Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of $\mathrm{GL}_g(\mathbb{Z})$ and $\mathrm{SL}_g(\mathbb{Z})$

TL;DR

The paper develops a general algebro-geometric framework to bordify moduli spaces of tropical objects, notably tropical curves and tropical abelian varieties, and proves that the tropical Torelli map extends to these bordifications. It analyzes the extension of Cartan-type bi-invariant differential forms to infinity, enabling new unstable cohomology classes for $GL_g(\mathbb{Z})$ and $SL_g(\mathbb{Z})$ and providing a geometric proof of Borel’s stable cohomology in certain ranges. A key innovation is the construction of bordifications via wonderful blow-ups of polyhedral data (via PLCs and BLCs) and the introduction of canonical integrals, motives, and graph-complex interpretations that connect to top-weight cohomology and to the Borel-Serre compactification. The work also develops a robust combinatorial and geometric framework for polyhedral complexes and their cohomology, with explicit applications to moduli of tropical curves and tropical abelian varieties, and it suggests broader applicability to other graph complexes and arithmetic groups. Overall, the results bridge tropical geometry, cohomology of arithmetic groups, and motivic perspectives, yielding new invariants and a unifying geometric approach to stability phenomena in cohomology.

Abstract

We construct bordifications of the moduli spaces of tropical curves and of tropical abelian varieties, and show that the tropical Torelli map extends to their bordifications. We prove that the classical bi-invariant differential forms studied by Cartan and others extend to these bordifications by studying their behaviour at infinity, and consequently deduce infinitely many new non-zero unstable cohomology classes in the cohomology of the general and special linear groups $\mathrm{GL}_g(\mathbb{Z})$ and $\mathrm{SL}_g(\mathbb{Z})$. In particular, we obtain a new and geometric proof of Borel's theorem on the stable cohomology of these groups. In addition, we completely determine the cohomology of the link of the moduli space of tropical abelian varieties within a certain range, and show that it contains the stable cohomology of the general linear group. In the process, we define new transcendental invariants associated to the minimal vectors of quadratic forms, and also show that a certain part of the cohomology of the general linear group $\mathrm{GL}_g(\mathbb{Z})$ admits the structure of a motive. In an appendix, we give an algebraic construction of the Borel-Serre compactification by embedding it in the real points of an iterated blow-up of a projective space along linear subspaces, which may have independent applications.

Figures (12)

• Figure 1: Known ranges for which the cohomology of $\mathrm{GL}_g(\mathbb Z)$ has been completely determined SouleSL3LeeSzczarbaElbazVincentGanglSoule. An entry $H^i$ in the table indicates that $H^i(\mathrm{GL}_g(\mathbb Z);\mathbb R)$ is non-zero and of rank $1$. All non-zero classes are explained by theorem \ref{['thm: introcohomSLGL']} except for the two boxed entries.
• Figure 2: Compactly-supported cohomology of $\mathcal{P}_g/\mathrm{GL}_g(\mathbb Z)$ in the known ranges SouleSL3LeeSzczarbaElbazVincentGanglSoule. An entry $H_c^i$ in the table indicates that $H_c^i(\mathrm{GL}_g(\mathbb Z);\mathbb R)$ is non-zero and of rank $1$. All non-trivial classes are explained by theorem \ref{['thm: introcohomSLGL']} except for the two boxed entries.
• Figure 3: An entry in row $g$ and column $n$ equals $\dim H^n(\mathrm{SL}_g(\mathbb Z);\mathbb R)$; blank entries are zero. All non-zero entries in this table are explained by the previous theorem except for the three boxed entries. These are discussed in §\ref{['sect: RecentProgress']}. Some partial computations are known in $g=8$ by GL8.
• Figure 4: The decomposition of the link $L \mathcal{P}^{\mathrm{rt}}$ into Voronoï cells viewed inside the projective space $\mathbb P^2(\mathbb R)$ of symmetric $2\times 2$ matrices. The space $|L\mathcal{A}_2^{\mathrm{trop}}|$ is the quotient of the closed simplex $\sigma_{Q}$ by the action of $\Sigma_3$, and is the one-point compactification of the interior $|L\mathcal{A}^{\circ,\mathrm{trop}}|$ at the cusp. The space $|L\mathcal{A}_2^{\mathrm{trop},\mathcal{B}}|$ has an additional half-interval at the cusp.
• Figure 5: Left: a polyhedral linear configuration in $\mathbb P^2$. Right: two polyhedra $\sigma_1$, $\sigma_2$ are glued together along the common face $\sigma_{12}=\sigma_1 \cap\sigma_2$. An algebraic differential form $\omega$ defines two forms $\omega_i= \omega|_{\sigma_i}$, for $i=1, 2$, which coincide on $\sigma_{12}$. The form $\omega$ has poles along a subscheme $\mathcal{X}$, which, as depicted, may not necessarily meet the topological realisation $\sigma_1 \cup_{\sigma_{12}} \sigma_2$.

Theorems & Definitions (215)

• Theorem 1.1
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Example 2.1
• Definition 2.3
• Definition 2.4
• Definition 2.5