Kahler geometry on toric manifolds, and some other manifolds with large symmetry

By Simon Donaldson. 2008

172 citations recorded by the source corpus.

This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2d2fe8c4a1c94de81c1bacf259122b87fd738a38

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Stability, birational transformations and the Kahler-Einstein problem (2010, unknown): “If X is a suitable small deformation of the Mukai-Umemura manifold X0, then Tian showed in [14] that X does not admit a Kahler-Einstein metric, but since X0 has a Kahler-Einstein metric (as explained in [6]), there is a sequence ωi on X satisfying the condition above.”
Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, methodology): “Donaldson [22] proved the existence of Kähler–Einstein metric on X0 using Tian’s ̨-invariant.”
Compact moduli space of Kahler-Einstein Fano varieties (2014, background): “s, is proved 2who also (inspiringly) showed that the degenerations are not GIT (semi)stable (cf., also recent reﬁnement [WX]) K-MODULI OF FANO VARIETIES 3 in [OSS]. This extends the picture of [Tia], [Don1] for the commonly studied “Mukai-Umemura 3-fold” case, and the general result by Szekelyhidi [”
Moment Measures (2013, methodology): “For instance, see Bunch and Donaldson [5] or Doran, Headrick, Herzog, Kantor and Wiseman [11] for a numerical approximation of ψ in the case where μ is the uniform probability measure on a hexagon centered at the origin in R2.”
Greatest lower bounds on Ricci curvature for toric Fano manifolds (2009, background): “denote by {xti}, and y∞ ∈ ∂△, such that lim ti→R(X△) D˜u0(xti ) = y∞ (10) This follows easily from the properness of ˜u0 and compactness of △. We now use the key relation (See [7] Lemma 3.3, and also [1] page 29) 0 = Z Rn Dw(x)e−wdx = Z Rn ((1 −t)Du˜0 +tDu)e−wdx 5 Since Z Rn Due−wdx = Z Rn Dudet(uij”