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Logarithmic Hodge Complexes And Smooth Open Weights: Examples

1 · Prerequisites

2 · Summary

This companion to logarithmic-hodge-complexes-and-smooth-open-weights runs the smallest noncompact example of the logarithmic construction. The punctured line has a weight-two logarithmic class takes $U = C^{\times} \subset P^{1}$ with boundary the two points $0$ and $\infty$ : the weight spectral sequence of the logarithmic complex has only the column $a = 0$ with $H^{0}$ and $H^{2}$ of $P^{1}$ and the column $a = 1$ with $H^{0} ({0, \infty}; Q) (- 1) = Q (- 1)^{2}$ , the single nonzero differential sends a pair of residues to their sum, and its kernel is the line spanned by $(1, - 1)$ , so ${Gr}_{2}^{W} H^{1} (U; Q) = Q (- 1)$ while all other weight-graded pieces of $H^{1}$ vanish. The class $(2 π i)^{- 1} d z / z$ , whose normalized rational residue coordinates at $0$ and $\infty$ are $1$ and $- 1$ , is the rational generator. The example therefore exhibits a cohomology class whose weight $2$ is strictly larger than its cohomological degree $1$ , with Hodge type $(1, 1)$ ; it is a leaf on which nothing else depends. Its verification uses the full logarithmic theorem, including the rational comparison and the Hodge-compatible identification of the weight-graded cohomology.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedOpen item page →

The punctured line has a weight-two logarithmic class

Example

Assume AC (The Axiom of Choice). For $U = C^{\times} \subset P^{1}$ , $H^{1} (U, Q) = Q (- 1)$ , of weight $2$ and type $(1, 1)$ . In particular its weight is strictly larger than its cohomological degree.

Facts & Assumptions

Given: The Axiom of Choice; the smooth quasiprojective curve $U = C^{\times} = P^{1} (C) ∖ {0, \infty}$ with its SNC compactification $j : U ↪ X = P^{1}$ , boundary $D = {0} ⊔ {\infty}$ (Simple normal crossings divisors and simultaneous normal crossings position).

[F1]

The strata of the compactification. $D^{[0]} = X = P^{1}$ , $D^{[1]} = D$ is the disjoint union of the two points $0$ and $\infty$ , and $D^{[a]} = \emptyset$ for $a \geq 2$ ; all are smooth projective complex varieties of pure dimensions $1$ and $0$ . The two projective coordinate charts identify the analytic space of $P^{1}$ with the Riemann sphere, homeomorphic to $S^{2}$ by Stereographic projection identifies the Riemann sphere with the unit two-sphere. Under $C ≅ R^{2}$ , For $n \geq 1$ , radial normalisation is a deformation retraction of $R^{n} ∖ {0}$ onto $S^{n - 1}$ gives the deformation retraction $H (z, t) = ((1 - t) + t / ∣ z ∣) z$ onto $S^{1}$ , so inclusion induces a homology isomorphism by Singular homology is invariant under deformation retracts.

[F2]

Cohomology of spheres (Homology of spheres, Cohomology over a field is dual to homology over that field). Over $Q$ , $H_{0} (S^{1}) = H_{1} (S^{1}) = Q$ and $H_{n} (S^{1}) = 0$ otherwise, and the same pattern holds for $S^{2}$ in degrees $0$ and $2$ ; over the field $Q$ evaluation identifies $H^{n} (-; Q)$ with the full linear dual of $H_{n} (-; Q)$ . Hence $H^{0} (S^{1}; Q) = H^{1} (S^{1}; Q) = Q$ with higher groups zero, and $H^{0} (S^{2}; Q) = H^{2} (S^{2}; Q) = Q$ with $H^{1} (S^{2}; Q) = 0$ ; a point and a two-point space have cohomology $Q$ and $Q^{2}$ in degree $0$ and zero in positive degrees.

[F3]

The logarithmic mixed Hodge structure (Logarithmic construction of smooth-open mixed Hodge structures). For the SNC compactification of [F1] the logarithmic complex endows $H^{n} (U; Q)$ with its weight filtration $W$ , of weights at least $n$ , satisfying $W_{n} H^{n} (U; Q) = im (H^{n} (X; Q) \to H^{n} (U; Q))$ , with weight spectral sequence $E_{1}^{- a, q} = H^{q - 2 a} (D^{[a]}; Q) (- a) \Rightarrow H^{q - a} (U; Q)$ degenerating at $E_{2}$ and ${Gr}_{m}^{W} H^{n} (U; Q) ≅ E_{2}^{n - m, m}$ ; the local residue normalization of that construction assigns to the closed logarithmic form $d z / z$ on $U$ the rational class $(2 π i)^{- 1} d z / z \in H^{1} (U; Q)$ , whose normalized rational residue coordinates at $0$ and at $\infty$ are $1$ and $- 1$ ; the ordinary analytic residue of the normalized form is $(2 π i)^{- 1}$ times the corresponding coordinate.

[F4]

Tate twist and type (Rational mixed Hodge structures). The Tate structure $Q (- 1)$ is the pure rational Hodge structure of weight $2$ and type $(1, 1)$ , concentrated in $H^{1, 1}$ ; a mixed Hodge structure whose weight-graded pieces are ${Gr}_{2}^{W} = Q (- 1)$ and ${Gr}_{m}^{W} = 0$ for $m \neq 2$ is pure of weight $2$ and type $(1, 1)$ .

[A1]

AC is assumed and is consumed only through [F3] and its suppliers; the finite computations below are choice-free. (The Axiom of Choice)

Verification

technique · direct: compute both sides of the weight spectral sequence of the logarithmic complex for $\mathbb P^1$ and identify the surviving degree-one term with the Tate twist

1.1F1F2givenalgebra

The two cohomologies. By [F1] and [F2], $H^{0} (X; Q) = Q$ , $H^{1} (X; Q) = 0$ , $H^{2} (X; Q) = Q$ ; $H^{0} (D^{[1]}; Q) = Q^{2}$ and $H^{1} (D^{[1]}; Q) = 0$ ; and $H^{0} (U; Q) = Q$ , $H^{1} (U; Q) = Q$ , $H^{2} (U; Q) = 0$ , the generator being the meridian class dual to the circle.

2.1F3step 1.1algebra

The logarithmic generator. By the residue normalization of [F3] the class $(2 π i)^{- 1} d z / z$ is a nonzero rational class in $H^{1} (U; Q)$ pairing by $1$ with the positively oriented meridian, so it generates the one-dimensional space $H^{1} (U; Q)$ of step 1.1; its normalized rational residue coordinates at $0$ and $\infty$ are $1$ and $- 1$ , because in the coordinate $w = 1 / z$ at infinity the form $d z / z$ equals $- d w / w$ .

3.1F3step 1.1step 2.1algebra

The weight spectral sequence. Only $a = 0, 1$ contribute. The $a = 0$ column has $E_{1}^{0, 0} = Q (0)$ , $E_{1}^{0, 1} = 0$ , $E_{1}^{0, 2} = H^{2} (P^{1}; Q) = Q (- 1)$ , pure of weight $2$ and type $(1, 1)$ ; the $a = 1$ column has $E_{1}^{- 1, 2} = Q (- 1)^{2}$ and is otherwise zero. The sole possible nonzero differential is $d_{1} : E_{1}^{- 1, 2} \to E_{1}^{0, 2}$ , a pure Hodge morphism. Since the sequence degenerates at $E_{2}$ and $H^{2} (U) = 0$ by step 1.1, its cokernel is zero, so its rank is one. The global closed logarithmic form of step 2.1 has normalized residue vector $(1, - 1)$ , which must survive this differential because it represents an actual cohomology class; hence this vector lies in its kernel. A rank-one map from a two-dimensional space has one-dimensional kernel, so $\ker d_{1} = Q (1, - 1) ≅ Q (- 1)$ . Equivalently, in a rational target generator normalized to the nonzero image of $(1, 0)$ , $d_{1} (r_{0}, r_{\infty}) = r_{0} + r_{\infty}$ . Thus $E_{2}^{- 1, 2} = Q (- 1)$ , $E_{2}^{0, 2} = 0$ , $E_{2}^{0, 0} = Q (0)$ , and all remaining terms vanish. This computes the required weight-graded piece without assuming an unproved Gysin normalization.

4.1F3F4step 2.1step 3.1algebra

The mixed Hodge structure on $H^{1} (U, Q)$ . By [F3] the weight-graded piece ${Gr}_{m}^{W} H^{1} (U; Q)$ is $E_{2}^{1 - m, m}$ , so by step 3.1 it is $Q (- 1)$ for $m = 2$ and vanishes for every other $m$ ; in particular $W_{1} H^{1} (U; Q) = 0$ and $W_{2} H^{1} (U; Q) = H^{1} (U; Q)$ , which also agrees with the weights-at-least- $1$ and image formulas of [F3] and with $H^{1} (X; Q) = 0$ from step 1.1. By [F4] the class supported in weight $2$ with ${Gr}_{2}^{W} = Q (- 1)$ makes $H^{1} (U; Q)$ pure of weight $2$ and type $(1, 1)$ , so $H^{1} (U, Q) = Q (- 1)$ ; the classical generator $(2 π i)^{- 1} d z / z$ of step 2.1 is a rational class of this structure.

5.1A1F4step 2.1step 4.1given∎

Conclusion and boundaries. The weight of $H^{1} (U, Q)$ is $2$ , strictly larger than the cohomological degree $1$ , and the only nonzero Hodge number is $h^{1, 1} = 1$ . The space $U$ is nonempty and disconnected cases do not arise; $H^{0}$ and $H^{2}$ of $U$ are handled by step 1.1 and carry weights $0$ and at most $2$ ; the two residue coordinates are the two components of $D$ , and the kernel computation is finite linear algebra over $Q$ ; no choice beyond [A1] is used.

Remarks

The example is a leaf. It consumes the full logarithmic mixed Hodge theorem, including the normalized rational comparison, the shifted weight spectral sequence and its Hodge-compatible abutment; the pure type $(1, 1)$ conclusion requires that full supplier, rather than only its vector-space spectral sequence. The ordinary analytic residues of $d z / z$ are $(1, - 1)$ ; after multiplication by $(2 π i)^{- 1}$ the normalized rational coordinates still are $(1, - 1)$ , while the analytic residues themselves are scaled. AC is carried explicitly in the statement because it is inherited through the theorem.