Angular pair-of-pants decompositions of complex varieties

TL;DR

This paper introduces a new class of torically hyperbolic varieties and constructs pair-of-pants decompositions for them, generalizing classical decompositions for hyperbolic surfaces and establishing homotopy equivalences via angle maps.

Contribution

It defines torically hyperbolic varieties, constructs pair-of-pants decompositions using angle sets, and proves homotopy equivalences extending previous work, with explicit applications to complete intersections.

Findings

Angle map is a homotopy equivalence for hyperplane complements.

Homotopy equivalence extends to Kummer coverings.

Decomposition aligns with tropical geometry techniques.

Abstract

We define the notion of torically hyperbolic varieties and we construct pair-of-pants decompositions for these in terms of angle sets of essential projective hyperplane complements. This construction generalizes the classical pair-of-pants decomposition for hyperbolic Riemann surfaces. In our first main theorem, we prove that the natural angle map associated to an essential projective hyperplane complement is a homotopy equivalence, extending earlier work of Salvetti and Bj\"orner-Ziegler. By a topological argument, we further show that the angle map for a finite Kummer covering of an essential projective hyperplane complement is likewise a homotopy equivalence. We then explain how these local building blocks can be glued along the dual intersection complex of a semistable degeneration. Using the theory of Kato-Nakayama spaces, we prove that the resulting space is homotopy equivalent to the original algebraic variety. We make this explicit for complete intersections in projective space using techniques from tropical geometry.