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Birationality of Berglund-Hübsch-Krawitz mirrors. (English) Zbl 1395.14034
One very explicit way to construct mirror pairs of Landau-Ginzburg models is due to the construction of Berglund-Hübsch-Krawitz. Given a weighted polynomial $$W$$ with $$n+1$$ monomials in $$n+1$$ variables and a certain finite group $$G$$, there is a mirror dual Landau-Ginzburg model $$(W^T,G^T)$$ which is essentially given by transposing a square matrix whose entries are the exponents of $$W$$.
If the weights sum up to the degree of $$W$$, then $$\{W = 0\} \subset [ \mathbb P/G]$$ and $$\{W^T = 0\} \subset [ \mathbb P^T/ G^T]$$ give mirror Calabi-Yau varieties by work of A. Chiodo and Y. Ruan [Adv. Math. 227, No. 6, 2157–2188 (2011; Zbl 1245.14038)].
It is known that two Calabi-Yau varieties given by $$(W,G)$$ and $$(W',G)$$ are related by a smooth deformation, if they are homogeneous with respect to the same weights and share the same group $$G$$. The main result of this article is that the respective mirrors $$\{W^T = 0\}$$ and $$\{W'{}^T = 0\}$$ inside $$[\mathbb P^T/G^T]$$ are birational.
This result is proven describing $$[\mathbb P/G]$$ and $$[\mathbb P^T/G^T]$$ as toric orbifolds. Actually, the author describes the underlying coarse moduli spaces which are toric varieties, but for orbifolds the corrisponding lattices and fans are the same, see the work of L. A. Borisov et al. [J. Am. Math. Soc. 18, No. 1, 193–215 (2005; Zbl 1178.14057)]. Using this, the Berglund-Hübsch-Krawitz mirror construction can be understood in terms of the toric mirror construction by V. V. Batyrev [J. Algebr. Geom. 3, 493–535 (1994; Zbl 0829.14023)]. The author gives an explicit way to obtain the fan $$\Sigma$$ of $$\mathbb P/G$$. Then $$\{W = 0\}$$ is given as the zero set of a rational function on $$\mathbb P/G$$, again explicitly given by a sum of $$n+1$$ points in the (toric) lattice. Applied to the two mirrors given by $$(W^T,G^T)$$ and $$(W'{}^T,G)$$, the author shows that these lie in (birational) toric orbifolds given as the zero set of the same equation, hence they are birational.
The author gives as an example three quite differently looking hypersurfaces, which are mirror dual to the Fermat quintic threefold. By the main result they are birational.

##### MSC:
 14J33 Mirror symmetry (algebro-geometric aspects) 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14E05 Rational and birational maps
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##### References:
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