Stability, energy functionals, and Kahler-Einstein metrics

By Duong H. Phong, Jacob Sturm. 2002

91 citations recorded by the source corpus.

An explicit seminorm $||f||_{#}$ on the vector space of Chow vectors of projective varieties is introduced, and shown to be a generalized Mabuchi energy functional for Chow varieties. The singularities of the Chow varieties give rise to currents supported on their singular loci, while the regular parts are shown to reproduce the Mabuchi energy functional of the corresponding projective variety. Thus the boundedness from below of the Mabuchi functional, and hence the existence of K\"ahler-Einstein metrics, is related to the behavior of the current $[Y_s]$ and the seminorm $||f||_{#}$ along the orbits of $SL(N+1,{\bf C})$.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/af687a0b1bb25fc76b8ea869bb6200c87aa416f9

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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].”
Lectures on Stability and Constant Scalar Curvature (2008, methodology): “A different and perhaps simpler proof based on comparing the derivatives of both sides along one-parameter subgroups can be found in [106].”
Calabi flow and projective embeddings (2008, background): “Suppose that Aut(X,L) is discrete and that the class c1(L) contains a metric ω of constant scalar curvature.”
Scalar curvature and projective embeddings, II (2004, background): “On the other hand, Phong and Sturm [14], [15] have sharpened some of the arguments in [6].”
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] involving the “Chow norm” and an action on Chow vectors.”
Scalar curvature, moment maps, and the Deligne pairing (2002, background): “),σZ(ξ)) = ΩM(σZ(ξ),σZ(ξ)) by Theorem 1, and this last 9 expression is given by the right hand side of (5.5) in view of Deligne’s formula (2.9). (Alternatively, (5.5) has also been proved directly in [PS], formula (3.5)). Consider now the exact sequence of holomorphic vector bundles: 0 → TX→ ι∗TPN →”
Test configurations for K-stability and geodesic rays (2006, background): “Let V be a finite dimensional vector space, Z ⊆ P(V ) a smooth subvariety, and B ∈ gl(V ).”
The Monge-Ampère operator and geodesics in the space of Kähler potentials (2005, background): “6) The asymptotic behavior of F 0 ωφ0 along geodesics in Hk is known to be closely related to Chow-Mumford stability (see [31], and also [20], [21],[23]).”