Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities

By Xiuxiong Chen, Simon Donaldson, Song Sun. 2012

436 citations recorded by the source corpus.

This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8abfc10e48181d417332feb79cc16182ccb3c92e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Sasaki–Einstein metrics and K–stability (2015, methodology): “The properties of the twisted Ding functional in this setting follow calculations analogous to those in the compact Kähler case that was studied in [30] (see Ding-Tian [36], Berndtsson [10], Chen-Donaldson-Sun [24] for earlier work), using some additional identities that are valid on cones.”
Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II (2015, background): “In terms of the notion of K-stability for polarized affine varieties formulated in [11], and suppose the results of [2, 7, 8, 9] extend to this case, we can say W is K-semistable and C(Y ) is K-stable.”
Special test configurations and $K$-stability of Fano varieties (2011, positive): “1In the recent remarkable works, Tian and Chen-Donaldson-Sun announced proofs of this conjecture [Tiaa], [CDSa], [CDSb], [CDSc].”
On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, methodology): “p 1@x@ 1/ D e 1 : Now one can construct the special test configuration as in [11, 53].”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, positive): “As for the converse it was ﬁnally settled very recently in the deep works by Chen-Donaldson-Sun [ CDS15a, CDS15b, CDS15c ] and Tian [ Tia15 ], independently, in the case when X is smooth.”
On the moduli of Kähler-Einstein Fano manifolds (2012, methodology): “on a Fano manifold is proved to be equivalent to the Kstability. The proof of stability of KE Fano manifolds is established by [Mab08], [Mab09], [Ber12] and for the converse direction, please consult [CDS12a], [CDS12b], [CDS13], [Tia12b] which discuss after the Donaldson’s proposal of a new continui”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, methodology): “1 provides another result which would be required in adapting the methods of [9, 7, 8, 51] to the Aubin continuity method.”
On the constant scalar curvature Kähler metrics (I)—A priori estimates (2017, methodology): “One conspicuous and memorable feature of CDS’s proof is the heavy use of CheegerColding theory on manifold with Ricci curvature bounded from below.”