The Kato class on compact manifolds with integral bounds on the negative   part of Ricci curvature
  We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci curvature. Moreover, this provides quantitative statements about the cohomology group, contrary to results by Elworthy and Rosenberg. 
Christian Rose, Peter Stollmann
math.DG