Derived categories of BHK mirrors

We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund-Hübsch duality. Our method is a variant of the so-called Landau-Ginzburg/Calabi-Yau correspondence of Calabi-Yau orbifolds with an involution that does not preserve the volume form. We deduce a version of mirror duality for the fixed loci of the involution, which are beyond the Calabi-Yau category and feature hypersurfaces of general type.The above theorem provides many examples of Calabi-Yau mirror pairs unknown before. These statements turn into ordinary cohomology statement whenever crepant resolutions on the two sides exist.Crepant resolution conjecture [28]. Finally, very recently, Hull, Israel and Sarti used mirror symmetry for K3 surfaces to form "non-geometric backgrounds" in the physics paper [21].1.10. Contents. In §2 we recall terminology briefly. In §3 we recall some basic definitions about Berglund-Hübsch invertible polynomials. In §4 we treat orbifold cohomology, its σorbifold variant, and we prove the compatibility result (5) stated above. In §5 we prove all the relevant statements at the level of Landau-Ginzburg state spaces. In §6 we derive the corresponding geometric versions stated above, see in particular §6.3 with some examples. Relation to K3 surfaces is treated in §6.4; we compare to the approach of [2] in Example 6.4.3.Higher dimensional Borcea-Voisin mirror theorem is deduced in §6.5.Aknowledgements. We are grateful to Alfio Ragusa, Francesco Russo and Giuseppe Zappalà for organising Pragmatic 2015, where this work started. We are grateful to London 2. Terminology 2.1. Conventions. We work with schemes and stacks over the complex numbers. All schemes are Noetherian and separated. By linear algebraic group we mean a closed subgroup of GL m (C) for some m. We often use strict Henselizations in order to describe a stack or a morphism between stacks locally at a closed point: by "local picture of X at the geometric point x ∈ X" we mean the strict Henselization of X at x.2.2. Notation. We list here notation that occurs throughout the entire paper.V K the invariant subspace of a vector space V linearized by a finite group K; P(w w w) the quotient stack [(C n \ 0 0 0)/G m ] with w w w-weighted G m -action; Z(f ) the variety defined as zero locus of f ∈ C[x 1 , . . . , x n ].Remark 2.2.1 (zero loci). We add the subscript P(w w w) when we refer to the zero locus in P(w w w) of a polynomial f which is w w w-weighted homogeneous. In this way we haveRemark 2.2.2 (graphs and maps). Given an automorphism α of X we write Γ α for the graph X → X × X. However, to simplify formulae, we often abuse notation and use α for the graph Γ α as well as the automorphism. In this way, the diagonal ∆ : X → X × X will be often written as id X or simply id. Berglund-Hübsch polynomialsThe setup presented here is due to Berglund-Hübsch [3]. We also refer to [4,15, 16,24,22]. It can be motivated as the simplest generalization of Gree...

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“…The setup presented here naturally yields a reformulation in toric geometry. We refer to [6] and [18].…”

Section: 4mentioning

confidence: 99%

Semi-Calabi–Yau orbifolds and mirror pairs

Chiodo

Kalashnikov

Veniani

2020

Advances in Mathematics

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“…Furthermore, V is open in XΣ. Hence, the result will now follow from Corollary 4.7 of [FK16a] assuming the conditions are satisfied. Phrased geometrically, Corollary 4.7 of loc.…”

Section: Blowups and Resolutionsmentioning

confidence: 93%

“…Our strategy to prove this result is to compare the spacesṼ := tot(−K X Σ star i ) and V := tot(−K X Σ i ) using geometric invariant theory. This may not be done directly, but can be done after partially compactifying V. The desired equivalence will then be obtained by constructing said partial compactification XΣ of V which can instead be compared withṼ using Corollary 4.7 of [FK16a].…”

Section: Blowups and Resolutionsmentioning

confidence: 99%

“…We are now squarely in the context of Section 4 of [FK16a]. Namely, we have two fans:Σ and the fan Σ star i forṼ.…”

Section: Blowups and Resolutionsmentioning

confidence: 99%

“…In the case of Berglund-Hübsch-Krawitz mirror duality, this question was answered on the level of birational equivalence in [Bor13,Sho14,Kel13,Cla14]. Derived equivalence of the Berglund-Hübsch-Krawitz mirrors to hypersurfaces in the same Gorenstein toric Fano variety was proven in [FK16a] from an examination of the secondary fan associated to the anti-canonical linear system. The analogous question for Batyrev-Borisov mirrors was solved for both birational equivalence [Li16,Cla17a] and derived equivalence [FK17].…”

Section: Introductionmentioning

confidence: 99%

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Equivalences of families of stacky toric Calabi-Yau hypersurfaces

Doran

Favero

Kelly

2018

Proc. Amer. Math. Soc.

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Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

Keating

2021

Math. Ann.

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We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn–Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian $$S^1 \times \Sigma _g$$ S 1 × Σ g in $${\mathbb {C}}^3$$ C 3 , distinguished by soft invariants for any genus $$g \ge 2$$ g ≥ 2 ; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn–Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms.

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Derived categories of BHK mirrors

Cited by 17 publications

References 37 publications

Semi-Calabi–Yau orbifolds and mirror pairs

Semi-Calabi–Yau orbifolds and mirror pairs

Equivalences of families of stacky toric Calabi-Yau hypersurfaces

Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

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