Szego kernels and a theorem of Tian

By Steve Zelditch. 2000

639 citations recorded by the source corpus.

We give a simple proof of Tian's theorem that the Kodaira embeddings associated to a positive line bundle over a compact complex manifold are asymptotically isometric. The proof is based on the diagonal asymptotics of the Szego kernel (i.e. the orthogonal projection onto holomorphic sections). In deriving these asymptotics we use the Boutet de Monvel-Sjostrand parametrix for the Szego kernel.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bb930b82dc115646db969c76efb0d72e13e20d41

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics (2008, background): “potentials) as these metrics are dense in the smooth topology (see [35], [32], [44], [6] ).”
Berezin-Toeplitz Operators, a Semi-Classical Approach (2003, result): “The existence of this asymptotic expansion was proved by Zelditch [17], by using the result of Boutet de Monvel and Sjöstrand.”
Growth of balls of holomorphic sections and energy at equilibrium (2008, background): “When φ has strictly positive curvature, the celebrated Bouche-Catlin-Tian-Zelditch theorem ([Bo90, Cat99, Tia90, Zel98]) asserts that β ( µ, kφ ) admits a full asymptotic expansion in the space of smooth volume forms, with the probability measure µ eq ( X, φ ) as the dominant term.”
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 .”
The log term of Szego Kernel (2004, methodology): “also [26]) states that there exists a symbol s ∈ Sn(X × X × R) of the type”
Projections in several complex variables (2008, methodology): “Zelditch (1998, [34]) adapted a result of Boutet de Monvel-Sjöstrand for the asymptotics of the Szegö kernel on a strictly pseudoconvex boundary to establish the complete asymptotic expansion of the Bergman kernel for a high power of a holomorphic line bundle with positive curvature.”
Lower bounds on the Calabi functional (2005, background): “The essential ingredient in our proof is the asymptotic expansion for the “density of states” function due to Yau, Tian, Zelditch,Liu, Catlin and Ruan [20], [24],[13], [4], [18].”
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 [13] (see also [34, 49, 51]).”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, result): “1 for a precise statement) reminiscent to the celebrated results by Catlin [18], Ruan [74], Tian [86] and Zelditch [89] in the case v = 1.”