Lemma local version

LEMMA 2.4.3 Let .x 2 X; / be the germ of a log canonical pair where X has dimension n, and let P ai Di be a local decomposition. Assume that KX and D1; D2; : : : ; Dk are Cartier. If D n P ai D n d is the local complexity, then the following hold. (1) 0. (2) If < 1, then, after possibly reordering D1; D2; : : : ; Dk, there is an integer m n 2 0 such that .X; D1 C D2 C C Dm/ is log smooth and D1 C D2 C C Dm: (3) If < 3 2 , then either X is smooth at x or has a cAl singularity at x. (paper) (Brown et al., 2018, p. 942)