Bergman kernels on degenerations

2023/04/04 by Linsheng Wang, Wang, Linsheng, Shengxuan Zhou +1 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2304.01943

openalex publication_date 2023/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the fiberwise Bergman kernel for a flat family of polarized varieties over a Riemann surface, which extends the classical Bergman kernel defined on the reduced fibers. We establish the continuity of the fiberwise Bergman kernel and provide a result on uniform convergence for the Fubini-Study currents. As a consequence, we show that the fiberwise Bergman kernel on test configurations exhibits continuity, and the Fubini-Study currents converge uniformly.

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• Characterization of fiberwise bimeromorphism and specialization of bimeromorphic types I: locally Moishezon case

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