Boundary asymptotics of the relative Bergman kernel metric for curves
arXiv:2005.11826 · doi:10.1007/s00526-022-02347-9
Abstract
We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit coefficients. In general the Bergman kernels on a given cuspidal family do not always converge to that on the regular part of the limiting surface, which is different from the nodal case. It turns out that information on both the singularity and complex structure contributes to various asymptotic behaviors of the Bergman kernel. Our method involves the classical Taylor expansion for Abelian differentials and period matrices.
arXiv comment30 pages, final version to appear in Calculus of Variations and Partial Differential Equations
References in corpus (12)
- Invariance of Plurigenera
- Effectiveness of Demailly's strong openness conjecture and related problems
- Extension of holomorphic functions defined on non reduced analytic subvarieties
- New characterizations of plurisubharmonic functions and positivity of direct image sheaves
- Faltings delta-invariant and semistable degeneration
- Complex structure degenerations and collapsing of Calabi-Yau metrics
- Curvature semipositivity of relative pluricanonical systems
- BCOV invariants of Calabi--Yau manifolds and degenerations of Hodge structures
- Rigidity theorem by the minimal point of the Bergman kernel
- Riemannian geometry of Kahler-Einstein currents III: compactness of Kahler-Einstein manifolds of negative scalar curvature
- Canonical bundle formula and degenerating families of volume forms
- Convergence of Bergman measures towards the Zhang measure