Primitive Calabi-Yau Threefolds

TL;DR

This paper investigates primitive Calabi-Yau threefolds, focusing on their smoothability and classification, and explores how these fundamental structures relate to the broader moduli space of Calabi-Yau threefolds.

Contribution

It extends previous work on Calabi-Yau threefolds by providing new criteria for smoothability and classification of primitive Calabi-Yau threefolds, and discusses their role in the moduli space.

Findings

Results on smoothability of certain Calabi-Yau threefolds

Restrictions on the classification of primitive Calabi-Yau threefolds

Speculations on connecting moduli spaces of Calabi-Yau threefolds

Abstract

A primitive Calabi-Yau threefold is a non-singular Calabi-Yau threefold which cannot be written as a crepant resolution of a singular fibre of a degeneration of Calabi-Yau threefolds. These should be thought as the most basic Calabi-Yau manifolds; all others should arise through degenerations of these. This paper first continues the study of smoothability of Calabi-Yau threefolds with canonical singularities begun in the author's previous paper, ``Deforming Calabi-Yau Threefolds.'' (alg-geom 9506022). If X' is a non-singular Calabi-Yau threefold, and f:X'->X is a contraction of a divisor to a curve, we obtain results on when X is smoothable. We then discuss applications of this result to the classification of primitive Calabi-Yau threefolds. Combining the deformation theoretic results of this paper with those of ``Deforming Calabi-Yau Threefolds,'' we obtain strong restrictions on the class of primitive Calabi-Yau threefolds. We also speculate on how such work might yield results on connecting together all moduli spaces of Calabi-Yau threefolds.