Kähler Metrics with Cone Singularities Along a Divisor
By Simon Donaldson. 2011
255 citations recorded by the source corpus.
We develop some analytical foundations for the study of Kahler metrics with cone singularities in codimension one. The main result is an analogue of the Schauder theory in this setting. In the later parts of the paper we discuss connections with the existence problem for Kahler–Einstein metrics,in the positive case.
Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics
View Paper: https://www.semanticscholar.org/paper/27594deaeb394f4d422e345d03d926866f853a40
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Geometry and Topology
In Nuclear and High Energy Physics
Citations in the corpus
- 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].”
- Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “There has been much recent work on the study of Kähler-Einstein metrics with cone angles along a divisor [14].”
- Kähler–Einstein metrics and volume minimization (2016, methodology): “For special test configurations, the (log-)CM weight and (log-)Ding invariant both coincide with the (log-)Futaki invariant (see [25,22]): log-CM(X ,D;L) = −δ n+1L̄n+1 (n + 1)(−KX −D)n = log-Ding(X ,D;L).”
- Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, background): “Similarly, if one also fixes a Q−divisor ∆ on X, with normal crossings, one can more generally define the log K-polystability of (X,L) wrt ∆ as before [31, 40, 45], but phrased in terms of the corresponding log Donaldson-Futaki invariants defined by”
- Kahler-Einstein metrics with edge singularities (2011, methodology): “Recently, Donaldson [24] proposed using these metrics in a similar way to construct smooth Kähler–Einstein metrics on Fano manifolds by deforming the cone angle of Kähler–Einstein metrics of positive curvature, and more generally to relate this approach to the much-studied obstructions to existence ”
- A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem (2011, methodology): “ A generalized Bando-Mabuchi theorem. As pointed out to me by Robert Berman, Theorems 6.1 and 6.3 lead to versions of the Bando-Mabuchi theorem for ’twisted Kähler-Einstein equations’, [20], [3], and [14]. Let θbe a positive (1,1)-current that can be written θ= i∂∂ψ¯ with ψa metric on a R-line bundl”
- Kähler–Ricci flow, Kähler–Einstein metric, and K–stability (2015, methodology): “About the second ingredient, very recently Datar and Székelyhidi [11] have adapted the results of [4] to the Aubin-Yau continuity path, which gives a new proof of the theorem of Chen-Donaldson-Sun.”
- Testing log K-stability by blowing up formalism (2011, unknown): “ (usual) DonaldsonFutaki invariant of (X,L), DF(X,L) is defined as 2(b 0a 1 −b 1a 0). Here we deliberately add a constant 2 so that the formula in the following definitions is simplified. Definition 3.2 ([Don11]). The logarithmic Donaldson-Futaki invariant of a log test configuration ((X,D),L) with cone ”
- Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, methodology): “X; ̨D ; KX=C/ as the log-Futaki-invariant (see [23] and also [27]) on the central fibre as follows: Fut.”
- K‐Stability and Kähler‐Einstein Metrics (2012, unknown): “As observed in [22, 31], adapting arguments from [25], we can define the Futaki invariant fM1;.”