Tropical cohomology via reductions of tropical varieties

Technical Summary: Tropical Cohomology via Reductions of Tropical Varieties

Problem Statement
The paper addresses the construction of tropical spectral sequences that relate tropical cohomology to the cohomology of maximally degenerate algebraic varieties. Previous work by Itenberg-Katzarkov-Mikhalkin-Zharkov (IKMZ) established an isomorphism between tropical cohomology and the weight-graded quotients of the limit mixed Hodge structure for smooth realizable tropical varieties. This was later generalized by Amini-Piquerez to abstract compact tropical homology manifolds. However, the existing constructions of these tropical spectral sequences rely on explicit double complexes derived from polyhedral structures and require tropical Poincaré-Verdier duality. This duality condition is closely tied to the "maximally degenerate" assumption in complex algebraic geometry and is not required in the classical Steenbrink spectral sequence for limit mixed Hodge structures. Consequently, the tropical constructions are technically distinct from their complex-geometric counterparts, and the assumption of a specific polyhedral structure (N-unimodular) was previously only proven for tropicalizations in projective space ($\mathbb{P}^n$), limiting the generality of the resulting isomorphisms.

Methodology
The author proposes a new construction of tropical spectral sequences that mimics Steenbrink's geometric monodromy-weight spectral sequence for limit mixed Hodge structures, rather than relying on the explicit double complexes used in prior tropical literature. The core methodological innovation is the introduction of reductions of tropical varieties.

1. Reductions of Tropical Varieties: The paper defines a "strictly semi-stable reduction" ($X_\infty$) for a tropical variety $X$ regular at infinity. This is constructed by taking the closure of the cone $\mathbb{R}_{>0} \cdot (X \times \{1\})$ within a tropical toric variety associated with a refined fan structure. This reduction $X_\infty$ serves as the tropical analog of the special fiber in a semi-stable reduction of an algebraic family.
2. Log Tropical Forms and Filtrations: The author introduces log tropical holomorphic forms and a weight filtration on the sheaf of these forms. This allows for the construction of a spectral sequence converging to the tropical cohomology of $X$, where the $E_1$-terms are expressed in terms of the tropical cohomology of the strata of the reduction $X_\infty$.
3. Lattice Refinement: To ensure the existence of the necessary polyhedral structures for these reductions, the paper proves that for any finite union of $\mathbb{Q}$-rational polyhedra regular at infinity in a smooth proper tropical toric variety, there exists a $1/k\mathbb{N}$-unimodular $\mathbb{Q}$-rational polyhedral complex structure for some integer $k$. This generalizes previous results limited to $\mathbb{P}^n$.
4. Gauss-Manin Connections: The paper constructs a "tropical Gauss-Manin connection" using relative log tropical forms. It demonstrates that the residue of this connection corresponds to the eigenwave action (a cap product operation) on tropical cohomology.

Key Contributions and Results

• Theorem 1.1 (Spectral Sequence Construction): The paper establishes a spectral sequence for a tropical variety $X$ regular at infinity with an $N$-unimodular structure. The $E_1$-terms are given by the tropical cohomology of the intersections of components of the strictly semi-stable reduction $X_\infty$. The differentials are composed of pullback maps and Gysin maps (up to signs). This construction parallels Steenbrink's work and avoids the direct reliance on tropical Poincaré-Verdier duality in the construction phase.
• Proposition 1.3 / Theorem 2.7 (Existence of Structures): The author proves the existence of $1/k\mathbb{N}$-unimodular polyhedral complex structures for tropical varieties regular at infinity in smooth proper tropical toric varieties. This justifies the assumptions required for the spectral sequence construction in a broader setting than just projective space.
• Theorem 2.13 (Generalized Isomorphism): As a consequence of the new construction and the existence of unimodular structures, the isomorphism between tropical cohomology and the weight-graded quotients of the limit mixed Hodge structure (previously known for $\mathbb{P}^n$) is generalized to families of algebraic varieties in any smooth proper toric variety.
• Theorem 1.4 / Theorem 4.13 (Eigenwave and Gauss-Manin): The paper shows that the dual of the eigenwave action on tropical cohomology coincides with the residue of the tropical Gauss-Manin connection. This provides a tropical analog to the complex-geometric result where the monodromy action is represented by the Gauss-Manin connection.

Significance
The paper claims to provide a "simpler construction" of tropical spectral sequences that aligns more closely with the classical Steenbrink construction in complex algebraic geometry. By introducing the concept of tropical reductions, the author bridges the technical gap between tropical and complex spectral sequences, removing the need for the specific "maximally degenerate" duality assumptions in the construction phase. Furthermore, by proving the existence of unimodular structures in general smooth proper toric varieties, the paper extends the applicability of the IKMZ isomorphism beyond projective space, validating the use of tropical cohomology as a robust invariant for a wider class of degenerating families. The identification of the eigenwave action with the residue of the tropical Gauss-Manin connection further unifies the understanding of monodromy in the tropical setting with its complex counterpart.