CY Principal Bundles over Compact K\"ahler Manifolds

TL;DR

This paper proves the existence of Calabi-Yau principal bundles over compact Kähler manifolds that support liftings of all line bundles, enabling uniform differential systems for period integrals, and establishes their rigidity when the structure group is a torus.

Contribution

It demonstrates the existence of CY bundles over Kähler manifolds supporting all line bundle liftings and proves their rigidity for algebraic torus groups.

Findings

Existence of CY bundles over Kähler manifolds supporting all line bundle liftings.

Existence of CY bundles in general when Picard group is discrete.

Rigidity of CY bundles with algebraic torus as principal group.

Abstract

A CY bundle on a connected compact complex manifold $X$ was a crucial ingredient in constructing differential systems for period integrals in [LY], by lifting line bundles from the base $X$ to the total space. A question was therefore raised as to whether there exists such a bundle that supports the liftings of all line bundles from $X$, simultaneously. This was a key step for giving a uniform construction of differential systems for arbitrary complete intersections in $X$. In this paper, we answer the existence question in the affirmative if $X$ is assumed to be K\"ahler, and also in general if the Picard group of $X$ is assumed to be discrete. Furthermore, we prove a rigidity property of CY bundles if the principal group is an algebraic torus, showing that such a CY bundle is essentially determined by its character map.