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Derived categories of BHK mirrors. (English) Zbl 1444.14076
P. Candelas et al. observed a mathematical phenomenon (later named by Brian Greene as “mirror symmetry”) where two Calabi-Yau hypersurfaces in weighted-projective 4-spaces come in pairs with flipped Hodge numbers [Nucl. Phys., B 341, No. 2, 383–402 (1990; Zbl 0962.14029)]. P. Berglund and T. Hübsch gave a mirror construction for quasi-smooth hypersurfaces in a weighted projective space [Nucl. Phys., B 393, No. 1–2, 377–391 (1993; Zbl 1245.14039)]. This proposal had a drawback – it was unable to accommodate the later enriched theory of P. Candelas et al. [Nucl. Phys., B 450, No. 1–2, 267–290 (1995; Zbl 0896.14023)]. Fortunately, a toric mirror construction due to V. V. Batyrev (and Batyrev-Borisov) saved the day and provided pivotal construction for future work on mirror symmetry [J. Algebr. Geom. 3, No. 3, 493–535 (1994; Zbl 0829.14023)]. In 2007, Berglund-Hübsch’s mirrors revived in a series of articles by H. Fan et al. in their study of Landau-Ginzburg mirror symmetry [Ann. Math. (2) 178, No. 1, 1–106 (2013; Zbl 1310.32032)]. Soon later, M. Krawitz formulated a Berglund-Hübsch mirror symmetry statement [“FJRW rings and Landau-Ginzburg mirror symmetry”, Preprint, arXiv:0906.0796] and A. Chiodo and Y. Ruan went on to prove that the Berglund-Hübsch-Krawitz (BHK) mirrors form a mirror pair at the level of Chen-Ruan orbifold cohomology [Adv. Math. 227, No. 6, 2157–2188 (2011; Zbl 1245.14038)].
Both Batyrev-Borisov mirrors and Berglund-Hübsch-Krawitz mirrors had evidence of being correct ones, however they are not necessarily isomorphic. So one needs a mathematical notion of ‘equivalence’ such that different mirrors are ‘equivalent’. The answer is the derived category of coherent sheaves (in the complex geometry side) and the derived Fukaya category (in the symplectic side) in light of Kontsevich’s homological mirror symmetry (HMS) conjecture. The paper under review work in this direction: the authors proved that HMS conjecture holds for Berglund-Hübsch-Krawitz mirror pencils in projective spaces. The proof is based on earlier works of P. Seidel [Homological mirror symmetry for the quartic surface. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1334.53091)] and N. Sheridan [Invent. Math. 199, No. 1, 1–186 (2015; Zbl 1344.53073)].
Reviewer: Yalong Cao (Chiba)

MSC:
14J33 Mirror symmetry (algebro-geometric aspects)
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
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