$L^2$ extension of holomorphic functions for log canonical pairs

In a general $L^2$ extension theorem of Demailly for log canonical pairs, the $L^2$ criterion with respect to a measure called the Ohsawa measure determines when a given holomorphic function can be extended. Despite the analytic nature of the Ohsawa measure, we establish a geometric characterization of this analytic criterion using the theory of log canonical centers from algebraic geometry. Along the way, we characterize when the Ohsawa measure fails to have generically smooth positive density, which answers an essential question arising from Demailly’s work.