Scalar curvature and projective embeddings, II

By Simon Donaldson. 2004

166 citations recorded by the source corpus.

We prove that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H. Luo. This gives, as a Corollary, the uniqueness of constant scalar curvature Kahler metrics in a given rational cohomologyclass. The proof uses results in the literature on the asymptotics of the Bergman kernel. The arguments are presented in a general framework involving moment maps for two different group actions.

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

View Paper: https://www.semanticscholar.org/paper/e389fdd8eaf9d90e54f42ddd5413228fc885601a

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Citations in the corpus

Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, background): “Proof of Theorem 1 Here outline the proof of Theorem 1 from the Introduction, which follows from a straightforward extension of the arguments of [35, 55, 61, 75] that has been given in the special cases (i), (ii), (iv).”
Hamiltonian 2-forms in Kähler geometry, III extremal metrics and stability (2005, result): “Building on earlier work by Bando–Mabuchi [6], Chen [9], Donaldson [16] and others, Chen and Tian have established the following uniqueness result and necessary condition for existence of an extremal Kähler metric.”
Geometry of Kähler metrics and foliations by holomorphic discs (2005, background): “2 After we finished the first draft of this paper, we learned that S. K. Donaldson proved this theorem in the case of projective manifolds without holomorphic vector fields [ 13 ].”
Existence of approximate Hermitian-Einstein structures on semi-stable bundles (2010, background): “Donaldson [12] has shown that Chow-Mumford stability implies the lower boundedness of the K-energy.”
Calabi flow and projective embeddings (2008, methodology): “This time, the Kempf–Ness function is a normalised version of what is called the “F0-functional” by some authors (it is the function denoted ˜ Z in [ 15 ]).”
Space of Kähler metrics III – On the lower bound of the Calabi energy and geodesic distance (2006, background): “After we announced our work [12], S. K. Donaldson[ 17 ] proved a similar lower bound in the algebraic settings.”
The Monge-Ampère operator and geodesics in the space of Kähler potentials (2005, background): “D G ] 1 M ay 2 00 5 THE MONGE-AMPÈRE OPERATOR AND GEODESICS IN THE SPACE OF KÄHLER POTENTIALS1 D.H. Phong∗ and Jacob Sturm† ∗ Department of Mathematics Columbia University, New York, NY 10027 † Department of Mathematics Rutgers University, Newark, NJ 07102”
Growth of balls of holomorphic sections and energy at equilibrium (2008, methodology): “It also corresponds to the functional I in [Che00, Don05a ] and to minus F0 !0 on p.59 of Tian’s book [TiaBook], where it is proved that � 7→ E(�) is non-decreasing and concave on smooth psh weights.”
A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry (2013, unknown): “by a variant, introduced in [7], of Donaldson’s L-functional, [16].”
Positivity of direct image bundles and convexity on the space of Kähler metrics. (2006, methodology): “In section 5 we relate our results to the study of variations o f Kähler metrics onZ, cf [14], [1], [7], [8] [15] [5].”