Benoit Bertrand: Betti numbers of tropical subvarieties of $\mathbb T \mathbb P^n$

Abstract: I will present some results from my joint paper with E. Brugallé and L. López de Medrano "Planar Tropical Cubic Curves of Any Genus, and Higher Dimensional Generalisations" in which we study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in \(\mathbb T \mathbb P^n\). I will first describe the case of curves before stating our more general results. We provide a lower bound for the maximal value of the top Betti number, which naturally depends on the dimension and degree, but also on the codimension. In particular, when the codimension is large enough, this lower bound is larger than the maximal value of the corresponding Hodge number of complex algebraic projective varieties of the given dimension and degree. In the case of surfaces, we extend our study to other tropical homology groups.