  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.
