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Tropical Hodge theory for fans

Matthieu Piquerez (École Polytechnique - France)
Tuesday, May 18, 2021 - 12:00pm
via Zoom
Please pay attention to the unusual time.

Abstract: Many combinatorial objects have properties analog to those from Hodge theory, for instance the hard Lefschetz theorem. In the classical case, Hodge theory states some nice properties of the cohomology ring of some complex varieties. One can wonder if such a cohomological point of view could explain these properties in the combinatorial case. The first question is what is a combinatorial variety. Tropical varieties are a natural answer. Then we might ask what is smoothness. This should be a local property, i.e., it should concerns fans. Up to now, the way to define smoothness was to use Bergman fans, which are fans associated to matroids. In this second talk, I will explain a more abstract answer which extends this notion of smoothness that we develop with Amini in [arXiv:2105.01504]. I will also present some tools that let us create a wide class of fans, called the shellable fans, which verify smoothness and the hard Lefschetz theorem.

The talk will start with a pre-seminar at 11:30am:

Title: Combinatorial Hodge theory

Abstract: Fifty years ago, McMullen stated the g-conjecture: this is a list of inequalities which characterize entirely the number of faces of simplicial polytopes. Stanley discovered how Hodge theory can finish to solve this conjecture: one can associate a variety to a (sufficiently nice) simplicial polytope, and the necessity of the g-conjecture can be translated in terms of Hodge theoretic properties of the cohomology of the variety. For more complicated combinatorial objects, for instance for fans associated to matroids, such a variety does not exist in general, which make combinatorial analogs of properties of Hodge theory harder to prove for these objects. In this talk, after a detailed presentation of this context, I will explain how tropical varieties and tropical Hodge theory can help to establish these properties.


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