Logarithmic cotangent bundles, Chern‐Mather classes, and the Huh‐Sturmfels involution conjecture

Abstract

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 schön assumptions. As the second application, we prove an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was conjectured by Huh and Sturmfels in 2013.

Maximum Likelihood Estimation for Unrooted 3-Leaf Trees: An Analytic Solution for the CFN Model

“…Despite these limitations, solutions can nonetheless be obtained using tools from numerical algebraic geometry which return theoretically correct solutions with probability one (see, e.g., [22, 15, 8, 9]). Moreover, the number of solutions, called the maximum likelihood (ML) degree , can be computed using Grobner basis techniques [20] and methods from singularity theory [29, 25].…”