On the global moduli of Calabi-Yau threefolds

TL;DR

This paper explores the global structure of Calabi-Yau threefold moduli spaces, focusing on their Picard groups and line bundles, with explicit calculations for certain orbifold resolutions of elliptic curve products.

Contribution

It initiates a program to describe Calabi-Yau moduli spaces globally and computes their Picard groups and line bundles for specific examples.

Findings

Verified that a power of the Hodge line bundle is trivial in studied cases

Calculated Picard groups for several Calabi-Yau threefolds

Confirmed recent theoretical claims about line bundles in these moduli spaces

Abstract

In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle, and the closely-related Bagger-Witten line bundle. We do this here for several Calabi-Yau's obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16] by noting that a power of the Hodge line bundle is trivial -- even though in most of these cases the Picard group is infinite.