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Log motivic Gysin isomorphisms

This paper constructs Gysin isomorphisms within an axiomatic motivic framework for fs log schemes by formulating purity transformations for log smooth morphisms and demonstrating that these transformations are isomorphisms for specific non-strict morphisms.

Original authors: Doosung Park

Published 2026-07-20

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Doosung Park

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to understand the shape of a city. In the world of mathematics, there is a field called algebraic geometry, which treats shapes (like curves and surfaces) as if they were built from equations. For a long time, mathematicians have had a powerful toolkit called "motivic homotopy theory." Think of this toolkit as a way to take a complex shape, break it down into its most basic building blocks, and study how those blocks fit together, much like a physicist studying atoms to understand a solid object.

One of the most useful tools in this toolkit is the "Gysin isomorphism." You can think of this as a magical translation device. If you have a shape with a smaller shape stuck inside it (like a circle drawn on a sphere), the Gysin isomorphism tells you that the "vibe" or mathematical essence of the whole sphere, once you remove the circle, is exactly the same as the "vibe" of the tube of space immediately surrounding that circle. It's like saying that if you know how a donut is built around a hole, you automatically know how the whole donut relates to the empty space where the hole used to be. This works beautifully for standard shapes, but what happens when the shapes get "logarithmic"? In the world of log geometry, shapes have extra "logarithmic" data attached to them, like invisible tags or labels that tell you how the shape behaves near its edges or singularities. These tags make the shapes much more complex, and the old translation devices often break down when you try to use them on these new, tagged shapes.

This paper, written by Doosung Park, is about fixing those broken translation devices for the world of logarithmic shapes. The author tackles a specific problem: when you have a "log smooth" shape (a shape with these special tags that behaves nicely) and you try to remove a smaller tagged shape from inside it, the usual way of defining "what's left" fails because the tags get in the way. The paper proves that even in this messy, non-standard situation, you can still build a perfect translation device. By inventing a new kind of mathematical space called "divided log spaces" (which act like a flexible, stretchy fabric that can handle these tricky tags), the author shows that the Gysin isomorphism still holds true. They demonstrate that the relationship between the whole shape, the removed part, and the surrounding "tube" of space remains perfectly balanced, even when the shapes are not strictly aligned in the traditional sense. This discovery is a crucial step toward building a complete "six-functor formalism" for log schemes, which is essentially a master rulebook for how to move, transform, and compare these complex logarithmic shapes in a consistent way.

Technical Summary: Log Motivic Gysin Isomorphisms

Problem Statement
The paper addresses the construction of Gysin isomorphisms within the axiomatic motivic setting for fine and saturated (fs) log schemes. While Morel and Voevodsky established Gysin isomorphisms for smooth schemes in $\mathbb{A}^1$ -homotopy theory, extending this to fs log schemes presents a fundamental obstruction: for a closed immersion $Z \to X$ of log schemes, the open complement $X - Z$ does not necessarily exist within the category of fs log schemes if the immersion is not strict.

Previous attempts to resolve this involved local log blow-ups to achieve strictness, but this approach lacked global functoriality and required complex Zariski covers. The paper seeks to formulate a natural isomorphism:
$\Sigma^\infty(X/X - Z) \simeq \Sigma^\infty(N_Z X / N_Z X - Z)$
in the motivic stable homotopy category $SH(S)$, even when $Z \to X$ is a non-strict closed immersion.

Methodology
The author employs the theory of divided log spaces, developed in prior work [15], to overcome the limitations of the category of fs log schemes.

Divided Log Spaces: The paper utilizes the category $lSpc/B$ of divided log spaces over a base $B$ . These are dividing Zariski sheaves on the category of fs log schemes ($lSch/B$) that satisfy axioms inspired by algebraic spaces. Crucially, in $lSpc/B$, the open complement of a closed immersion always exists, and log smooth morphisms admit well-defined blow-ups and normal bundles.
Extension of Motives: The stable motivic homotopy category $SH$ (or a general sheaf of presentably symmetric monoidal $\infty$ -categories $\mathcal{T}$ ) is extended from $lSch/B$ to $lSpc/B$. This extension relies on the equivalence of $\infty$ -categories of sheaves on $lSch/B$ and $lSpc/B$ (Proposition 2.2.1) and utilizes the technique of partial adjoints (Liu-Zheng [7]) to establish the six-functor formalism (existence of $f^\sharp$ , base change, projection formula) for log smooth morphisms in $lSpc/B$.
Deformation to the Normal Cone: The core technical tool is the deformation to the normal cone construction adapted to divided log spaces. The proof reduces the general case to the case of strict closed immersions in log smooth schemes by:
- Using Zariski covers to reduce to log schemes admitting fan charts.
- Employing a base change along a log smooth morphism $g: S' \to S$ (constructed via saturation properties in §3.3) such that the pullback functor is conservative and the resulting morphism is saturated.
- Utilizing the concept of pans (lattice subsets derived from fans) and their subdivisions to handle the combinatorial aspects of the log structures.

Key Contributions and Results

Log Gysin Isomorphism (Theorem 3.4.1): The paper establishes a natural isomorphism in $\mathcal{T}(S)$ for any closed immersion $Z \to X$ of log smooth divided log spaces:
$M_S(X/X - Z) \simeq M_S(N_Z X / N_Z X - Z)$
This result holds without requiring the immersion to be strict, resolving the existence problem of the open complement via divided log spaces.
Thom Transformations (Theorem 4.2.5): As a direct application, the paper proves that the Thom transformation $Th(p, a) := p^\sharp a^*$ is an equivalence of $\infty$ -categories for any log smooth morphism $p: X \to S$ with a section $a: S \to X$ that is a closed immersion. This generalizes the classical Thom isomorphism to the log setting.
Purity for Non-Strict Morphisms (Theorem 5.2.11): The paper investigates the "purity" of morphisms, defined by the isomorphism $f^\sharp \simeq f^* \Sigma_f$ (where $\Sigma_f$ is the purity transformation). While purity is known for strict smooth morphisms, the paper proves that specific non-strict proper morphisms of fs log schemes are pure.
- Specifically, Theorem 1.4.2 demonstrates that the projection $f: W \times_{\mathbb{A}^1} S \to S$ is pure, where $W$ is a specific compactification of $\mathbb{A}^2_{\mathbb{N}} \to \mathbb{A}^1_{\mathbb{N}}$ constructed by gluing specific log schemes ( $V_1, V_2, V_3$ ).
- This result relies on a detailed analysis of cofiber sequences involving open and closed immersions within the constructed log scheme $W$ .

Significance and Claims
The paper claims that these results provide the necessary foundational blocks for the motivic six-functor formalism for fs log schemes.

The Gysin isomorphism and the purity of specific non-strict morphisms are described as essential for establishing the full six-functor formalism in subsequent works [16] and [18].
The use of divided log spaces allows for a unified treatment of strict and non-strict morphisms, avoiding the technical complications of local blow-ups and ensuring functoriality.
The work extends the scope of motivic homotopy theory to include the rich geometry of log schemes, particularly those arising in compactifications and toroidal embeddings, where non-strict morphisms are prevalent.

The author explicitly notes that the paper is a continuation of [17] and focuses on constructing these isomorphisms and transformations within the axiomatic framework, deferring the full development of the six-functor formalism to future publications.

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