Semi-Calabi–Yau orbifolds and mirror pairs

Abstract

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...

Borcea–Voisin mirror symmetry for Landau–Ginzburg models

“…Indeed, one of the first predictions of mirror symmetry is the rotation of the Hodge diamond. Such a statement is one of the main results of Chiodo-Kalashnikov-Veniani in [9] for the Borcea-Voisin orbifolds similar to the form we consider here. In other words, they prove the following corollary in case…”

“…In some sense these theorems are LG analogues of results in both the work of Artebani-Boissière-Sarti in [2] and of Chiodo-Kalashnikov-Veniani in [9] simultaneously, which we discuss below. Other than the examples computed by the last author in [44] this is the first time we are aware that this particular LG model has been considered in the literature.…”

“…We can do the same with W 2 obtaining a similar isomorphism. Putting these together with (9) and (11), we obtain the isomorphism of state spaces predicted by the LG/CY correspondence.…”

“…Remark 5. 9. Recall that for an element ⌈m ; g⌋ from an LG state space, m represents a monomial together with a volume form.…”

“…In [9] Chiodo-Kalashnikov-Veniani undertake to prove a mirror statement for the geometric model (i.e. for orbifold BV models), again using the LG model as a guide.…”