Scalar Curvature and Stability of Toric Varieties

By Simon Donaldson. 2002

923 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e4fd2e37d9efb1c520657ebf8fd579f8025e8163

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On stability and the convergence of the Kähler-Ricci flow (2006, background): “Several specific notions of stability have been by now proposed in the literature [13] [14] [24] [27] [29] [33].”
On the constant scalar curvature Kähler metrics, general automorphism group (2018, background): “Alternatively, Donaldson [27] already proved the existence of weak minimizers of K-energy among toric invariant metrics (c.”
K-stability of constant scalar curvature Kähler manifolds (2008, methodology): “In this paper we are concerned with Donaldson’s algebraic K-stability [6].”
The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality (2012, background): “Another importance of the conjecture lies in the fact that it supplies a new approach to the Yau’s conjecture [53] on the equivalence of existence of Kähler-Einstein metric for Fano manifolds and certain algebro-geometric stability condition, which is refined and extended by Tian [44] and Donaldson”
DELIGNE PAIRINGS AND THE KNUDSEN-MUMFORD EXPANSION (2006, neutral): “Then h0e −φ is a metric on L0 and 〈h0e −φ, ..., hn〉 = 〈h0, ..., hn〉 · e −ψ (2.5) where ψ : B → C is the function ψ = ∫ X/B φ · ω1 ∧ · · · ∧ ωn (2.6) and ωj = − √ −1 2π ∂∂̄ log hj is the curvature of hj .”
LOG CANONICAL THRESHOLDS OF SMOOTH FANO THREEFOLDS (2008, background): “For example, if the variety X is smooth, then it does not admit a Kähler–Einstein metric if • either the group Aut(X) is not reductive (see [122]), • or the tangent bundle of X is not polystable with respect to −KX (see [114]), • or the Futaki character of holomorphic vector fields on X does not van”
Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, background): “As conjectured by Donaldson and shown in [7] the corresponding iteration Hk,m on Hk converges, in the double scaling limit where m/k → t, to the (normalized) Kähler-Ricci flow on H(−KX).”
Asymptotically conical Calabi-Yau manifolds, III (2012, background): “(i) There exists a test configuration [7, 28] π : (Y,L) → C with general fibre (Y,L) such that • there exists a homeomorphism F from Y0 onto the central fibre Y0 such that if v denotes the vertex of the projective cone Y0, then F |Y0\{v} is biholomorphic onto its image, and • L is the Q-line bundle ”
Regularity of weak minimizers of the K-energy and applications to properness and K-stability (2016, methodology): “f L contains a Ka¨hler metric with constant scalar curvature if and only if (X,L) is stable in an appropriate sense, inspired by Geometric Invariant Theory. In the formulation introduced by Donaldson [Do] the stability in question was formulated as K-polystability, but in view of an example in [ACGT”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, methodology): “We prove our result in the general setting of relative Kähler test con gurations (introduced by Sjöström Dyrefelt [40] and Dervan–Ross [28] independently) which contains the usual projective test con gurations of Donaldson [29] as a special case.”