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A mixed Hodge extension without a rational splitting

Example

A rational mixed Hodge structure can be an extension of two pure structures and still admit no splitting over $Q$ . On $H = Q e_{0} \oplus Q e_{1}$ put $W_{- 3} = 0$ , $W_{- 2} = W_{- 1} = Q e_{1}$ , $W_{0} = H$ , and $F^{1} = 0$ , $F^{0} = C (e_{0} + i e_{1})$ , $F^{- 1} = H_{C}$ . This is a mixed Hodge structure sitting in a short exact sequence $0 \to Q (1) \to H \to Q (0) \to 0$ which has no splitting in the category of rational mixed Hodge structures.

Facts & Assumptions

Given: The rational vector space $H = Q e_{0} \oplus Q e_{1}$ (Rational mixed Hodge structures), the increasing filtration $W_{- 3} = 0$ , $W_{- 2} = W_{- 1} = Q e_{1}$ , $W_{0} = H$ and the decreasing filtration $F^{1} H_{C} = 0$ , $F^{0} H_{C} = C (e_{0} + i e_{1})$ , $F^{- 1} H_{C} = H_{C}$ , with $i \in C$ satisfying $i^{2} = - 1$ and conjugation $σ (z) = \overset{ˉ}{z}$ (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ).

[F1]

A rational mixed Hodge structure is the datum of a finite increasing $W$ and a finite decreasing $F$ such that on every ${Gr}_{n}^{W} H_{C}$ the induced filtrations are $n$ -opposed, i.e. split into the pieces $H_{n}^{p, q}$ with $\overline{H_{n}^{p, q}} = H_{n}^{q, p}$ ; a morphism preserves $W$ and $F$ (Rational mixed Hodge structures).

[F2]

The Tate structure $Q (r)$ is the pure structure of weight $- 2 r$ with $Q (r)_{C} = C$ concentrated in type $(- r, - r)$ ; in particular $Q (1)$ has weight $- 2$ and type $(- 1, - 1)$ , while $Q (0)$ has weight $0$ and type $(0, 0)$ (Rational mixed Hodge structures).

[F3]

On a graded piece ${Gr}_{n}^{W} H_{C}$ the induced filtration $F$ is the image of $F \cap W_{n}$ and $\overset{ˉ}{F} = σ (F)$ ; the summands are $H_{n}^{p, q} = F^{p} {Gr}_{n}^{W} \cap {\overset{ˉ}{F}}^{q} {Gr}_{n}^{W}$ (Rational mixed Hodge structures).

Verification

technique · direct

1.1givenF1F2F3algebra

The weight filtration is finite with ${Gr}_{- 2}^{W} = Q e_{1}$ , ${Gr}_{- 1}^{W} = 0$ and ${Gr}_{0}^{W} = H / Q e_{1} ≅ Q e_{0}$ . On ${Gr}_{- 2}^{W}$ the induced $F$ has $F^{0} = 0$ and $F^{- 1} = {Gr}_{- 2}^{W}$ , since $F^{0} \cap W_{- 2} = C (e_{0} + i e_{1}) \cap C e_{1} = 0$ ; the conjugate filtration satisfies ${\overset{ˉ}{F}}^{- 1} = {Gr}_{- 2}^{W}$ and ${\overset{ˉ}{F}}^{0} = 0$ for the same reason, so the pieces are $H^{- 1, - 1} = {Gr}_{- 2}^{W}$ and nothing else, and $H^{- 1, - 1}$ is spanned by the image of $e_{1}$ , which is rational. On ${Gr}_{0}^{W}$ the image of $F^{0} \cap W_{0} = C (e_{0} + i e_{1})$ is all of ${Gr}_{0}^{W}$ because $e_{0} + i e_{1} \mapsto e_{0} \neq 0$ , and $F^{1} = 0$ ; the conjugate ${\overset{ˉ}{F}}^{0} = σ (F^{0}) = C (e_{0} - i e_{1})$ likewise has image all of ${Gr}_{0}^{W}$ , so the pieces are $H^{0, 0} = {Gr}_{0}^{W}$ and nothing else. Both graded pieces are therefore $n$ -opposed, with the rational structure of $e_{1}$ and $e_{0}$ respectively, so $(H, W, F)$ is a rational mixed Hodge structure; its graded pieces are $Q (1)$ and $Q (0)$ in the convention [F2].

2.1givenF1F2step 1.1algebra

Let $ι ⁣ : Q (1) \to H$ be the rational inclusion $1 \mapsto e_{1}$ , whose complexification is $C \to H_{C}$ and let $π ⁣ : H \to Q (0)$ be the $Q$ -linear map $e_{0} \mapsto 1$ , $e_{1} \mapsto 0$ . Both preserve the filtrations: $ι$ sends the weight- $(- 2)$ structure into $W_{- 2} H = Q e_{1}$ and $F^{- 1} Q (1)_{C} = C$ into $F^{- 1} H_{C} = H_{C}$ , while $F^{0} Q (1)_{C} = 0$ ; and $π$ satisfies $π (W_{- 1} H) = 0 = W_{- 1} Q (0)$ and $π (W_{0} H) = Q (0) = W_{0} Q (0)$ , $π (F^{0} H_{C}) \subseteq C = F^{0} Q (0)_{C}$ and $π (F^{1} H_{C}) = 0$ . Hence $0 \to Q (1) \overset{ι}{\to} H \overset{π}{\to} Q (0) \to 0$ is a short exact sequence of rational mixed Hodge structures, with $\ker π = Q e_{1}$ .

3.1givenF1step 2.1algebra∎

Every $Q$ -linear section $s$ of $π$ satisfies $s (1) = e_{0} + q e_{1}$ with $q \in Q$ , because $π (e_{0} + q e_{1}) = 1$ and $π (e_{1}) = 0$ . If $s$ were a morphism of mixed Hodge structures, it would satisfy $s_{C} (F^{0} Q (0)_{C}) \subseteq F^{0} H_{C} = C (e_{0} + i e_{1})$ ; since $F^{0} Q (0)_{C} = C$ and $s_{C} (1) = e_{0} + q e_{1}$ , there would be $λ \in C$ with $e_{0} + q e_{1} = λ (e_{0} + i e_{1})$ . Comparing the coefficients of $e_{0}$ and $e_{1}$ in the basis $(e_{0}, e_{1})$ gives $λ = 1$ and $q = i$ , which is impossible for $q \in Q$ . Hence the extension does not split in the category of rational mixed Hodge structures. The obstruction is visible directly in the $(0, 0)$ -direction: $F^{0} H_{C} \cap {\overset{ˉ}{F}}^{0} H_{C} = C (e_{0} + i e_{1}) \cap C (e_{0} - i e_{1}) = 0$ , while a rational section must send $1$ into a rational vector of $F^{0}$ , which is then also fixed by conjugation and hence would have to lie in this zero intersection.

Source notes

Deligne II gives the theory, the Tate object and the morphism condition; the two-dimensional witness with $F^{0} = C (e_{0} + i e_{1})$ was not located verbatim in the fetched texts, so the statement and verification are recorded as generated, with the definitional background cited. The closest statement-level analogue is the one-dimensional example on the punctured line, El Zein, Example 1.3. No choice principle is used: the only quantifiers run over elements of the displayed two-dimensional space.

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Sources

Pierre Deligne, Theorie de Hodge II, Publ. Math. IHES 40 (1971) 5-57 (standard reference, not scraped)
Claire Voisin, Hodge structures and the topology of algebraic varieties, author-hosted lecture notes (Harvard CMSA revision) (standard reference, not scraped)
Andrzej Weber, Notes on Mixed Hodge Structures, Srni January 2005 (author-hosted, 15 pp.) (standard reference, not scraped)
F. El Zein, Hodge Structures and Mixed Hodge Structures, University of Pennsylvania course notes (June 24, 2010) (standard reference, not scraped)