Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program

TL;DR

This paper advances the understanding of the Doran-Harder-Thompson conjecture by employing a modified Gross-Siebert program, tropical geometry, and deformation theory to relate Calabi-Yau and Fano varieties through mirror symmetry.

Contribution

It proves one direction of the Doran-Harder-Thompson conjecture using a novel approach combining tropical geometry and deformation theory within the Gross-Siebert framework.

Findings

Established a link between fibration structures and degenerations in mirror pairs.

Applied tropical geometry to analyze Calabi-Yau degenerations.

Utilized modern deformation theory to support the conjecture.

Abstract

The Doran-Harder-Thompson "gluing/splitting" conjecture unifies mirror symmetry conjectures for Calabi-Yau and Fano varieties, relating fibration structures on Calabi-Yau varieties to the existence of certain types of degenerations on their mirrors. This was studied for the case of Calabi-Yau complete intersections in toric varieties by Doran, Kostiuk and You for the Hori-Vafa mirror construction. In this paper we prove one direction of the conjecture using a modified version of the Gross-Siebert program. This involves a careful study of the implications within tropical geometry and applying modern deformation theory for singular Calabi-Yau varieties.