  Using compactifications in the logarithmic cotangent bundle, we obtain a
formula for the Chern classes of the pushforward of Lagrangian cycles under an
open embedding with normal crossing complement. This generalizes earlier
results of Aluffi and Wu-Zhou. The first application of our formula is a
geometric description of Chern-Mather classes of an arbitrary very affine
variety, generalizing earlier results of Huh which held under the smooth and
schon assumptions. As the second application, we confirm an involution formula
relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was
conjectured by Huh and Sturmfels in 2013.
