# Schwarz lemmas via the pluricomplex Green's function

## TL;DR

This paper introduces a new Schwarz lemma for holomorphic maps into symmetric products of Riemann surfaces, utilizing the pluricomplex Green's function, and extends the approach to other related lemmas.

## Contribution

It presents a novel proof of Schwarz lemmas employing the pluricomplex Green's function, offering a new function-theoretic method in complex analysis.

## Key findings

- Established a Schwarz lemma for mappings into symmetric products
- Developed a new proof technique using pluricomplex Green's function
- Extended the approach to multiple Schwarz lemmas

## Abstract

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green's function. We also prove several other Schwarz lemmas using this function.