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Daily Overview |
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AG4: Algebraic Geometry Location: A701 Session Chair: Mateusz Michalek Session Chair: Botong Wang | |
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Toric variety bundles and their applications 1: HEP Vaud; 2: EPFL Toric variety bundles are fibrations whose fiber is a toric variety; more concretely, they are equivariant (partial) compactifications of principal torus bundles. They arise naturally in spherical geometry as toroidal horospherical varieties, and in logarithmic Gromov-Witten theory as irreducible components of logarithmic expansions. In this talk, I will report on recent developments concerning toric variety bundles. I will first give a combinatorial description of these objects, then give combinatorial descriptions of their intersection theory and of the positivity conditions satisfied by divisors on them. I will conclude with an application of this theory: I will show that the integral of a Lorentzian polynomial over a Minkowski linear combination of convex bodies in the positive orthant is itself a Lorentzian polynomial. As a corollary, we obtain an analogue of the Alexandrov-Fenchel inequalities for such mixed integrals. If time permits, I will also present a similar application to discrete valuations: Stanley's non-negativity for certain classes of weighted Ehrhart series. | |
