Canonical bundle formula in a field of positive characteristic field

Over the field of complex numbers, Kodaira formulates a canonical bundle formula for relatively minimal elliptic fibration in terms of $j$-function of the elliptic curves and singularities of fibers. Over the field of positive characteristic, Bombieri and Mumford give a canonical bundle formula in terms of Euler number of the surface. In the talk, we try to give a canonical bundle formula of Kodaira type over the field of positive characteristic. This is a joint work with Yi Gu.