# On the index of harmonic maps from surfaces to complex projective spaces

## TL;DR

This paper improves lower bounds on the index of complex isotropic harmonic maps from surfaces to complex projective spaces by estimating the dimensions of holomorphic sections of specific line bundles.

## Contribution

It introduces refined estimates for the index of harmonic maps by analyzing holomorphic sections, extending previous bounds for maps from various surfaces.

## Key findings

- Improved lower bounds on the index for harmonic maps from spheres and tori.
- Extended bounds to higher genus surfaces.
- Enhanced understanding of the relationship between holomorphic sections and harmonic map stability.

## Abstract

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces.