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Tropical Hodge theory and applications

Abstract : In this thesis, we prove that the tropical cohomology of a smooth projective tropical variety verifies several symmetry properties: namely, tropical analogs of the Kähler package composed of the Poincaré duality, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations and the monodromy-weight conjecture. We also give some applications.In the local case, we construct a wide family of fans, called tropically shellable fans, whose canonical compactifications verify the Kähler package. We show that the tropical cohomology computes their Chow rings and some quotients of the Stanley-Reisner rings of simplicial complexes which are of particular interest in combinatorics.In the global case, the proof of the main theorem mentioned above uses interesting objects as the existence of some good triangulations and specific versions of tropical analogs of the Deligne spectral sequence, the Steenbrink spectral sequence and the monodromy operator also known as the tropical eigenwave operator.As an application of our results, we get a generalization of the work of De Concini-Procesi and Feichtner-Yuzvinski about wonderful compactifications to the case of toric compactifications induced by unimodular subfans of Bergman fans.In another direction, we prove a tropical Hodge conjecture for smooth projective varieties admitting a rational triangulation: the tropical Hodge classes coincide with the kernel of the monodromy restricted to parts of bidegree (p,p).Finally, we provide a generalization of Symanzik polynomials in higher dimensions. In dimension one, these polynomials appear in combinatorics, in physics and recently in asymptotic Hodge theory. They have many known properties that are still valid in our generalization. This is a first step to understand the asymptotic of some data on degenerating families of complex varieties in any dimension. We also provide a complete description of the exchange graph of independent sets of any matroid.
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Submitted on : Tuesday, December 21, 2021 - 3:44:06 PM
Last modification on : Sunday, June 26, 2022 - 3:28:42 AM
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Matthieu Piquerez. Tropical Hodge theory and applications. Algebraic Geometry [math.AG]. Institut Polytechnique de Paris, 2021. English. ⟨NNT : 2021IPPAX073⟩. ⟨tel-03499730⟩

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