Greatest lower bounds on the Ricci curvature of Fano manifolds

By G'abor Sz'ekelyhidi. Compositio Mathematica, 2009

104 citations recorded by the source corpus.

Abstract On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric ω∈c1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin’s continuity path for finding Kähler–Einstein metrics. We show that on P2 blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f6105584631c534d092a13ae8bee708e5c87f87b

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, background): “[19], [23], [24]), but for the reader’s convenience, I will sketch the proof of this proposition.”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “In this setting, Székelyhidi [50] defined the greatest lower bound on the Ricci curvature R(X) = {supβ : equation (2) is solvable}.”
Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities (2012, background): “[23] Gábor Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, Com- pos.”
Kähler–Einstein metrics along the smooth continuity method (2015, background): “It follows from the work of the second author [44] that R(M) is in fact independent of the choice of α.”
On the constant scalar curvature Kähler metrics (II)—Existence results (2020, background): “It is not hard to see the relation between the invariant introduced in [57] and the invariant introduced above when restricted to the canonical Kähler class in Fano manifold, where we take [χ] to be the first Chern class in (1.”
A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry (2013, background): “5 lea d to versions of the BandoMabuchi theorem for ’twisted Kähler-Einstein equations’, [29], [3], and [15].”
Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, background): “f. Given the “toric dictionary” above the theorem follows immediately from Theorem 2.18, apart from the global regularity of the solutions, which in turn follows from Theorem 1.5 in [12]. As shown in [52] when Xis any smooth (and not necessary toric) Fano manifold and ∆ = 0 the sup over all r∈[0,1[ ”
Kähler–Einstein metrics and volume minimization (2016, unknown): “X/〈eξ〉 = (S, Δ)), we define the greatest lower bound of the Ricci curvature (see [56,55]: β(S,Δ) = sup{t > 0;∃ an orbifold Kähler metric ω with [ω] = 2πc1(−K(S,Δ)) and Ric(ω) ≥ tω}.”
Sasaki–Einstein metrics and K–stability (2015, unknown): “On the other hand it is easy to show in an identical way to the argument in [74] that if (X, ξ, (1 − t)ατ ) admits a solution of the twisted equation, then so does (X, ξ, (1 − t)α̃τ ) for any α̃τ in the same transverse cohomology class as ατ .”
The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality (2012, methodology): “J. 137 (2007), no. 1, 159–184 [39] Sun, S. Note on K-stability of pairs, arXiv:1108.4603 [40] Szekelyhidi, G. Greatest lower bounds on the Ricci curvature of Fano manifolds, Compositio Math.”