Lelong Numbers of $m$-Subharmonic Functions Along Submanifolds

TL;DR

This paper investigates the singularities of m-subharmonic functions along submanifolds in compact Kähler manifolds, establishing bounds on their growth and pole strength, extending classical results like Siu's theorem.

Contribution

It introduces a maximal growth rate for m-subharmonic functions along submanifolds, generalizing Siu's theorem to this setting and characterizing pole behavior.

Findings

m-subharmonic functions have at worst log poles along submanifolds when codimension is less than m

the pole strength is constant along the submanifold

a maximal growth rate depending only on m and the codimension is established

Abstract

We study the possible singularities of an $m$-subharmonic function $\varphi$ along a complex submanifold $V$ of a compact K\"ahler manifold, finding a maximal rate of growth for $\varphi$ which depends only on $m$ and $k$, the codimension of $V$. When $k < m$, we show that $\varphi$ has at worst log poles along $V$, and that the strength of these poles is moveover constant along $V$. This can be thought of as an analogue of Siu's theorem.