---
title: Logarithmic de Rham Cohomology
url: https://www.emergentmind.com/topics/logarithmic-de-rham-cohomology-c013fbb7-55cb-43bb-896d-118123055592
type: topic
---

# Logarithmic de Rham Cohomology

Logarithmic de Rham cohomology is a comprehensive theoretical framework extending the classical de Rham cohomology to spaces equipped with logarithmic structures, notably fine or fs log schemes and pairs consisting of a smooth variety with a normal crossings divisor. It appears as a central object in comparisons among topological, algebraic, and $p$-adic cohomology theories, facilitating a deeper understanding of cycle classes, period maps, and duality in positive and mixed characteristic. Recent advances introduce powerful derived and prismatic techniques, provide key inputs for $p$-adic Hodge theory, and clarify the Hodge-to-de Rham spectral sequence in the logarithmic context.

## 1. Logarithmic de Rham Complex: Construction and Basic Properties

Given a smooth algebraic variety $X$ over a field $k$ and a normal crossings divisor $D \subset X$, the logarithmic de Rham complex $\Omega_X^\bullet(\log D)$ is defined as the subcomplex of the meromorphic de Rham complex $\Omega_X^\bullet(*D)$ consisting of differential forms with at most simple poles along $D$ and whose differentials also have at most simple poles. Locally, if $(U; x_1,\ldots,x_n)$ are coordinates with $D = \{x_1 \cdots x_r = 0\}$, then
\[
\Omega^1_U(\log D) = \bigoplus_{i=1}^r \mathcal{O}_U \cdot \frac{dx_i}{x_i} \oplus \bigoplus_{j=r+1}^n \mathcal{O}_U \cdot dx_j,
\]
with higher degree forms defined by exterior powers. The logarithmic complex sits naturally as a subcomplex,
\[
\Omega_X^\bullet(\log D) \subset \Omega_X^\bullet(*D).
\]
The hypercohomology $\mathbb{H}^n(X,\Omega_X^\bullet(\log D))$ defines the logarithmic de Rham cohomology of $(X,D)$, denoted $H^n_{dR,\log}(X)$ [2303.09932]. This construction is functorial in morphisms of pairs and compatible with various filtrations, most notably the Hodge and conjugate filtrations.

## 2. Logarithmic de Rham–Witt and Derived Cohomologies

For fine log schemes over a perfect field of positive characteristic, the logarithmic de Rham–Witt complex $W_r\Omega_{X,\log}^\bullet$ provides a universal enhancement encoding both de Rham and crystalline information. This complex can be constructed via the Décalage operator $L\eta_p$ acting on pro-systems of log-crystalline complexes:
\[
W_r\Omega_{X,\log}^\bullet = H^\bullet\left(L\eta_p(R\Gamma_{r+1})\right), \qquad W\Omega_{X,\log}^\bullet = R\!\!\!\lim_r\, W_r\Omega_{X,\log}^\bullet.
\]
It comes equipped with Frobenius ($F$), Verschiebung ($V$), and restriction ($R$) operators, satisfying the familiar Dwork–Cartier relations [1808.09120]. For log-smooth schemes of log-Cartier type, this construction is naturally isomorphic to the Hyodo–Kato and Matsuue log de Rham–Witt complexes, preserving all additional structures.

The comparison with log-crystalline cohomology is provided by a canonical morphism
\[
\tau: W\Omega_{X,\log}^\bullet \to A\Omega_X \otimes^L_{A_\text{inf}} W(k),
\]
which is a quasi-isomorphism; at the $W_1$-level, it specializes to the log Cartier isomorphism.

The derived perspective, via log derived de Rham cohomology, uses the cotangent complex of log structures (Gabber–Illusie, Beilinson) and produces comparison theorems:
\[
dR_f^{\log} \xrightarrow{\simeq} R\Gamma_{\mathrm{crys}}((N \to B)/(M \to A))
\]
for G-lci morphisms of prelog rings, with explicit control over filtrations (Hodge, conjugate) and induced spectral sequences [1204.6560].

## 3. Spectral Sequences, Hodge Theory, and Degeneration

The logarithmic de Rham complex admits a natural stupid (Hodge) filtration,
\[
F^p\Omega_X^\bullet(\log D) = 0 \rightarrow \cdots \rightarrow 0 \rightarrow \Omega_X^p(\log D) \rightarrow \Omega_X^{p+1}(\log D) \rightarrow \cdots,
\]
which induces a spectral sequence
\[
E_1^{p,q} = H^q(X,\Omega_X^p(\log D)) \Rightarrow H^{p+q}_{dR,\log}(X).
\]
A foundational result, due to Deligne–Illusie and extended by Kato and Hablicsek to the logarithmic case, asserts that this spectral sequence degenerates at $E_1$ provided $(X,D)$ lifts to $W_2(k)$ when $\operatorname{char}(k) = p > \dim X$. The degeneration is proved geometrically via twisted derived intersections and formality theorems for the Frobenius-pushforward of the log complex [1503.00177].

## 4. Comparison Theorems, Prismatic Cohomology, and $p$-adic Hodge Theory

Logarithmic de Rham cohomology is a key player in $p$-adic comparison isomorphisms. For smooth rigid analytic spaces $X$ with strict simple normal crossings divisor $D$, there is a canonical comparison of log de Rham and $B_{dR}^+$-pro-étale cohomology,
\[
H^i_{\mathrm{proet}}(X \setminus D, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} B_{dR}^+ \cong H^i_{dR}(X, D) \otimes_k B_{dR}^+,
\]
compatible with filtrations and Galois actions. The logarithmic Hodge–de Rham spectral sequence degenerates and yields a Hodge–Tate-type decomposition [1801.01779].

Logarithmic prismatic cohomology provides a further canonical bridge: for log-smooth schemes over bounded prelog prisms, there is a comparison
\[
R\Gamma_{\Prism}(X/A) \otimes_A^{L} A/I \cong R\Gamma_{dR}\bigl((X, M_X)/(A/I, M_A)\bigr).
\]
Natural filtrations (Nygaard and Hodge) coincide under the comparison; the graded pieces are log differentials, recovering Kato's Cartier isomorphism and the Hodge–Tate comparison [2306.00364].

## 5. Logarithmic Classes, Cycle Theory, and Duality

The logarithmic subspace of de Rham cohomology is characterized as the image of $H^n(X, \Omega_X^\bullet(\log D))$ in $H^n_{dR}(X)$. Remarkably, for $X$ smooth over a field of characteristic zero, cycle classes of codimension $d$ are precisely those of Hodge type $(d,d)$ arising from logarithmic classes. Conversely, any such logarithmic class comes from an algebraic cycle, forming a bridge to the Tate and Hodge conjectures [2303.09932]. The p-adic analytic analogue, established via rigid-analytic spaces and formal models, mirrors the algebraic case.

In positive characteristic, logarithmic de Rham–Witt sheaves support the study of wild ramification and class field theory. A perfect duality is established:
\[
H^i(U, W_m\Omega_{X,\log}^r(D))_\mathrm{disc} \times \varprojlim_{D' \leq D} H^{d+1-i}(X, W_m\Omega_{X,\log}^{d-r}(D'))_\mathrm{pro} \to \mathbb{Z}/p^m\mathbb{Z}
\]
for $X$ smooth proper over a finite field and $D$ an effective divisor, generalizing Serre–Grothendieck duality and underpinning higher dimensional class-field theory [1611.08720].

## 6. D-Module Theory and Logarithmic Comparison Theorems

The logarithmic comparison theorem (LCT) asks when the inclusion $\Omega_X^\bullet(\log D) \to \Omega_X^\bullet(*D)$ is a quasi-isomorphism. Using D-module theory, LCT holds for locally quasihomogeneous free divisors $D$ and is characterized by the isomorphism $D_X \otimes_{V_X} \mathcal{O}_X(D) \simeq \mathcal{O}_X(*D)$ in $D^b(D_X)$. For normal crossings divisors, explicit calculations confirm this, recovering topological cohomology. The failure of LCT for more general, non-quasihomogeneous free divisors highlights the subtlety of the Koszul property and the complexity of logarithmic stratifications [2303.05417].

## 7. Advanced Structures: Chiral de Rham, Log-Jet Spaces, and Extended Symmetries

The chiral de Rham complex admits a logarithmic counterpart $\Omega_X^{ch}(\log D)$, a sheaf of vertex algebras with weight-zero subspace $\Omega_X^\bullet(\log D)$. This enhancement encodes additional topological and superconformal structures, especially for log Calabi–Yau pairs. The $(q, y)$-character of its cohomology interpolates between the $\chi_y$-genus and Euler characteristic. Furthermore, the log chiral complex admits a birational description in terms of the sheaf of log forms on a suitably modified log-jet space, providing a geometric model for the interplay between jets, log structures, and vertex algebraic symmetry [2510.04515].

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**Selected References:**
- "Logarithmic de Rham--Witt complexes via the Décalage operator" [1808.09120]
- "p-adic derived de Rham cohomology" [1204.6560]
- "Duality for relative logarithmic de Rham-Witt sheaves and wildly ramified class field theory over finite fields" [1611.08720]
- "Hodge theorem for the logarithmic de Rham complex via derived intersections" [1503.00177]
- "Logarithmic Comparison Theorems" [2303.05417]
- "Logarithmic prismatic cohomology II" [2306.00364]
- "Logarithmic jets and the chiral de Rham complex of a pair" [2510.04515]
- "De Rham logarithmic classes and Tate conjecture" [2303.09932]

Source: https://www.emergentmind.com/topics/logarithmic-de-rham-cohomology-c013fbb7-55cb-43bb-896d-118123055592