Angular pair-of-pants decompositions of complex varieties

TL;DR

The paper develops a higher-dimensional analogue of the classical pair-of-pants decomposition by introducing torically hyperbolic varieties and building the topology from angle sets attached to essential hyperplane complements. It shows that the local angle blocks are captured by homotopy-equivalent angle spaces and that these blocks can be glued along the dual intersection complex via Kato-Nakayama spaces to recover the topology of the original variety. Finite Kummer coverings extend the framework, and tropical geometry provides explicit toric degenerations, enabling complete-intersection cases to be handled concretely. The approach yields explicit, computable decompositions and a homotopy-theoretic description of the variety in terms of a colimit of angle data, with potential for computing fundamental groups and higher homotopy structures in tropical settings.

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 Björner-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.

Figures (18)

• Figure 1: A representation of how pair-of-pants decomposition of compact Riemann surfaces arise naturally from maximal degenerations. Here one first degenerates the given surface $X$ to a semistable curve $\mathcal{X}_{s}$ using a one-parameter degeneration $\mathcal{X}\to U$. One then recovers the original surface by consider the real oriented blow-up or Kato-Nakayama space of the singular curve.
• Figure 3: The completed angle set of $V(1+x+y+z)\subset (\mathbb{C}^{\times})^{3}$ and the corresponding angle set of the initial degeneration $V(1+x+y)\subset(\mathbb{C}^{\times})^{3}$. Note that the second is a subcomplex of the first. This corresponds to an inclusion of Kato-Nakayama spaces $Z^{\log}_{\sigma}\subset Z^{\log}$.
• Figure 4: The polyhedral complex studied in \ref{['exa:P1Minus4Points']}. It is the tropicalization of a line $\mathbb{P}^{1} \setminus \{p_1, \dots, p_4\}$ suitably embedded into the $2$-dimensional torus in $\mathbb{P}^2$.
• Figure 5: Left: a tropically smooth elliptic curve. Right: a pair of pants decomposition of the intersection of an elliptic curve with the torus in $\mathbb{P}^2$.
• Figure 6: The left image gives the polyhedral complex $\Sigma$. The right gives the homotopy colimit of the functor that assigns a circle everywhere.

Theorems & Definitions (236)

• Theorem A
• Remark 1.1
• Corollary 1.2
• Remark 1.3
• Remark 1.4
• Example 1.5
• Theorem B
• Remark 1.6
• Theorem C
• Theorem D