Separately meromorphic mappings into compact Kähler manifolds

On the classification of rank-two representations of quasiprojective fundamental groups

“…Going back to all of X, we conclude by Hartogs principle [Shi86,Shi94,Sic69] that the a ij are meromorphic along smooth components of the divisor. By the more classical Hartogs principle, they are meromorphic everywhere.…”

“…To show the meromorphy of a ij along D and to calculate their maximum order of poles, it suffices to restrict to curves given by z i = p i fixed for i > 1. This is because if the integral is finite, then it is finite for almost all curves, and we have a Hartogs principle for separately almost everywhere meromorphic functions [Shi86,Shi94,Sic69]. The slices in the other directions are automatically almost everywhere holomorphic, indeed for any z 1 = 0.…”