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On a notion of anticanonical class for families of convex polytopes
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On a notion of anticanonical class for families of convex polytopes
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The purpose of this note is to give a generalization of the statement that the anticanonical class of a (smooth) projective toric variety is the sum of invariant prime divisors, corresponding to the rays in its fan (or facets in its polytope), to some other classes of varieties with algebraic group actions. To this end, we suggest an analogue of the notion of anticanonical class (of a compact complex manifold) for linear families of convex polytopes. This is inspired by the Serre duality for smooth projective varieties as well as the Ehrhart-Macdonald reciprocity for rational polytopes. The main examples we have in mind are: (1) The family of polytopes normal to a given fan (which corresponds to the case of toric varieties). (2) The family of Gelfand-Zetlin polytopes (which corresponds to the case of the flag variety). (3) The family of Newton-Okounkov polytopes for a (smooth) group compactification.
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