Modular Tropical Toric Reflexive Sheaves

• Modular tropical toric reflexive sheaves are defined by tropical toric reflexive sheaves whose underlying matroid is modular, integrating fan filtrations with combinatorial exactness.
• They generalize classical toric reflexive sheaves by employing piecewise-linear maps to Bergman fans and establishing Harder–Narasimhan slope theory via matroidal filtrations.
• The framework exhibits proto-abelian and proto-exact structures, linking tropical combinatorics with algebraic geometry through stability, restriction, contraction, and modular generating functions.

Modular tropical toric reflexive sheaves are tropical toric reflexive sheaves whose underlying matroid is modular; in the current formulation, they form the full subcategory $\mathbf{MTRS}\subset \mathbf{TRS}_\bullet^\Sigma$ associated to a complete rational fan $\Sigma$, where objects consist of a simple pointed $\mathbb{T}$-matroid together with ray-indexed decreasing chains of flats. More broadly, the topic lies at the intersection of tropical toric geometry, matroidal exactness, stability theory in non-additive categories, and the older algebro-geometric study of toric reflexive sheaves whose moduli generate modular forms (Jun et al., 9 Sep 2025, Gholampour et al., 2012).

1. Tropical toric reflexive sheaves and their toric data

Fix a complete rational fan $\Sigma$ in a lattice $L\cong \mathbb{Z}^n$, with dual lattice $\Lambda=\mathrm{Hom}(L,\mathbb{Z})$, and the tropical toric variety $\mathrm{trop}(X_\Sigma)$. A tropical toric reflexive sheaf of rank $r$ is a tuple

$$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$

where $N$ is a simple pointed $\Sigma$0-matroid of rank $\Sigma$1 on a pointed ground set, and for each ray $\Sigma$2, $\Sigma$3 is a decreasing chain of flats of the underlying matroid, eventually equal to the full ground set for $\Sigma$4 and to the basepoint for $\Sigma$5 (Jun et al., 9 Sep 2025).

This definition is the tropical counterpart of Klyachko-type filtration data. In the same framework, a tropical toric vector bundle is the special case in which, for every maximal cone $\Sigma$6, there exists a multiset $\Sigma$7 and a basis $\Sigma$8 of the underlying matroid such that the filtrations $\Sigma$9 are recovered from the inequalities $\mathbb{T}$0. Rank $\mathbb{T}$1 objects are tropical line bundles, and these correspond to integral vectors $\mathbb{T}$2 (Jun et al., 9 Sep 2025).

A closely related formulation describes tropical toric vector bundles as piecewise-linear maps

$\mathbb{T}$3

to the Bergman fan of a loop-free matroid $\mathbb{T}$4, or equivalently as maps $\mathbb{T}$5 satisfying circuit and apartment-compatibility conditions. In that language, tropical toric vector bundles are also called toric matroid bundles, and they carry equivariant $\mathbb{T}$6-classes, Chern classes, positivity notions, and a vanishing of higher cohomologies result (Kaveh et al., 2024).

The relation between these viewpoints is structural rather than merely notational. The matroidal filtration model emphasizes strict subobjects, kernels, cokernels, and slope theory; the Bergman-fan model emphasizes piecewise-linear toric data, characteristic classes, and section-counting. This suggests that modular tropical toric reflexive sheaves should be read as the filtration-theoretic enlargement of toric matroid bundles in which the underlying matroid is required to be modular.

2. Modularity in the matroidal sense

For a matroid $\mathbb{T}$7 with lattice of flats $\mathbb{T}$8, a pair of flats $\mathbb{T}$9 is modular when

$\Sigma$0

where $\Sigma$1 and $\Sigma$2 denote meet and join in $\Sigma$3. A flat is modular if it forms a modular pair with every flat, and a matroid is modular if all of its flats are modular. Equivalently, the simplification of each connected component is either free or the matroid of a finite projective geometry (Jun et al., 9 Sep 2025).

A modular tropical toric reflexive sheaf is therefore a tropical toric reflexive sheaf

$\Sigma$4

whose underlying matroid $\Sigma$5 is modular. No additional condition is imposed on the filtrations beyond the requirement that they are chains of flats in that modular lattice (Jun et al., 9 Sep 2025).

This condition is stable under the basic matroid operations relevant for exactness. Restriction to a flat $\Sigma$6 identifies $\Sigma$7 with the interval $\Sigma$8, and contraction by $\Sigma$9 identifies $L\cong \mathbb{Z}^n$0 with the interval $L\cong \mathbb{Z}^n$1; intervals in a modular lattice are modular. Consequently, admissible restriction and contraction procedures preserve modularity (Jun et al., 9 Sep 2025).

The modular condition also has a geometric antecedent in the tropical splitting theory of toric matroid bundles. Over $L\cong \mathbb{Z}^n$2, splitting holds for tropical toric vector bundle classes whose matroid extends to a modular matroid, whereas the Vamos matroid furnishes a non-splitting example. In that setting, modularity is the analogue of the building-theoretic property that forces simultaneous compatibility of filtrations (Kaveh et al., 2024). This suggests that, in the tropical reflexive setting, modularity is not an incidental restriction but the precise combinatorial hypothesis that restores the behavior expected from classical exact and stability theories.

3. Proto-exact and proto-abelian structure

The category $L\cong \mathbb{Z}^n$3 is built from morphisms

$L\cong \mathbb{Z}^n$4

where $L\cong \mathbb{Z}^n$5 is a morphism of simple $L\cong \mathbb{Z}^n$6-matroids and $L\cong \mathbb{Z}^n$7, subject to the compatibility condition

$L\cong \mathbb{Z}^n$8

for every ray $L\cong \mathbb{Z}^n$9 and every integer $\Lambda=\mathrm{Hom}(L,\mathbb{Z})$0. Composition is given by

$\Lambda=\mathrm{Hom}(L,\mathbb{Z})$1

and isomorphisms are exactly those for which $\Lambda=\mathrm{Hom}(L,\mathbb{Z})$2 is an isomorphism of $\Lambda=\mathrm{Hom}(L,\mathbb{Z})$3-matroids, with inverse $\Lambda=\mathrm{Hom}(L,\mathbb{Z})$4 (Jun et al., 9 Sep 2025).

Restriction and contraction are defined by flats. If $\Lambda=\mathrm{Hom}(L,\mathbb{Z})$5 and $\Lambda=\mathrm{Hom}(L,\mathbb{Z})$6 is a flat of $\Lambda=\mathrm{Hom}(L,\mathbb{Z})$7, then restriction is

$\Lambda=\mathrm{Hom}(L,\mathbb{Z})$8

and contraction is

$\Lambda=\mathrm{Hom}(L,\mathbb{Z})$9

where $\mathrm{trop}(X_\Sigma)$0 corresponds to $\mathrm{trop}(X_\Sigma)$1 under the lattice identification for $\mathrm{trop}(X_\Sigma)$2. These operations induce the kernel and cokernel descriptions in $\mathrm{trop}(X_\Sigma)$3: the kernel is restriction to the preimage of the basepoint, and the cokernel is contraction by the image (Jun et al., 9 Sep 2025).

Admissible monomorphisms are morphisms that factor as restriction to a flat followed by an isomorphism; admissible epimorphisms are morphisms that factor as contraction by a flat followed by an isomorphism. With these classes, $\mathrm{trop}(X_\Sigma)$4 is proto-exact in the sense of Dyckerhoff–Kapranov, and because it also has kernels, cokernels, and satisfies the relevant mono/epi criterion, it is proto-abelian in the sense of André. The modular subcategory $\mathrm{trop}(X_\Sigma)$5 inherits both structures (Jun et al., 9 Sep 2025).

This categorical structure is part of a broader non-additive exactness theory for matroids over idylls. For every perfect idyll $\mathrm{trop}(X_\Sigma)$6, the category $\mathrm{trop}(X_\Sigma)$7 is proto-exact and proto-abelian; valuated matroids arise when $\mathrm{trop}(X_\Sigma)$8. The tropical toric reflexive category is then obtained by enriching simple $\mathrm{trop}(X_\Sigma)$9-matroids with compatible flat filtrations indexed by the rays of $r$0 (Jun et al., 9 Sep 2025).

4. Degree, slope, and Harder–Narasimhan theory

Fix a smooth toric variety $r$1 with constant polarization

$r$2

For a tropical toric reflexive sheaf $r$3, the rank is the matroid rank $r$4, and the degree is defined by summing over rays and filtration jumps, weighted by the polarization and by the integer indices $r$5 appearing in the filtrations (Jun et al., 9 Sep 2025).

Tensoring with a tropical line bundle shifts the filtrations, and Khan–Maclagan’s degree formula satisfies

$r$6

Under constant polarization, principal divisors have degree zero, so degree is invariant under the character twists that occur in the isomorphism notion of $r$7 (Jun et al., 9 Sep 2025).

The slope is

$r$8

for nonzero $r$9. Stability and semistability are then defined in the proto-abelian sense: a nonzero object is semistable if every strict subobject has slope at most that of the whole object, and stable if every nonzero strict subobject has strictly smaller slope (Jun et al., 9 Sep 2025).

In $$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$0, strict subobjects of $$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$1 are restrictions $$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$2 to flats $$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$3. Their intersections and sums are explicitly matroidal: $$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$4 For modular underlying matroids, Khan–Maclagan’s slope inequality yields the strong slope inequality required in Li’s proto-abelian slope theory, and degree is additive on short exact sequences under the modularity assumptions used in that framework (Jun et al., 9 Sep 2025).

Jun–Sistko–Wright verify the remaining hypotheses of Li’s theorem for $$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$5: the category is small and proto-abelian, every object is Artinian and Noetherian because chains of strict subobjects correspond to chains of flats in a finite ground set, rank and degree are additive on short exact sequences, and rank zero forces the zero object. Hence every nonzero modular tropical toric reflexive sheaf admits a unique Harder–Narasimhan filtration with semistable factors of strictly decreasing slopes. They also show that Khan–Maclagan’s combinatorially constructed Harder–Narasimhan filtration coincides with this categorical slope filtration (Jun et al., 9 Sep 2025).

5. Classical toric reflexive sheaves, filtrations, and polyhedra

The tropical theory sits atop a long toric-sheaf tradition in which reflexive equivariant sheaves are controlled by filtrations, posets, and polyhedra. In the affine case, finitely generated $$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$6-graded reflexive modules on a normal affine toric variety are equivalent to finite-dimensional vector spaces endowed with full filtrations indexed by the rays of the defining cone, and the graded pieces are recovered by intersecting the filtration steps along the primitive ray generators (Perling, 2011). Perling’s further analysis of lcm-lattices, vector space arrangements, hyperplane arrangements, and combinatorial Betti numbers suggests a natural bridge from toric reflexive sheaves to matroidal and tropical combinatorics.

A global polyhedral reformulation replaces Klyachko filtrations by Weil decorations

$$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$7

factoring through $$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$8 and satisfying

$$\mathcal{F}=(N,\{F^\rho_\bullet\}_{\rho\in\Sigma(1)}),$$9

On a smooth projective toric variety, this description leads to a constructible sheaf $N$0 on the moment polytope $N$1, an identification

$N$2

and a spectral sequence whose $N$3-page is built from the reduced cohomology of polyhedral complements $N$4 (Altmann et al., 2024). This makes the cohomology of toric reflexive sheaves explicitly polyhedral.

A complementary cohomological criterion uses Weil decorations to prove acyclicity. For a torus-linearised reflexive sheaf $N$5 on a smooth projective toric variety, “nefly decorated” means that the image of the decoration lies in $N$6, equivalently that each decorated divisor is nef. In that case $N$7 is acyclic, and more generally the paper constructs a canonical resolution of $N$8 by split line bundles determined by the decoration (Altmann et al., 28 May 2025). This is especially relevant for tropical interpretation because positivity of the sheaf is reduced to the placement of finitely many polyhedra inside the nef cone.

Weighted and stacky surfaces provide the link to stability and modular counting. On the weighted projective plane $N$9, toric torsion-free sheaves are described by stacky $\Sigma$00-families, and rank $\Sigma$01 toric locally free sheaves of type I are encoded by integers $\Sigma$02, positive integers $\Sigma$03, and projective points $\Sigma$04, with stability reduced to the triangle inequalities

$\Sigma$05

The resulting moduli problem is therefore cut out by explicit integral polyhedral inequalities (Gholampour et al., 2012).

6. Modularity phenomena and the meaning of “modular”

In the classical toric-sheaf literature, “modular” appears through generating functions rather than through modular lattices. For stable rank $\Sigma$06 locally free sheaves on $\Sigma$07 with $\Sigma$08, the generating functions of topological Euler characteristics can be expressed in terms of Hurwitz class numbers and give rise to modular forms of weight $\Sigma$09. This generalizes Klyachko’s computation on $\Sigma$10 and is consistent with $\Sigma$11-duality predictions from physics (Gholampour et al., 2012).

The mechanism is combinatorial. After fixing Chern data, the generating function becomes a sum over triples $\Sigma$12 satisfying divisibility conditions and the triangle inequalities, weighted by an explicit quadratic form. The relevant triples are identified with reduced positive definite binary quadratic forms of fixed discriminant, and the resulting coefficients are Hurwitz class numbers (Gholampour et al., 2012). In this sense, modularity emerges from counting lattice points in a polyhedral region with quadratic weights.

The tropical theory uses the same kind of raw material—filtrations by flats, admissible weight data, and categorical stability—but reorganizes it around matroidal exactness. No theorem in the cited tropical papers identifies the Harder–Narasimhan generating series of $\Sigma$13 with modular forms. A plausible implication is that the adjective “modular” now has two distinct but adjacent meanings: first, the underlying matroid is modular; second, the broader toric-reflexive-sheaf tradition exhibits modular-form behavior when stability data are summed over polyhedral parameter spaces. The weighted-projective-plane formulas suggest that any future enumerative theory for modular tropical toric reflexive sheaves would likely involve the same combination of lattice inequalities, quadratic forms, and theta-type generating functions (Jun et al., 9 Sep 2025, Gholampour et al., 2012).

The current subject is therefore best understood as an overview. Tropical toric reflexive sheaves provide a matroidal and proto-abelian replacement for equivariant reflexive sheaves; modularity of the underlying matroid supplies the exactness and slope inequalities needed for Harder–Narasimhan theory; and the classical toric literature shows that polyhedral stability data can produce modular generating functions. The common substrate across these developments is a rigid combinatorial geometry of fans, filtrations, flats, and polyhedra.