Logarithmic De Rham Cohomology

• Logarithmic de Rham cohomology is a framework that extends classical de Rham theory to log schemes, effectively handling singular divisors and arithmetic structures.
• It utilizes complexes of logarithmic differentials and derived constructions to bridge analytic, crystalline, and prismatic cohomology theories.
• Comparison theorems and duality results validate its role in relating Hodge, $p$-adic, and de Rham–Witt complexes, thereby enhancing both theoretical insights and computational techniques.

Logarithmic de Rham cohomology studies sheaf-theoretic and derived invariants attached to varieties or log schemes endowed with logarithmic structures, extending the classical de Rham cohomology theory to accommodate singularities (such as divisors with normal crossings) and arithmetic structures relevant to $p$-adic geometry, crystalline theory, and comparison results. At its core are logarithmic analogs of the de Rham complex, the associated cohomology, and derived constructions, with deep connections to cycle classes, Hodge theory, $p$-adic Hodge theory, and modern prismatic and Witt cohomological frameworks.

1. Definitions and Fundamental Complexes

A log-scheme, in the sense of Kato, is a scheme $\underline X$ equipped with a sheaf of monoids $M_X \to \mathcal O_X$; it is called fine if locally $M_X$ admits a chart by a fine monoid. For a pair $(X, D)$, with $D \subset X$ a reduced divisor (often normal crossings), the logarithmic de Rham complex $\Omega_X^\bullet(\log D)$ is defined via sheaves of differential forms with at worst logarithmic singularities along $D$, i.e., generated locally by $dx_i/x_i$ along $D$ and by $dy_j$ transversely (Castro-Jiménez et al., 2023, Hablicsek, 2015, Miyatani, 2014).

In the algebraic setting, the logarithmic de Rham subcomplex $\Omega_{X,\log}^\bullet \subset \Omega_X^\bullet$ consists Zariski-locally of forms generated by wedge products of $d f/f$, for units $f \in \mathcal O_X^\times$; its cohomology,

$H_{\dR,\log}^*(X) := H^*\big(X_{\et},\Omega_{X,\log}^\bullet\big),$

embeds naturally in the full de Rham cohomology (Bouali, 2023).

Over a log-scheme $(X, M)$, the sheaf of logarithmic Kähler differentials $\Omega^1_{X/S,\log}$ is locally free with generators $d\log(m)$ for $m \in M_X$ and $df$ for $f \in \mathcal O_X$, subject to standard relations. The log de Rham complex then arises by iterated exterior powers.

2. Comparison Theorems and Geometric Realizations

Logarithmic de Rham cohomology is robust under comparison with other cohomological theories:

• Analytic Case and the Logarithmic Comparison Theorem (LCT): For a complex manifold $X$ with free, locally quasi-homogeneous divisor $D$, the inclusion $\Omega_X^\bullet(\log D) \to \Omega_X^\bullet(*D)$ is a quasi-isomorphism; the logarithmic and meromorphic de Rham cohomology both compute the cohomology of the complement $U=X-D$ (Castro-Jiménez et al., 2023).
• Crystalline and Higher-Level Theories: Through the log jet complex construction, one identifies log crystalline cohomology of higher level with the hypercohomology of a logarithmic de Rham-type complex, under suitable local freeness and lifting hypotheses (Miyatani, 2014). The formal Poincaré lemma holds modulo $p$ for the logarithmic jet complex:

$$H^i\big(L\dot\Omega_{X/S}^{(m),\bullet}\big) = 0 \ (i>0)$$

under the conditions $p\cdot\mathcal O_X=0$, log-smoothness, and flatness.

• Derived and Prismatic Approaches: The derived logarithmic de Rham complex, constructed via suitable resolutions in the category of prelog rings, naturally compares to log crystalline cohomology for log-lci maps,

$\dR_f \xrightarrow{\simeq} R\Gamma_{\rm crys}(f),$

supporting filtered $E_\infty$-structures and period ring realizations (e.g., $A_{\rm cris}$) (Bhatt, 2012). The log prismatic cohomology framework further provides a canonical φ-equivariant isomorphism:

$$R\Gamma_{\log\mathrm{dR}}(X/A/I\!, M_A) \cong R\Gamma_{\mathrm{prism, log}}(X/A, M_A) \otimes^L_{A, \varphi_A} A/I,$$

with an explicit Nygaard filtration and compatible functoriality (Koshikawa et al., 2023).

3. Degeneration and Hodge-Theoretic Properties

The Hodge-to-de Rham spectral sequence for $(X,D)$ with normal crossings divisor is given by

$$E_1^{p,q} = H^q\big(X, \Omega_X^p(\log D)\big) \implies H^{p+q}_{\mathrm{dR}}(X;\log D).$$

The E$_1$-degeneration is known under liftability to $W_2(k)$ (Kato) and can be proven via derived intersection methods, which relate the formality of the log de Rham complex under Frobenius to the triviality of an associated Azumaya algebra class (Hablicsek, 2015).

4. Logarithmic de Rham–Witt Complexes and Arithmetic Duality

For log schemes (especially in finite characteristic), the de Rham–Witt complex $W_n\Omega^\bullet_X$ admits a logarithmic variant, $W_n\Omega_X^\bullet(\log D)$, defined as the $W_n\mathcal O_X$-submodule of $j_* W_n\Omega^\bullet_U$ generated étale-locally by forms $d\log [x_1]_n \wedge \cdots \wedge d\log [x_r]_n$ (where $x_i$ are invertible functions on the open complement). These sheaves possess Frobenius ($F$), Verschiebung ($V$), and Cartier operators with exact sequences linking various pole structures (Jannsen et al., 2016).

• Duality: For $X$ of dimension $d$ over a perfect field, with normal crossing divisor $D$, there is a perfect duality pairing between the cohomology of $W_n\Omega_U^r(\log D)$ and a projective limit over interpolating modules $W_n\Omega_X^{d-r}|D'$ (Jannsen et al., 2016). This duality can be applied to the study of wild ramification in class field theory.
• Décalage Formalism: The strict saturated log Dieudonné algebra structure on the de Rham–Witt complex allows recovery of classical results (Hyodo–Kato, Matsuue), monodromy operators, and explicit comparison with $A_{\inf}$-cohomology (Yao, 2018).

5. Cycle Theory, Log Classes, and Applications

Logarithmic de Rham cohomology identifies and characterizes De Rham cycle classes:

• Every cycle class $[Z]$ in $H^{2d}_{\dR}(X)$ is logarithmic.
• Logarithmic cohomology classes of type $(d,d)$ (i.e., in the intersection $H^{2d}_{\dR,\log}(X) \cap F^d$) arise as De Rham classes of cycles of codimension $d$.
• There is vanishing outside even bidegrees, and these correspondences extend to $p$-adic analytic settings, underpinning the Tate conjecture for varieties with good reduction (Bouali, 2023).

6. Examples and Computational Illustrations

• No-log case: When $M_X = \mathcal O_X^\times$ and $m=0$, the logarithmic de Rham complex reduces to the classical de Rham complex and recovers the classical results (e.g., Berthelot–Ogus Poincaré lemma) (Miyatani, 2014).
• Simple normal crossings: In $X = \mathbb C^n$ with $D=\{x_1\cdots x_r=0\}$,

$$\Omega_X^1(\log D) = \bigoplus_{i=1}^r \mathcal O_X \cdot \frac{dx_i}{x_i} \oplus \bigoplus_{j=r+1}^n \mathcal O_X \cdot dx_j,$$

yields explicit computations matching topological cohomology (e.g., $H^k((\mathbb C^*)^r;\mathbb C)$) (Castro-Jiménez et al., 2023).

• Hyperplane arrangements and weighted homogeneous curves: The log complex agrees with well-known combinatorial and geometric results (e.g., Brieskorn’s and Terao’s theorems) (Castro-Jiménez et al., 2023).
• Derived prismatic context: In log affine settings with free monoid charts, graded pieces of log prismatic cohomology align with log Hodge–Tate decomposition and Hyodo–Kato components (Koshikawa et al., 2023).

7. Structural Features and Future Directions

Logarithmic de Rham cohomology acts as a bridge across Hodge, crystalline, prismatic, and $p$-adic theories, with formal properties ensuring compatibility with base change, filtrations (Hodge, Nygaard), and functoriality at both the complex and derived levels (Koshikawa et al., 2023, Bhatt, 2012). Open questions remain regarding local freeness of higher-level jet complexes in general, with conjectural extensions of base-change and duality theorems outside the zero-characteristic or vanishing-$p$ regime (Miyatani, 2014). The compatibility with period ring constructions, duality in ramified and irregular settings, and deeper stack-theoretic or prismatic perspectives represent current directions of ongoing research.