On the connectedness of the standard web of Calabi-Yau 3-folds and small transitions

TL;DR

This paper proves that all complete intersection Calabi-Yau 3-folds in product of projective spaces are connected via projective conifold transitions, and introduces primitive small transitions to study their properties.

Contribution

It provides a detailed proof of the connectedness of Calabi-Yau 3-folds through conifold transitions and introduces primitive small transitions for further analysis.

Findings

All complete intersection Calabi-Yau 3-folds are connected through the standard web.

Primitive small transitions are characterized by specific conditions on singularities.

Calabi-Yau 3-folds with certain small resolutions have only ODPs as singularities.

Abstract

We supply a detailed proof of the result by P.S. Green and T. H$\ddot{\text{u}}$bsch that all complete intersection Calabi--Yau 3-folds in product of projective spaces are connected through projective conifold transitions (known as the standard web). We also introduce a subclass of small transitions which we call primitive small transitions and study such subclass. More precisely, given a small projective resolution $\pi : \widehat{X} \rightarrow X$ of a Calabi--Yau 3-fold $X$, we show that if the natural closed immersion $Def(\widehat{X}) \hookrightarrow Def(X)$ is an isomorphism then $X$ has only ODPs as singularities.