Calabi-Yau components in general type hypersurfaces

TL;DR

This paper explores the structure of general type hypersurfaces, constructing open covers via tropical geometry, and shows how their holomorphic forms relate to Calabi-Yau hypersurfaces, advancing understanding of mirror symmetry.

Contribution

It introduces a method to approximate holomorphic forms on general type hypersurfaces with Calabi-Yau components using tropical geometry techniques.

Findings

Holomorphic n-forms are supported on unique open components after normalization.

Open covers approximate Calabi-Yau hypersurfaces as parameters grow large.

Lagrangian fibers are asymptotically special Lagrangian.

Abstract

For a one-parameter family of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers using tropical geometry. We show that after normalization, each holomorphic n-form is approximately supported on a unique open component and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our results is a step toward understanding mirror symmetry for general type manifolds.