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Asymptotics of Partial Density Functions for Divisors.

Julius Ross1, Michael Singer2

  • 11Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK.

Journal of Geometric Analysis
|March 7, 2019
PubMed
Summary

This study analyzes the asymptotic behavior of a partial density function related to vanishing sections of hermitian line bundles. We prove its smooth expansion and characterize a "forbidden region" with error-function behavior.

Keywords:
Bergman kernelEquilibrium setForbidden regionInterface asymptotics

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Area of Science:

  • Complex Geometry
  • Differential Geometry
  • Analysis on Manifolds

Background:

  • Investigating the asymptotic properties of geometric objects is crucial for understanding their behavior.
  • Partial density functions and vanishing sections are key concepts in complex and algebraic geometry.
  • Hermitian line bundles and divisors play fundamental roles in defining geometric structures.

Purpose of the Study:

  • To analyze the asymptotic behavior of the partial density function for sections of a positive hermitian line bundle.
  • To prove the existence of a distributional asymptotic expansion that becomes smooth after real blow-up.
  • To characterize the "forbidden region" where the density function is exponentially small and its boundary behavior.

Main Methods:

  • Asymptotic analysis of the partial density function.
  • Utilizing properties of hermitian line bundles and divisors.
  • Employing techniques from real blow-up and distributional analysis.
  • Investigating symmetries and invariance under group actions.

Main Results:

  • The partial density function exhibits a smooth asymptotic expansion after a suitable real blow-up.
  • The existence of a "forbidden region" (R) is confirmed, where the density function is exponentially small.
  • The density function demonstrates "error-function" behavior across the boundary of the forbidden region.

Conclusions:

  • The study provides a detailed understanding of the asymptotic behavior of partial density functions in complex geometry.
  • The findings have implications for studying functions associated with divisors in Kähler manifolds.
  • This work contributes to the analysis of geometric objects and their singularities.