Gauss--Bonnet for CSM classes of very affine manifolds, and tropical isotopies

Abstract: The CSM class of a very affine manifold $U$ is represented by the rank drop locus of a general tuple of torus invariant 1-forms on it. This equality holds in the homology of any toric compactification $X\supset U$. It was proved for sch\"on $U$ by Huh, and later for all $U$ in the homology of $X=CP^n$ by Maxim--Rodriguez--Wang--Wu, using Ginsburg's interpretation of CSM classes as Lagrangian cycles. We deduce this identity in full generality from properties of affine characteristic classes, give an explicit sufficient condition of general position for the 1-forms, and use it to extend the identity to non-torus invariant 1-forms. Along the way, we characterize very affine varieties of ML degree 0, and give a useful criterion for a family of varieties to have a constant tropical fan (``tropical isotopy theorem''). Our main idea is a technique to study very affine varieties without compactifying them. As a result we do not have to deal with singularities occuring at the boundary of the compactification, which would require less constructive methods (such as resolution of singularities). This is what enables us to give constructive genericity conditions in this Gauss--Bonnet theorem.