On Normal Forms of Complex Points of codimension 2 submanifolds

TL;DR

This paper develops a linear algebra framework for classifying quadratic parts of complex points in codimension two submanifolds, providing complete normal forms under certain conditions and a full classification in low dimensions.

Contribution

It introduces a linear algebra approach to normal forms of complex points and extends classification results without additional assumptions in low dimensions.

Findings

Complete normal form descriptions under nondegeneracy assumptions

Full classification in low dimensions

Framework applicable to quadratic parts of complex points

Abstract

In this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold. Assuming some extra nondegenericity and using the result of Hong, complete normal form descriptions can be given, and in low dimensions, we obtain a complete classification without any extra assumptions.