Strange duality and polar duality

We describe a relation between Arnold’s strange duality and a polar duality between the Newton polytopes which is mostly due to M. Kobayashi. We show that this relation continues to hold for the extension of Arnold’s strange duality found by C. T. C. Wall and the author. By a method of Ehlers-Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. We generalize this method to the isolated complete intersection singularities embraced in the extended duality. We use this to explain the duality of characteristic polynomials of the monodromy discovered by K. Saito for Arnold’s original strange duality and extended by the author to the other cases. Introduction In [Eb] we gave a survey on some new features of Arnold’s strange duality and the extension of it found by C. T. C. Wall and the author [EW]. This duality can be considered as a two-dimensional analogue of the Mirror Symmetry of Calabi-Yau threefolds. Among other things, we discussed a duality of the characteristic polynomials of the monodromy operators of the singularities discovered by K. Saito [S]. We showed that Saito’s duality continues to hold for our extension of Arnold’s strange duality. V. Batyrev [B] showed that the Mirror Symmetry of Calabi-Yau hypersurfaces in toric varieties is related to the polar duality between their Newton polytopes. M. Kobayashi [Kob] observed that Arnold’s strange duality corresponds to a duality of weight systems and this in turn is related to a polar duality between certain polytopes. These polytopes are slightly modified Newton polytopes of the singularities. In this paper we consider this relation more closely. We give a more precise and simpler formulation of Kobayashi’s result. We consider this correspondence for our extension of Arnold’s strange duality. This extension embraces isolated complete intersection singularities (abbreviated ICIS in the sequel). We define Newton diagrams and Newton polyhedra for these singularities and show that Kobayashi’s correspondence continues to hold for our extension of the duality. By a method found by A. N. Varchenko [V] and independently by F. Ehlers [Eh], the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. A generalization of this method to the principal monodromy of non-degenerate complete intersection singularities has been given by M. Oka [O]. We show that for all the isolated complete intersection singularities embraced in the duality with one exception, the Ehlers-Varchenko method can be applied more directly. We use this to show