Polar decompositions of complex varieties and tropical homotopy theory

Abstract

We take the first steps towards developing a tropical homotopy theory, which we use to construct explicit finite CW complex structures on smooth complex varieties. Our approach decomposes a variety into a set of radial and angular parts, a construction we call polar decomposition. We derive a combinatorial formula for the $f$-vector of the resulting CW complex in terms of a weighted sum of local angle $f$-vectors, and we determine these $f$-vectors explicitly for toric varieties and complete intersections in terms of Stirling numbers. To construct these decompositions, we introduce the notion of a rational tropical manifold, which we view as a tropical manifold with a non-reduced structure. The local topology of these is dictated by Kummer coverings of affine linear spaces in a complex torus, as opposed to the affine linear spaces in the theory of tropical manifolds. Examples of such manifolds include the Fermat and Dwork hypersurfaces. Using Kato-Nakayama spaces, we associate a diagram of topological spaces to any rational tropical manifold. We then construct a cofibrant replacement of this diagram using sealed Kato-Nakayama spaces to show that the homotopy colimit is homotopy equivalent to a general nearby fiber. To construct CW complex structures, we then show that the spaces in the diagram can be replaced by their angle sets. More precisely, for any Kummer covering of a very affine linear space, we prove that the angle map is a homotopy equivalence, extending earlier work of Salvetti and Björner--Ziegler. We construct compatible cellulations for these angle spaces and show that these give rise to an explicit CW complex structure on the nearby fibers. For complete intersections, we make these spaces and all the CW complex structures completely explicit using refined Coxeter arrangements.

arXiv comment82 pages, 17 Figures. Comments are welcome!