Homological mirror symmetry for generalized Greene–Plesser mirrors

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.

Homological mirror symmetry for Milnor fibers via moduli of A∞$A_\infty$‐structures

“…The large complex structure limits in Theorem 1.7 are different from those appearing in [71] and its generalizations [72, 73]. In his construction, Seidel removes the divisor $$\lbrace x_1x_2x_3=0\rbrace$$ from the Milnor fiber $$\check{V}$$ on the $$A$$‐side and considers the reducible singular variety $$\lbrace x_0x_1x_2x_3=0\rbrace$$ instead of $$Z$$ on the $$B$$‐side (cf.…”

Hodge numbers and Hodge structures for 3-Calabi–Yau categories

“…In case X b is smooth, the category A b is 3-Calabi-Yau. It is proved in [33] that (for the right choice of b) there is an equivalence between A b and a version of the Fukaya category of Z 2 . It would be interesting to check if their techniques can be applied to answer the third item of Question 3.12.…”

“…It would be again interesting to know if the techniques developed in [33] could be used to answer the third item of Question 3.15.…”

Constructing the relative Fukaya category

“…2 The proof of homological mirror symmetry for generalized Greene-Plesser mirrors goes through almost verbatim over a coefficient ring Λ k,K where k is a field of characteristic zero and K ⊂ R is a subgroup containing the image of κ(NE). The obstruction to extending the proof to the case of finite characteristic is that the deformation theory techniques used in [She20], on which [SS21] depends, use the characteristic-zero assumption in a fundamental way; however this could possibly be circumvented by using a different approach to the deformation theory (for example, [Sei15, Lemma 3.9] works for a field of arbitrary characteristic). The characteristic-zero assumption is also used in [SS21] in the verification of smoothness of the mirror, but this can presumably be remedied by modifying the 'MPCP condition' depending on the characteristic.…”