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New Calabi-Yau orbifolds with mirror Hodge diamonds. (English) Zbl 1266.14033
The article under review focuses on an aspect of the mirror symmetry conjecture – the relation between the stringy invariants of the pair of \(n\)-dimensional complex, possibly singular, Calabi-Yau varieties \((V,W)\): \[ E_{\mathrm{st}}(V;u,v) = (-u)^nE_{\mathrm{st}}(W;u^{-1},v). \] The aims are to find combinatorial and representation-theoretic formulas for the stringy invariants and to give new pairs of Calabi-Yau orbifolds satisfying this relation.
The author considers varieties \(X\) and \(X^*\) which are hypersurfaces in toric varieties associated with dual reflexive lattice polytopes \(P\) and \(P^*\), as in the construction of V. Batyrev and L. Borisov [Invent. Math. 126, No. 1, 183–203 (1996; Zbl 0872.14035)]. The additional assumption is that \(P\) is invariant with respect to the linear action of a finite group \(\Gamma\) on the lattice \(M\) by representation \(\rho\), and that \(X\) and \(X^*\) are invariant with respect to the induced action. Then equivariant stringy invariants \(E_{\mathrm{st},\Gamma}(X;u,v)\), which are polynomials in \(u,v\) with coefficients in the representation ring \(R(\Gamma)\), are defined. The main results of the paper is a general formula for the equivariant Hodge-Deligne polynomial of a non-degenerate hypersurface in a torus.
Theorem 4.10. This and the formula for the equivariant stringy invariant (given in Proposition 5.5) imply a representation-theoretic version of Batyrev-Borisov mirror symmetry, stated in terms of equivariant stringy invariants: \[ E_{\mathrm{st},\Gamma}(X;u,v) = (-u)^{d-1}\det(\rho)\cdot E_{\mathrm{st},\Gamma}(X^*;u^{-1},v). \] The proofs of these results rely on earlier works of the author [Adv. Math. 226, No. 6, 5268–5297 (2011; Zbl 1223.14059)] and [Adv. Math. 226, No. 4, 3622–3654 (2011; Zbl 1218.52014)]; they are purely combinatorial.
It follows that by taking quotients of crepant resolutions \(\widetilde{X}/\Gamma\) and \(\widetilde{X}^*/\Gamma\) by \(\Gamma \in \mathrm{SL}(M)\) one obtains pairs of orbifolds with mirror Hodge diamonds, which do not appear in the Batyrev and Borisov’s construction.
The last two sections of the paper are devoted to the study of special cases: the situation when \(P\) is a centrally symmetric reflexive polytope and the case of the Fermat quintic threefold.

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
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