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

Abstract

We extend our previous result on the behavior of the quadratic part of a complex points of a small ${\mathcal{C}}^2$-perturbation of a real $4$-manifold embedded in a complex $3$-manifold. We describe the change of the structure of a normal form of a complex point. It is an immediate consequence of a theorem clarifying how small perturbations can change the bundle of a pair of one arbitrary and one symmetric $2\times 2$ matrix with respect to an action of a certain linear group.