On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch

By Zhiqin Lu. 2000

287 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4ca8a55bd51a0a736593962d38bf80656d30b496

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

K-STABILITY OF FANO MANIFOLDS WITH NOT SMALL ALPHA INVARIANTS (2016, methodology): “For the definition of α(X) in this article, we use Demailly’s algebraic interpretation [Dem08] (see also [TY87, Zel98, Lu00]).”
A direct approach to Bergman kernel asymptotics for positive line bundles (2008, background): “The functions bj(x, x) contain interesting geometric information of X with the Kähler metric i∂∂̄φ/2, see Lu [11].”
Lectures on Stability and Constant Scalar Curvature (2008, methodology): “Of course, the underlying motivation from symplectic geometry remains, in particular the key insight of Donaldson, based on Lu’s formula [84], that the constancy of the scalar curvature should be closely related to the constancy of the density of states.”
Stability, energy functionals, and Kahler-Einstein metrics (2002, background): “Donaldson [D2] relies instead on the concept of balanced manifolds and Lu’s recent evaluation [ Lu1 ] of the Tian-Yau-Zelditch expansion for the Bergman kernel on positive line bundles.”
Berezin-Toeplitz Operators, a Semi-Classical Approach (2003, methodology): “In [12], Lu computed S0, S1, S2, S3 and S4 and with his method we can also compute the other coefficients.”
The Monge-Ampère operator and geodesics in the space of Kähler potentials (2005, background): “6) Here the Aj(ω) are smooth functions on X defined locally by ω which can be computed in terms of the curvature of ω by the work of Lu [18].”
On the asymptotic expansion of Bergman kernel (2004, methodology): “3 was essentially obtained in [9,14] by applying the peak section trick, and in [5] and [15] by applying the Boutet de Monvel–Sjöstrand parametrix for the Szegö kernel [4].”
The J-flow and stability (2013, methodology): “The following asymptotic expansion is analogous to the Tian-Zelditch-Lu [20, 26, 12] expansion of the usual Bergman kernel associated to the metric ω. Theorem 6.”
Generalized Bergman kernels on symplectic manifolds (2004, result): “1 forq = 0 and (6) generalize the results of [3,6,9,10] to the symplectic case, and we can view as an extension and refinement of the results of [2], [5, §5] about the density of states function of p,Φ .”
Infinite geodesic rays in the space of Kähler potentials (2002, result): “For a fixed K¨ahler metric this is indeed possible and gives rise to many interesting results ([1 2] , [17], [25]).”
The log term of Szego Kernel (2004, methodology): “In [19], the first author was able to prove that all ak’s are polynomials of the curvature and its derivatives.”
Balance point and Stability of Vector Bundles Over a Projective Manifold (2002, background): “[ Lu ]). From Remark 3.7 we know actually we can get a singular metric on torsion free sheaves as Bando and Siu did in [BS] such that B(x) ≡ Const.”
Space of Ricci Flows I (2009, unknown): “[92],[59]).”
Stability of extremal Kahler manifolds (2004, background): “In particular, by an argument similar to [10], an extremal Kähler metric in a fixed integral Kähler class on a projective algebraic manifold M will be shown to be unique† up to the action of the group Aut0(M).”
Scalar Curvature and Projective Embeddings, I (2001, background): “Now the asymptotic behaviour of the “density of states” function ρk(ω) as k → ∞, with ω fixed, has been studied by Catlin [1], Tian [10], Zelditch [12] and Lu [8].”
Scalar curvature, moment maps, and the Deligne pairing (2002, methodology): “To find the balanced basis, [D] first uses Lu ’s formula [L] for the coefficients in the Tian-Yau-Zelditch asymptotic expansion for the density of states to produce, for each k, an ordered basis s′ = {s′0, ..., s′N } of H 0 (Lk) which is almost balanced in the following sense: PN j=0 |s ′ j| 2”