Homological mirror symmetry for generalized Greene-Plesser mirrors

TL;DR

This paper proves homological mirror symmetry for a broad class of Calabi-Yau varieties, including Greene-Plesser pairs and certain complete intersections, expanding the understanding of mirror symmetry in algebraic geometry.

Contribution

It establishes HMS for all Greene-Plesser mirror pairs and extends results to certain complete intersections and non-Calabi-Yau mirrors, broadening the scope of mirror symmetry.

Findings

Proves HMS for Greene-Plesser mirror pairs.

Extends HMS to certain complete intersections.

Includes cases with non-Calabi-Yau mirrors and derived categories.

Abstract

We prove Kontsevich's homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev-Borisov's `dual reflexive Gorenstein cones' construction. In particular we prove HMS for all Greene-Plesser mirror pairs (i.e., Calabi-Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi-Yau complete intersections arising from Borisov's construction via dual nef partitions, and also for certain Calabi-Yau complete intersections which do not have a Calabi-Yau mirror, but instead are mirror to a Calabi-Yau subcategory of the derived category of a higher-dimensional Fano variety. The latter case encompasses Kuznetsov's `K3 category of a cubic fourfold', which is mirror to an honest K3 surface; and also the analogous category for a quotient of a cubic sevenfold by an order-3 symmetry, which is mirror to a rigid Calabi-Yau threefold.