Lagrangian skeleta of very affine complete intersections

TL;DR

This paper computes the Lagrangian skeleta of certain complete intersections in complex tori, generalizing previous hypersurface results, and uses this to establish homological mirror symmetry for Batyrev-Borisov pairs.

Contribution

It extends the understanding of Lagrangian skeleta to complete intersections, providing a new geometric framework for mirror symmetry beyond hypersurfaces.

Findings

Computed the skeleton of complete intersections in complex tori.

Described the decomposition of the skeleton into standard mirror pieces.

Established homological mirror symmetry for Batyrev-Borisov pairs.

Abstract

Let $Z^\circ$ be a complete intersection inside $(\mathbb{C}^*)^n$ that compactifies to a smooth Calabi-Yau subvariety $Z$ inside a Fano toric variety $X$. We compute the skeleton of $Z^\circ$ and describe its decomposition into standard pieces that are mirror to toric varieties, which generalises the existing results in the case of hypersurfaces. This set-up was first considered by Batyrev and Borisov, who used combinatorial techniques to construct a mirror pair $(Z,\check{Z})$ of such complete intersections. We use our main result to establish homological mirror symmetry for Batyrev-Borisov pairs in the large-volume limit.