# On normal forms of complex points of small $\mathcal{C}^{2}$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold

## TL;DR

This paper investigates how the normal forms of complex points on real 4-manifolds embedded in complex 3-manifolds change under small perturbations, providing a detailed understanding of their quadratic structure.

## Contribution

It characterizes the stability of normal forms of complex points under small perturbations and describes their quadratic normal forms in embedded real 4-manifolds.

## Key findings

- Normal forms are stable under small perturbations.
- Quadratic parts of normal forms can be explicitly described.
- Results apply to small $ ext{C}^2$-perturbations of real 4-manifolds.

## Abstract

The purpose of this paper is to give a better understanding of complex points up to quadratic terms of real codimension $2$ submanifolds embedded in a complex $3$-manifold. We answer the question how a normal form of a pair of one arbitrary and one symmetric $2\times 2$ matrix with respect to a certain linear group action changes under arbitrarily small perturbations. This result is then applied to describe the quadratic part of normal forms of complex points of small $\mathcal{C}^{2}$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold.