Framed duality of toric varieties and its mirror symmetric consequences

In this talk I will present an extension of polar duality, called framed duality, beyond the class of Fano toric varieties. The key idea is thinking of polar duality as a duality between toric varieties ``framed'' by their anti-canonical divisor and then allowing a more general ``framing'', in principle just given by an effective divisor. When restricted to a general section of the framing linear system, this construction gives back a Batyrev-type mirror symmetry between toric hypersurfaces (and complete intersections). This process, when restricted to Calabi-Yau complete intersections reduces precisely to Batyrev-Borisov duality, when considered for negative Kodaira dimension, produces Landau-Ginzburg mirror models closely related to those proposed by Givental, and, when considered for positive Kodaira dimension, suggests interesting improvements of Hori-Vafa mirror models, so getting a unified approach to Mirror Symmetry of toric complete intersections.