Calabi-Yau fibrations, simple K-equivalence and mutations

TL;DR

This paper explores Calabi-Yau fibrations and their derived equivalences, extending duality concepts to higher dimensions and providing evidence for the DK-conjecture through examples involving complex fibered Calabi-Yau varieties.

Contribution

It introduces a relative duality framework for Calabi-Yau pairs associated with homogeneous roofs and extends derived equivalence results to higher-dimensional fibrations.

Findings

Derived equivalence can lift from pairs to Calabi-Yau fibrations.

The DK-conjecture holds for certain simple K-equivalent maps from bundles of roofs.

An example of 8-dimensional Calabi-Yau varieties fibered in dual CY threefolds is constructed and proven derived equivalent.

Abstract

A homogeneous roof is a rational homogeneous variety of Picard rank 2 and index $r$ equipped with two different $\mathbb P^{r-1}$-bundle structures. We consider bundles of homogeneous roofs over a smooth projective variety, formulating a relative version of the duality of Calabi--Yau pairs associated to roofs of projective bundles. We discuss how derived equivalence of such pairs can lift to Calabi--Yau fibrations, extending a result of Bridgeland and Maciocia to higher-dimensional cases. We formulate an approach to prove that the $DK$-conjecture holds for a class of simple $K$-equivalent maps arising from bundles of roofs. As an example, we propose a pair of eight-dimensional Calabi--Yau varieties fibered in dual Calabi--Yau threefolds, related by a GLSM phase transition, and we prove derived equivalence with the methods above.