Recently, algebraic geometry in positive characteristic is well
developed. One of the important tool used in it is Frobenius map. We give a brief review on it and give characterization of ordinary abelian varieties, log Fano varieties (in characteristic zero) and kinds of singularities. These results are joint works with Hiromu Tanaka, Yoshinori Gongyo, Shinnosuke Okawa and Shunsuke Takagi.