Geometric smoothing by the Kähler-Ricci Flow

Eleonora Di Nezza; Vincent Guedj; Chinh Hoang Lu

Geometric smoothing by the Kähler-Ricci Flow

Differential Geometry 2026-07-09 v1 Analysis of PDEs Complex Variables

Authors: Eleonora Di Nezza , Vincent Guedj , Chinh Hoang Lu

Abstract

We study the geometric regularization of a positive closed current by the (twisted) K\"ahler-Ricci flow on a compact K\"ahler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete K\"ahler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.

Keywords

Kähler-Ricci flow geometric regularization positive closed current Arnold multiplicities Lelong numbers Kähler manifold

Cite

@article{arxiv.2607.08325,
  title  = {Geometric smoothing by the Kähler-Ricci Flow},
  author = {Eleonora Di Nezza and Vincent Guedj and Chinh Hoang Lu},
  journal= {arXiv preprint arXiv:2607.08325},
  year   = {2026}
}

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30 pages