Bergman kernels on degenerations
type: Item
instance of: scholarly article
title: Bergman kernels on degenerations
author: Linsheng Wang
author: Q5887732
published in: Mathematische Zeitschrift
publication date: 2024-10-23T00:00:00Z
review text: Bergman kernels on polarized manifolds play a crucial role in Kähler geometry as a local version of Hilbert polynomials. \textit{G. Tian} [J. Differ. Geom. 32, No. 1, 99--130 (1990; Zbl 0706.53036)] utilized his peak section technique to establish that for a line bundle \(L\) the Bergman metrics corresponding to \(L^{m}\) approach the original polarized metric as \(m\) tends to infinity in the \(C^{2}\)-topology. Subsequent to Tian's work, numerous papers have investigated the asymptotic behaviour of Bergman kernels and Bergman metrics on a single polarized manifold. Partial \(C^{0}\)-estimates are also an important problem in the study of Bergman kernels, which is to determine whether there exists a uniform positive lower bound for Bergman kernels on a specific class of polarized Kähler manifolds.\N\NThis paper explores Bergman kernels on a flat family of complex spaces, defining fiberwise Bergman kernels for flat families over Riemann surfaces, which extends the classical Bergman kernel defined on the reduced fibers. The authors establish the continuity of the fiberwise Bergman kernel and provide a result on uniform convergence for Fubini-Study currents. Finally, the authors show that the fiberwise Bergman kernel on test configurations exhibits continuity, and the Fubini-Study currents converge uniformly. The methodology involves sophisticated techniques from algebraic geometry and complex analysis, making significant contributions to our understanding of the asymptotic behaviour of Bergman kernels in degenerate settings.
reviewed by: Guokuan Shao
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 7930939
zbMATH Keywords: Bergman kernels
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: flat families
MaRDI profile type: Publication
cites work: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant
cites work: On regularization of plurisubharmonic functions on manifolds
cites work: Uniform \(K\)-stability, Duistermaat-Heckman measures and singularities of pairs
cites work: Q4263085
cites work: Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \boldmath2𝜋 and completion of the main proof
cites work: Extension of plurisubharmonic functions with growth control
cites work: Equidistribution for sequences of line bundles on normal Kähler spaces
cites work: On the asymptotic expansion of Bergman kernel
cites work: Numerical characterization of the Kähler cone of a compact Kähler manifold
cites work: Scalar curvature and projective embeddings. I
cites work: Scalar curvature and stability of toric varieties.
cites work: Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry
cites work: Q3322387
cites work: Q4143433
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. II
cites work: Q3215513
cites work: Bergman kernel along the Kähler-Ricci flow and Tian's conjecture
cites work: Ein einfacher Beweis des Kohärenzsatzes von Grauert. (A simple proof of Grauert's coherence theorem)
cites work: The currents defined by analytic varieties
cites work: Toward a numerical theory of ampleness
cites work: On compact complex analytic surfaces. I
cites work: On compact analytic surfaces. II
cites work: On compact analytic surfaces. III
cites work: Q4212603
cites work: Abstract Bergman kernel expansion and its applications
cites work: Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below
cites work: Superconnection and family Bergman kernels
cites work: Q5580138
cites work: Riemannian Geometry
cites work: Canonical coordinates and Bergmann metrics
cites work: The partial \(C^0\)-estimate along the continuity method
cites work: On a set of polarized Kähler metrics on algebraic manifolds
cites work: On Calabi's conjecture for complex surfaces with positive first Chern class
cites work: Kähler-Einstein metrics with positive scalar curvature
cites work: Partial \(C^0\)-estimate for Kähler-Einstein metrics
cites work: K‐Stability and Kähler‐Einstein Metrics
cites work: Tian's partial \(C^0\)-estimate implies Hamilton-Tian's conjecture
cites work: Q4399451
cites work: Some refinements of the partial \(C^0\) estimate
DOI: 10.1007/S00209-024-03610-9
description: scientific article; zbMATH DE number 7930939