Tropicalizations of locally symmetric varieties

TL;DR

The paper develops a rigorous framework for tropicalizations of locally symmetric varieties via admissible boundary decompositions and toroidal compactifications, connecting tropical BM-homology to the top-weight cohomology of arithmetic quotients. It defines X^{\Sigma,\mathrm{trop}} as the geometric realization of admissible diagrams of rational polyhedral fans, proves independence from the chosen input (Admissible collections), and constructs a spectral sequence linking boundary-group cohomology to top-weight cohomology. Special attention is given to types C and A, with detailed analyses of isotropic-subspace combinatorics, level structures on $\mathcal{A}_g$ and special unitary cases, and modular interpretations of the tropical moduli spaces as moduli of tropical abelian varieties or Hermitian forms. The work yields concrete cohomological computations, delineates links to boundary complexes, and establishes Hopf-algebra structures on weight-zero cohomology, highlighting doubling phenomena and enabling new connections between tropical geometry and the cohomology of arithmetic groups. Overall, the results illuminate how tropicalizations encode top-weight cohomology and arithmetic information, and they provide modular interpretations and spectral sequences that facilitate cohomology calculations in important cases like $\mathcal{A}_g$ with level structure and special unitary Shimura varieties.

Abstract

This paper provides a rigorous study of tropicalizations of locally symmetric varieties. We give applications beyond tropical geometry, to the cohomology of moduli spaces as well as to the cohomology of arithmetic groups. We study two cases in detail: the case of level structures on the moduli space $\mathcal{A}_{g}$ of abelian varieties, and the special unitary case.

Figures (3)

• Figure 1: The $E^{1}$-page of the spectral sequence \ref{['eq:spectral-sequence']} appearing in the proof of Proposition \ref{['prop:Ag-GL']} in the case when $g=8$. In the light gray area, given by the region above the parabola $q>d_{p}-p$, we have that $E_{p,q}^{1}=0$. In the light gray region, given by $p>8$, we know that $E_{p,q}^{1}=0$ by definition. The darker gray region, with $p=8$ and $22 \leq q \leq 28$, survives unchanged to the $E^{\infty}$ page since the quadrants to its left and right are zero. This is the region where $\operatorname{Gr}^W_{2d} H^{2d-(p+q)}(\mathcal{A}_g[m];\mathbb{Q}) \cong W_0 H^{p+q}_{c}(\mathcal{A}_{g}[m];\mathbb{Q})^{\vee}$ is determined by $H^{d-(g+q)}(\mathrm{GL}_{g}(\mathbb{Z})[m];\mathbb{Q}_{\mathrm{or}})^{\oplus \pi_{g,g,m}}$.
• Figure 2: The link of $A_2^{\mathrm{trop}}[2]$ and the 15 isotropic planes of $(\mathbb{Z}/2\mathbb{Z})^4$, each of which corresponds to a triangle.
• Figure 3:

Theorems & Definitions (97)

• Definition A
• Theorem B
• Theorem C
• Theorem D
• Remark 1.1
• Theorem E
• Theorem F
• Remark 2.1
• Definition 2.2
• Remark 2.3