Coherent sheaves in logarithmic geometry

1. Problem Statement

The paper addresses a fundamental gap in logarithmic geometry: the lack of a well-behaved abelian category of coherent sheaves that remains stable under simple normal crossing (SNC) degenerations.

• Context: When a smooth projective variety $X$ degenerates to a singular scheme $X_0$ with SNC singularities, the behavior of moduli spaces of coherent sheaves (e.g., Picard schemes, Hilbert schemes, Donaldson-Thomas moduli) becomes unpredictable. Standard moduli spaces often fail to be universally closed or fail to admit degeneration formulas.
• Existing Approaches: Previous work (e.g., Gieseker, Maulik-Ranganathan) constructs good degenerations by studying sheaves across all expansions (log modifications) and root stacks of $X_0$. However, these constructions lacked a unified homological algebra framework.
• The Core Question: Is there a well-behaved abelian category of coherent sheaves defined "across all expansions" of $X_0$ that allows for standard homological operations (kernels, cokernels, derived functors) and unifies various logarithmic moduli spaces?

2. Methodology

The authors introduce Logarithmic Coherent Sheaves defined within the logarithmic étale topology ($X_{\text{log-ét}}$). Their methodology relies on a "calculus of ups and downs" to reduce complex logarithmic computations to conventional algebraic geometry on specific alterations.

A. The Logarithmic Étale Topology

• They define the category of logarithmic coherent sheaves, denoted $\text{LCoh}(X)$, as coherent sheaves on the ringed topos $(X_{\text{log-ét}}, \mathcal{O}_{X_{\text{log-ét}}})$.
• The topology is generated by strict étale covers and logarithmic alterations (compositions of log modifications and root stacks).
• Key Challenge: Logarithmic alterations are not flat, meaning the standard pullback functor does not commute with kernels, threatening the abelian nature of the category.

B. The "Yoga of Ups and Downs"

To overcome the non-flatness of log alterations, the authors introduce an adjoint pair of functors between conventional sheaves on a log alteration $X'$ and logarithmic sheaves on $X$:

1. Up Functor ($\Uparrow$): Maps a conventional sheaf on $X'$ to a logarithmic sheaf on $X$ (via sheafification).
2. Down Functor ($\Downarrow$): Maps a logarithmic sheaf on $X$ back to a conventional sheaf on $X'$.

The central insight is that while $\Uparrow$ is not generally exact, it becomes well-behaved when restricted to a specific class of sheaves called Static Quasi-Coherent Sheaves.

C. Static Quasi-Coherent Sheaves

• Definition: A sheaf is static if it has Tor dimension $\le 1$ over the Artin fan of $X$ (or equivalently, over the stack of log structures $\text{Log}_X$).
• Role: Static sheaves behave like "flat" sheaves in the logarithmic context.
• Statification Theorem: Using results from Gabber and Thompson, the authors prove that for any coherent sheaf $F$, there exists a log modification $q: X' \to X$ such that $q^*F$ is static.
• Algorithm: They provide a combinatorial algorithm using Gröbner stratifications and tropical geometry to explicitly construct these statifications for schemes of finite type.

3. Key Contributions and Results

I. The Abelian Category of Logarithmic Coherent Sheaves

• Theorem A: The category $\text{LCoh}(X)$ is an abelian category.
• Characterization: An object in $\text{LCoh}(X)$ is coherent if and only if it is isomorphic to $\Uparrow F$ for some conventional coherent sheaf $F$ on a log alteration $X'$.
• Resolution of Descent Issues: The paper resolves subtleties regarding logarithmic étale descent (addressing questions by Molcho and Wise) by showing that the structure sheaf $\mathcal{O}_{X_{\text{log-ét}}}$ is coherent after a suitable log modification.

II. Computational Tools (The Calculus)

The authors develop a suite of tools to compute derived functors (pushforward, pullback, tensor, Hom) by reducing them to conventional calculations on static sheaves:

• Theorem E: Derived functors preserve logarithmic (quasi-)coherence.
• Theorem F (Derived Pushforward): If $F$ is static and $R^i f_* F$ remains static, then $R^i f_{\text{log-ét}, *} (\Uparrow F) \cong \Uparrow (R^i f_* F)$.
• Algorithm: By iteratively applying statification (Theorem C) and base change, one can compute any derived functor on a logarithmic coherent sheaf by reducing it to a sequence of conventional computations on root stacks and modifications.

III. Moduli of Invertible Sheaves and Chip Firing

The paper connects the theory of logarithmic coherent sheaves to the Logarithmic Picard Group ($\text{LogPic}$) and the combinatorics of chip firing.

• The Moduli Stack $M_1(C/S)$: The stack of invertible logarithmic coherent sheaves (log line bundles).
• The Morphism $\varpi$: They construct a morphism $\varpi: M_1(C/S) \to \text{LogPic}(C/S)$.
• Theorem G: $\varpi$ satisfies a universal mapping property for "descending logarithmic algebraic spaces." It acts as a categorical moduli space (analogous to a good moduli space).
• $\ell$-Equivalence and Chip Firing:
  • Two log line bundles are identified by $\varpi$ if they are $\ell$-equivalent.
  • This equivalence is detected by the Interval $T_I$ (a logarithmic analogue of the stack $[\mathbb{A}^1/\mathbb{G}_m]$ used in S-equivalence).
  • Significance: The paper interprets chip firing (a combinatorial operation on graphs) as the tropical shadow of $\ell$-equivalence. Specifically, moving between two points in the moduli space via the interval corresponds to applying chip firing moves.

4. Significance and Impact

1. Unified Framework: The paper provides the first rigorous abelian category for logarithmic coherent sheaves, unifying previously disparate constructions like logarithmic Quot spaces, logarithmic Donaldson-Thomas theory, and logarithmic Picard groups.
2. Homological Algebra: It establishes that standard homological algebra (derived categories, duality) can be performed in the logarithmic setting by reducing to static sheaves. This paves the way for a "Logarithmic Derived Category" (explored in a companion paper).
3. Bridging Geometry and Combinatorics: The identification of chip firing as a geometric equivalence relation ($\ell$-equivalence) offers a new geometric interpretation of tropical combinatorics. It suggests that tropical vector bundles defined via chip firing are the correct higher-rank generalizations of logarithmic line bundles.
4. Solving Open Problems: It answers specific questions posed by Molcho and Wise regarding the coherence of the logarithmic structure sheaf and the descent properties of logarithmic algebraic spaces.

Summary

This paper constructs a robust foundation for studying sheaves on degenerating varieties by defining logarithmic coherent sheaves. By introducing static sheaves and the up/down functor calculus, the authors enable the computation of derived functors in the logarithmic setting. Furthermore, they reveal a deep connection between the moduli of log line bundles and chip firing, interpreting the latter as a logarithmic version of S-equivalence. This work is essential for advancing logarithmic enumerative geometry and the study of moduli spaces in degeneration.