Small Nijenhuis cohomologies and their properties

A construction of small and large Nijenhuis cohomologies is presented. For small cohomologies an isomorphism theorem is proved, relating de Rham cohomologies to small Nijenhuis cohomologies for a nondegenerate operator. Poncaré's lemma is deduced as a consequence. For operators with singularities a theorem on connections between the logarithmic and Nijenhuis cohomology is established. The cohomologies of balls containing differentially nondegenerate singularities are calculated. Relations between the cohomologies of different operators in Nijenhuis pencils are established. Bibliography: 16 titles.