---
abstract: 'The present paper is dedicated to illustrating an extension of polar duality between Fano toric varieties to a more general duality, called *framed* duality, so giving rise to a powerful method of producing mirror partners of hypersurfaces and complete intersections in toric varieties, of any Kodaira dimension. In particular, the class of projective hypersurfaces and their mirror partners are studied in detail. Moreover, many connections with known Landau-Ginzburg mirror models, Homological Mirror Symmetry and Intrinsic Mirror Symmetry, are discussed.'
address: 'Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino'
author:
- Michele Rossi
bibliography:
- 'MILEA.bib'
title: An extension of polar duality of toric varieties and its consequences in Mirror Symmetry
---

Introduction {#introduction .unnumbered}
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Polar duality between reflexive polytopes gives the well known Batyrev duality between Fano toric varieties, inducing a mirror symmetry between generic sections of their anti-canonical divisors [@Batyrev94]. Borisov and Batyrev extended this duality to complete intersections described by a nef partition of the anti-canonical divisor of a Fano toric variety [@Borisov], [@BB96]. By thinking of Batyrev duality as a duality between toric varieties *framed* by their anti-canonical divisor, the present paper is devoted to show how *deforming* the Batyrev-Borisov 