Analytic equivalence of geometric transitions

TL;DR

This paper introduces the concept of analytic equivalence for geometric transitions, classifies these transitions into equivalence classes, and proposes canonical models using a new invariant called bi-degree to understand their properties.

Contribution

It defines analytic equivalence for geometric transitions, introduces bi-degree as a classification invariant, and establishes a framework for canonical models and their organization in the \\cy web.

Findings

Analytic equivalence classes correspond to arrows in the \\cy web.

Bi-degree effectively summarizes topological, geometric, and physical changes.

Framework for canonical models of geometric transitions is proposed.

Abstract

In this paper \emph{analytic equivalence} of geometric transition is defined in such a way that equivalence classes of geometric transitions turn out to be the \emph{arrows} of the \cy web. Then it seems natural and useful, both from the mathematical and physical point of view, look for privileged arrows' representatives, called \emph{canonical models}, laying the foundations of an \emph{analytic} classification of geometric transitions. At this purpose a numerical invariant, called \emph{bi--degree}, summarizing the topological, geometric and physical changing properties of a geometric transition, is defined for a large class of geometric transitions.