Logarithmic cotangent bundles, Chern-Mather classes, and the Huh-Sturmfels Involution conjecture

TL;DR

This paper develops a new formula for Chern classes of Lagrangian cycles in logarithmic cotangent bundles, generalizes previous results on Chern-Mather classes, and confirms a conjecture relating ML degrees and bidegrees.

Contribution

It introduces a formula for Chern classes via compactifications in the logarithmic cotangent bundle, extending earlier work and confirming the Huh-Sturmfels involution conjecture.

Findings

Derived a formula for Chern classes of Lagrangian cycles

Provided a geometric description of Chern-Mather classes for any very affine variety

Confirmed the involution formula relating ML degrees and ML bidegrees

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 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.