfb5eb41dd8b664791bcfef9a4efe

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In Applied Mathematics

1. Chi Li
2. Mattias Jonsson
3. Yuchen Liu
4. H. Blum
5. Robert J. Berman
6. S. Boucksom
7. Chenyang Xu
8. Song Sun

In Geometry and Topology

1. Chi Li
2. Mattias Jonsson
3. Yuchen Liu
4. H. Blum
5. S. Boucksom
6. Robert J. Berman
7. Ziquan Zhuang
8. Chenyang Xu

Citations in the corpus

• Kähler–Einstein metrics and volume minimization (2016, methodology): “We prove this estimate using the method in [26,47].”
• Geodesic rays and stability in the cscK problem (2020, unknown): “For more details, see [20, 44, 61].”
• Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, background): “,p) weighted algebraic tangent cone W metric tangent cone C(Y), combined with the improved volume bound for K-stable Fano varieties recently obtained by Liu [41] (after works of Berman [6], K. Fujita [25] and Li [38]) in term of normalized volume of valuations. Another motivation for the writing to ”
• A variational approach to the Yau–Tian–Donaldson conjecture (2015, result): “A 0 is equivalent to K-semistability, cf. [BHJ15a]. The case = 0 of Theorem 2.1 therefore says that K-semistability is also equivalent to Ding semistability, thereby showing that the main result of [Fuj15] is also valid assuming only Xto be K-semistable. 3. From uniform Ding stability to coercivity ”
• Thresholds, valuations, and K-stability (2017, background): “uation von Xcentered at a closed point. Here vol(cv) is the normalized volume of v, introduced by C. Li [Li18]. When X is a Q-Fano variety and L= K X, Theorem D generalizes the volume bounds found in [Fuj18] and [Liu18], in which Xis assumed K-semistable, so that (L) 1. These volume bounds were expl”