The Mori Program and Non-Fano Toric Homological Mirror Symmetry

TL;DR

This paper advances the understanding of Homological Mirror Symmetry for toric varieties by integrating the Mori program, proposing a refined conjecture, and proving a new case involving a non-nef anticanonical bundle.

Contribution

It introduces a refined conjectural framework linking semi-orthogonal decompositions in mirror symmetry and proves a new case for a specific toric surface.

Findings

Established a new case of Homological Mirror Symmetry for a non-nef anticanonical bundle toric surface.

Proposed a refined conjecture connecting semi-orthogonal decompositions in the $A$- and $B$-models.

Unified the Mori program with Homological Mirror Symmetry in the context of toric varieties.

Abstract

In the case of toric varieties, we continue the pursuit of Kontsevich's fundamental insight, Homological Mirror Symmetry, by unifying it with the Mori program. We give a refined conjectural version of Homological Mirror Symmetry relating semi-orthogonal decompositions of the $B$-model on toric varieties to semi-orthogonal decompositions on the $A$-model on the mirror Landau-Ginzburg models. As evidence, we prove a new case of Homological Mirror Symmetry for a toric surface whose anticanonical bundle is not nef, namely a certain blow-up of $\P^2$ at three infinitesimally near points.