Logarithmic Quot Spaces

• Logarithmic Quot spaces are logarithmic analogues of Grothendieck’s Quot schemes that parametrize quotient sheaves on logarithmic modifications with built-in tropical and boundary data.
• They enforce a transversality condition through logarithmic flatness to ensure algebraic coherence and moduli representability in degenerations and Donaldson–Thomas theory.
• The framework integrates root stacks, log alterations, and tropicalizations to yield finite type moduli spaces with canonical minimal transversalization and K-tropical balancing conditions.

Searching arXiv for papers on logarithmic Quot spaces and related logarithmic coherent sheaves. Logarithmic Quot spaces are logarithmic analogues of Grothendieck’s classical Quot schemes for simple normal crossings pairs and degenerations. They parameterize quotient sheaves not merely on a fixed underlying scheme, but on logarithmic modifications, expansions, and root stacks, subject to a transversality condition formulated as logarithmic flatness or algebraic transversality. In the trivial logarithmic case they recover the classical Quot scheme; in the logarithmic setting they retain multidegree, boundary, and tropical data needed for degenerations, logarithmic Donaldson–Thomas theory, and related moduli problems (Kennedy-Hunt, 2023).

1. Foundational setting and basic definitions

The standard input is a simple normal crossings pair $(X,D)$, where $X$ is a scheme or smooth projective variety and $D$ is a reduced simple normal crossings divisor. One equips $X$ with the divisorial logarithmic structure induced by $D$, obtaining a fine and saturated logarithmic scheme. The associated characteristic monoid is

$\overline{M}_X := M_X/\mathcal{O}_X^\times.$

In the framework of logarithmic geometry, the boundary strata of $(X,D)$ are encoded by the Artin fan and its cone complex, and logarithmic modifications correspond, strict-étale locally, to subdivisions of the associated cone complex (Kennedy-Hunt et al., 11 Aug 2025).

For logarithmic Quot theory, the central geometric condition is algebraic transversality, also described as logarithmic flatness. For a coherent sheaf $\mathcal E$ on $X$, algebraic transversality means flatness over the Artin fan $\mathsf A(X,D)$. For a subscheme $X$0, it means that the local equations cutting out the components of $X$1 form a regular sequence in $X$2 at every point; equivalently, $X$3 is flat over the Artin fan. On expansions $X$4, the corresponding condition is flatness over the broken Artin fan $X$5 (Kennedy-Hunt et al., 11 Aug 2025).

In the foundational construction of the logarithmic Quot space, one fixes a coherent sheaf $X$6 on a projective fs log scheme $X$7. Over an fs log scheme $X$8, a logarithmic surjection is represented by a pair $X$9, where $D$0 is a logarithmic modification and

$D$1

is a surjection of coherent sheaves such that both $D$2 and $D$3 are logarithmically flat and integral over $D$4. Two such families are identified if they agree after pullback to a common further logarithmic modification. The logarithmic Quot space is the stackification of this moduli problem; when $D$5, one obtains the logarithmic Hilbert space (Kennedy-Hunt, 2023).

A more categorical definition appears in the theory of logarithmic coherent sheaves. There one works on the small logarithmic étale site $D$6, whose covers are generated by strict étale morphisms together with log alterations, meaning composites of logarithmic modifications and root stacks. A logarithmic coherent sheaf on $D$7 is a coherent $D$8-module on $D$9, and a logarithmic Quot functor is then defined by surjections in the abelian category $X$0 with logarithmically flat quotient and prescribed multi-Hilbert polynomial along the strata of the SNC boundary (Dell et al., 6 Apr 2026).

2. Moduli problem, equivalence, and representability

The logarithmic Quot functor may be expressed in two closely related forms. In the expansion-based approach, it associates to a logarithmic base $X$1 equivalence classes of algebraically transverse quotients on logarithmic modifications of $X$2. In the log-coherent approach, for a morphism $X$3 of fs log schemes and a logarithmic coherent sheaf $X$4, one defines

$X$5

by sending $X$6 to isomorphism classes of surjections $X$7 in $X$8, where $X$9 is logarithmically flat over $D$0 and has prescribed multi-Hilbert polynomial $D$1 (Dell et al., 6 Apr 2026).

Equivalence of families is essential because a quotient may naturally live on different expansions or logarithmic modifications. The equivalence relation is defined by pullback to a common refinement: if two representatives agree after pulling back to a further logarithmic modification, they represent the same logarithmic quotient. This formulation is the logarithmic counterpart of regarding expansions as auxiliary models rather than part of the intrinsic moduli datum (Kennedy-Hunt, 2023).

Representability is established in a logarithmic sense. The foundational result constructs the logarithmic Quot space as a logarithmic algebraic space: it is separated, universally closed, and admits a strict-étale cover by Deligne–Mumford stacks with log structure. Over a tropical model of the space of tropical supports, the corresponding pullback is algebraic, and its fibers over tropical data are algebraic (Kennedy-Hunt, 2023).

Within the later categorical framework, representability is reformulated using the notion of a descending logarithmic algebraic space. For projective $D$2 with fs log structures, $D$3 locally Noetherian, and $D$4, the logarithmic Quot functor is a sheaf on the big log-étale site, is obtained as the log-étale sheafification of a colimit of classical Quot functors on a cofiltered diagram of log alterations, and is algebraic and locally of finite presentation over $D$5. In settings arising in degeneration theory and logarithmic Donaldson–Thomas theory, properness or projectivity is ensured under the boundedness and properness hypotheses inherited from suitable static alterations and root stacks (Dell et al., 6 Apr 2026).

A classical limit is built into the theory. When the logarithmic structure is trivial, the logarithmic étale site reduces to the usual étale site, logarithmic modifications do not produce new moduli, and the logarithmic Quot functor reduces to Grothendieck’s classical Quot scheme (Kennedy-Hunt, 2023).

3. Logarithmic coherent sheaves and the up/down calculus

A major structural advance is the rephrasing of logarithmic Quot theory inside an abelian category of logarithmic coherent sheaves. For a logarithmic scheme $D$6, the category $D$7 consists of coherent $D$8-modules on $D$9. In the locally Noetherian case, coherence is equivalent to finite presentation. The main theorem states:

“$\overline{M}_X := M_X/\mathcal{O}_X^\times.$0 is abelian. An $\overline{M}_X := M_X/\mathcal{O}_X^\times.$1-module $\overline{M}_X := M_X/\mathcal{O}_X^\times.$2 is coherent if and only if there exists a conventional coherent sheaf $\overline{M}_X := M_X/\mathcal{O}_X^\times.$3 on some log alteration $\overline{M}_X := M_X/\mathcal{O}_X^\times.$4 with $\overline{M}_X := M_X/\mathcal{O}_X^\times.$5.” (Dell et al., 6 Apr 2026)

This theorem gives a precise sense in which a logarithmic coherent sheaf is a compatible system of ordinary coherent sheaves across all expansions and root stacks, glued through the logarithmic étale topology. The site-theoretic limit description

$\overline{M}_X := M_X/\mathcal{O}_X^\times.$6

encodes that compatibility.

The key mechanism is the adjoint pair

$\overline{M}_X := M_X/\mathcal{O}_X^\times.$7

attached to each log alteration $\overline{M}_X := M_X/\mathcal{O}_X^\times.$8. This “yoga of ups and downs” reduces homological algebra on $\overline{M}_X := M_X/\mathcal{O}_X^\times.$9 to conventional sheaf theory on suitable alterations. Although $(X,D)$0 is neither left exact nor fully faithful in general, the main theorem identifies logarithmic coherent sheaves as those arising from coherent sheaves on suitable alterations (Dell et al., 6 Apr 2026).

Root stacks are part of this infrastructure. For invertible Kummer extensions one obtains root stacks

$(X,D)$1

which are proper and finitely presented and satisfy $(X,D)$2. Logarithmic alterations are composites of log modifications and root stacks, and therefore the logarithmic Quot theory records both expansion data and root-index data in a single formalism (Dell et al., 6 Apr 2026).

This categorical perspective also clarifies a frequent source of confusion. Logarithmic Quot spaces are not merely Quot schemes on one chosen expansion. Rather, they encode equivalence classes across all relevant expansions and root stacks, with the logarithmic étale topology enforcing descent. A plausible implication is that the moduli problem is intrinsically logarithmic even when its calculations are performed on specific auxiliary models.

4. Tropicalisation, multidegrees, and K-tropicalizations

A defining feature of logarithmic Quot spaces is the presence of a tropical or polyhedral shadow. In the foundational theory, each logarithmic quotient carries a tropical support, defined as an initial piecewise linear subdivision characterized by stabilizer subtori in star fans and constructed locally via Gröbner-theoretic initial degenerations. The moduli functor of tropical supports is represented by a piecewise linear space, and the logarithmic Quot space carries a strict morphism to this tropical moduli space; the fibers over tropical data are algebraic (Kennedy-Hunt, 2023).

In the log-coherent formulation, the discrete data of a quotient are encoded by a multi-Hilbert polynomial along the strata of the SNC degeneration,

$(X,D)$3

where the summands record contributions on components, double loci, and higher intersections, together with the gluing constraints along the boundary. The balancing of multidegrees is governed by the short exact sequence

$(X,D)$4

on a log alteration $(X,D)$5. In the comparison with logarithmic Picard groups, this sequence yields chip-firing equivalence classes as the combinatorial shadow of logarithmic $(X,D)$6-equivalence (Dell et al., 6 Apr 2026).

The boundedness theory strengthens this tropical picture by introducing K-tropicalization. For a logarithmic quotient on an expansion $(X,D)$7, the K-weight on a cell $(X,D)$8 is defined by the multigraded Hilbert polynomial

$(X,D)$9

or, in the toric case, by the Euler characteristic of the restriction to the stratum. These weights are compatible with subdivisions and descend canonically to the Gröbner stratification. The K-tropicalization thus refines ordinary tropicalization by recording data on all strata and by being sensitive to scheme structure, including embedded components and negative Euler characteristics (Kennedy-Hunt et al., 11 Aug 2025).

The K-theoretic balancing condition is derived from the K-theory of toric bundles and yields explicit linear constraints on the Euler characteristics of restrictions to boundary components. In the toric case, for every character $\mathcal E$0 one obtains

$\mathcal E$1

These relations impose strong finiteness properties and lead to the theorem that K-tropicalizations with fixed numerics occur in only finitely many combinatorial types and are parameterized by a finite-dimensional polyhedral complex (Kennedy-Hunt et al., 11 Aug 2025).

A common misconception is to treat tropicalization here as only cycle-theoretic. The K-tropicalization is explicitly designed to be sensitive to scheme structure rather than just Chow data, and the paper states that it has “the same relationship to K-theory as traditional tropicalization has to Chow” (Kennedy-Hunt et al., 11 Aug 2025).

5. Boundedness, minimal transversalization, and deformation theory

The boundedness problem for logarithmic Quot spaces is solved by combining canonical transversalization with the finiteness of K-tropicalizations. For a coherent sheaf $\mathcal E$2 on an snc pair $\mathcal E$3, there exists a canonical piecewise linear stratification $\mathcal E$4, equivalently a logarithmic space $\mathcal E$5, with the universal property that an snc logarithmic blowup $\mathcal E$6 makes the strict transform algebraically transverse if and only if $\mathcal E$7 factors through a logarithmic modification of $\mathcal E$8. This is the canonical minimal transverse modification (Kennedy-Hunt et al., 11 Aug 2025).

The geometric input behind this universal property is a blowdown criterion based on vertical triviality along exceptional divisors. If $\mathcal E$9 is the blowup of a stratum and a coherent sheaf $X$0 on $X$1 is algebraically transverse and vertically trivial along the exceptional divisor, then $X$2 descends to an algebraically transverse sheaf on $X$3. Organizing this descent through the Gröbner stratification produces the minimal transversalization (Kennedy-Hunt et al., 11 Aug 2025).

With this in hand, boundedness follows. For an snc pair $X$4 and an algebraically transverse coherent sheaf $X$5 on $X$6, every connected component of $X$7 is of finite type, hence proper; fixing a Hilbert polynomial yields a proper space. The proof uses: canonical minimal transversalization, finiteness of K-tropicalizations with fixed numerics, and ordinary finite-type Quot schemes on finitely many fixed expansions (Kennedy-Hunt et al., 11 Aug 2025).

Deformation theory fits naturally into the log-coherent category. For deformations of logarithmic coherent objects, the tangent and obstruction groups are computed by Ext on $X$8: [ T^1 \cong \operatorname{Ext}^1_{X_{\mathrm{log}}}(\mathcal Q,\mathcal Q),\qquad \mathrm{Obs}\in \operatorname{Ext}^2_{X