---
tag: DERJPQ04BQ
kind: concept
name: Nevanlinna theory in several variables
msc: [32H30, 32A22]
sources:
  - url: https://en.wikipedia.org/wiki/Nevanlinna_theory
---

## Summary

Nevanlinna theory in several variables is the value distribution theory of holomorphic maps whose source or target has dimension greater than one.
Its model case is a holomorphic curve $f \colon \C \to \mathbb{P}^n$, studied against hyperplanes, and more generally holomorphic maps from $\C^m$ to projective varieties, studied against divisors.
It extends the one-variable [[Nevanlinna theory|1EHH63VNAD]], with a characteristic function, counting functions and proximity functions, and first and second main theorems.
It is closely tied to [[Kobayashi hyperbolicity|HRPAX8V7HA]] and, through an analogy found by Vojta, to Diophantine approximation.

## Explanations

### L1 human

One-variable Nevanlinna theory measures how often a meromorphic function on the plane hits each value.
In several variables the questions are the same, for maps into higher-dimensional spaces such as projective space.
For a curve in $\mathbb{P}^n$, the role of the values is played by hyperplanes, and one asks how often the curve meets each hyperplane.
Cartan's second main theorem says, roughly, that a curve not contained in any hyperplane cannot avoid too many hyperplanes in general position.
For $n = 1$ it gives Picard's theorem: a nonconstant meromorphic function on $\C$ omits at most two values.