Bergman kernels on degenerations (Q6623328)

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.