# Expected number of critical points of random holomorphic sections over complex projective space

## TL;DR

This paper analyzes the asymptotic behavior of the expected number of critical points of Gaussian random holomorphic sections over complex projective space, providing explicit growth rates and distribution insights.

## Contribution

It offers explicit calculations of exponential growth rates for critical points of various indices and the distribution of critical values, advancing understanding in high-dimensional complex geometry.

## Key findings

- Explicit exponential growth rates for critical points of largest and diverging indices
- Distribution of critical values for smallest index critical points
- Asymptotic formulas for expected number of critical points regardless of index

## Abstract

We study the high dimensional asymptotics of the expected number of critical points of a given Morse index of Gaussian random holomorphic sections over complex projective space. We explicitly compute the exponential growth rate of the expected number of critical points of the largest index and of diverging indices at various rates as well as the exponential growth rate for the expected number of critical points (regardless of index). We also compute the distribution of the critical values for the expected number of critical points of smallest index.