Abstract: In this talk, I will present recent results obtained with Gilles Carron (Nantes Université) and Ilaria Mondello (Université Paris-Est Créteil). We work in the setting of closed Riemannian manifolds satisfying a so-called Kato bound, that is to say, the negative part of the greatest lower bound on the Ricci curvature lies in some suitable Kato class. I will explain how this assumption provides good analytic and geometric control on the manifold, which allowed us to establish that a closed Riemannian manifold satisfying a strong Kato bound with maximal first Betti number is necessarily diffeomorphic to a torus: this extends a result by Colding and Cheeger-Colding obtained in the context of a lower bound on the Ricci curvature.