Scalar curvature, moment maps, and the Deligne pairing

By Duong H. Phong, Jacob Sturm. 2002

60 citations recorded by the source corpus.

In this paper, we improve the bound for the moment map derivative proved by Donaldson in his recent proof of the Hilbert-Mumford stability of complex manifolds with constant scalar curvature. The proof depends on the identification of Donaldson's symplectic form with the curvature of a certain Deligne pairing.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08a45245638f57e6a010e08c12983124599936af

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016, background): “+ c) = E(˚) + cand R(˚+ c) = R(˚) Sc , respectively. 1.5. Energy functionals as Deligne pairings. The functionals above can be expressed using Deligne pairings, an observation going back at least to [PS04]. Note that any metric ˚2Hinduces a smooth metric 1 2 logMA(˚) on K X. The following identities”
The Monge-Ampère operator and geodesics in the space of Kähler potentials (2005, unknown): “Then φ̈ − |∂φ̇|(2)ωφ = |πNV |2 ≥ 0, where πNV is the normal component of the holomorphic vector field V on CPk generated by σ [22], and so ∂L ∂t ≥ |DL|(2)φ.”
An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, I (2004, methodology): “Such a difficulty comes up when we use the estimate of Phong and Sturm [21].”
Lectures on Stability and Constant Scalar Curvature (2008, methodology): “Moreover, Deligne’s formula for the curvature of the Deligne pairing, which plays a key role in the second step below, has an elementary proof (which is given in [106]) and one does not need to invoke Deligne’s theory [38] to implement the method (thus simplifying the presentation in [107]).”
Lower bounds on the Calabi functional (2005, methodology): “The author first saw that this could be done using the point of view developed in the work of Zhang [25], Phong and Sturm [ 15 ], [16 ]i nvolving the “Chow norm” and an action on Chow vectors.”
Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics (2014, background): “→∞ Tk,A= TA In particular, this shows that TA ≥ 0 and as explained above this concludes the proof of the theorem. In the light of the relation between the Mabuchi functional and Deligne pairings (see [34]) the previous theorem also admits the following geometric formulation. Considera holomorphicsub”
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs (2013, methodology): “Fixing a smooth reference metric φ 0 on L one can use the latter transformation formula to deﬁne the metric (cid:104) φ (cid:105) as long as φ has ﬁnite energy.”
K-stability of constant scalar curvature polarization (2008, methodology): “The purpose of this section is to show the following (see Phong and Sturm [20], eqn 7.29, for the leading term; see also [5], p.464–467): Lemma 4.8.”