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

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 $\pi :\hat{X}\to X$ of a Calabi--Yau 3-fold $X$, we show that if the natural closed immersion $Def(\hat{X})\hookrightarrow Def(X)$ is an isomorphism then $X$ has only ODPs as singularities.

v2: 24 pages, minor corrections, exposition improved, especially in Section 6; added references

Cited by in corpus (2)

• Gromov--Witten/Pandharipande--Thomas correspondence via conifold transitions
• Affine elliptic surfaces with type-A singularities and orbi-conifolds