Lagrangian skeleta of very affine complete intersections

TL;DR

This work computes the Lagrangian skeleton of open Batyrev–Borisov complete intersections $Z^lat$ in $(\mathbb{C}^*)^n$ by embedding them in toric ambient spaces and exploiting tropical geometry. The skeleton is described as a singular Lagrangian lying inside the FLTZ skeleton $\mathbb{L}_\Sigma$ and decomposes into pieces that mirror lower-dimensional toric varieties via a transversal-cone indexing; this yields a natural open-cover description by $\Lambda(\sigma) \simeq \mathbb{L}_{\Sigma/\sigma} \times C_\sigma$. The main analytic-geometric toolkit includes nef partitions, tropical set-ups, tempered adapted potentials (including strongly adapted and tented variants), and a tailoring procedure to localise calculations to dominant terms; together these enable a precise Liouville description and a gluing strategy. The paper proves a homological mirror symmetry statement for BBCI pairs in the large-volume limit, matching the wrapped Fukaya category of Z with the derived category of a B-side mirror $\check Z$, constructed as the transversal toric boundary in the dual toric data. Overall, the results extend HMS from hypersurfaces to higher codimension Calabi–Yau complete intersections in Fano toric varieties, providing a concrete skeleta-based bridge between A- and B-model data and a robust toolkit for further explorations in toric and tropical HMS. The work’s synthesis of tropical geometry, Liouville geometry, and microlocal sheaf theory yields a rich, computable framework for understanding open BBCIs and their mirrors in the large-volume regime.

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.

Figures (8)

• Figure 1: Dual nef partitions $(\Delta=\Delta_1+\Delta_2, \nabla=\nabla_1+\nabla_2)$ from Example \ref{['example:running-nef']}
• Figure 2: Tropical Batyrev--Borisov complete intersection from Example \ref{['example:running-trop']}
• Figure 3: Barycentric subdivision of the fan of the first Hirzerbruch surface
• Figure 4: Level set of a tented potential $\phi_1$ before smoothing and convexifying for $P$ dual to the fan from Figure \ref{['fig:bar-subdiv']} (tent points are in red and the "tents" have height $\varepsilon_1$)
• Figure 5: Deformations of the piecewise linear functions (from the setting associated to Fig. \ref{['fig:bar-subdiv']} and \ref{['fig:level-set']}) to make $\varphi$ look standard near the vertex $v=F_{\sigma_2}$ in Lemma \ref{['lemma:pre_isotopy_exists']}

Theorems & Definitions (317)

• Theorem 1.1: Main theorem
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4: BN, Corollary 3.17
• Definition 2.5
• Theorem 2.6: Borisov93