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Weights control restriction images into proper varieties

Statement

Assume AC. Let $U$ be smooth quasiprojective over $C$ , let $j : U ↪ X$ be any smooth projective compactification, and let $g : Y \to U$ be a morphism with $Y$ proper (possibly singular and reducible). Then, in every degree $n$ , $im (H^{n} (X, Q) \overset{(j g)^{*}}{\to} H^{n} (Y, Q)) = im (H^{n} (U, Q) \overset{g^{*}}{\to} H^{n} (Y, Q)) .$ Both images are pure Hodge substructures of weight $n$ . This includes the restriction of a smooth projective total space to a proper singular fiber through an open subvariety, which is the exact use in de Cataldo-Migliorini Proposition 8.3.2 of arXiv:math/0306030v1 (pp. 64-65).

Facts & Assumptions

Given: AC, a smooth quasiprojective $U$ with a smooth projective compactification $j : U ↪ X$ , and a morphism $g : Y \to U$ with $Y$ proper.

[F1]

Logarithmic construction of smooth-open mixed Hodge structures: the logarithmic mixed Hodge structure on $H^{n} (U, Q)$ has weights at least $n$ and satisfies $W_{n} H^{n} (U, Q) = im (j^{*} : H^{n} (X, Q) \to H^{n} (U, Q))$ for every smooth projective compactification, independently of the chosen $j$ ; for a non-SNC boundary the logarithmic theorem proves the same actual image formula by SNC domination and its modification-image supplier.

[F2]

Proper cohomology and functorial mixed Hodge weights: the proper variety $Y$ carries a functorial mixed Hodge structure on each $H^{n} (Y, Q)$ with $W_{n} H^{n} (Y, Q) = H^{n} (Y, Q)$ , and the pullback $g^{*} : H^{n} (U) \to H^{n} (Y)$ for the actual morphism $g : Y \to U$ with proper $Y$ and smooth quasiprojective $U$ is a mixed Hodge morphism for the logarithmic structure on the source.

[F3]

Strictness of mixed Hodge morphisms and Rational mixed Hodge structures: a mixed Hodge morphism $f$ is $W$ -strict, so $f (W_{a}) = f (H) \cap W_{a} K$ for every $a$ ; its image with the induced filtration is a mixed Hodge structure; and a morphism from a structure of weights at least $n$ to a structure of weights at most $n$ has pure image of weight $n$ .

[F4]

The Axiom of Choice: the standing hypothesis, inherited from the two cited theorems.

Proof

1.1F1F2F3givenalgebra

By [F3] applied to the mixed Hodge morphism $g^{*}$ and the strictness identity $g^{*} (W_{n} H^{n} (U)) = im g^{*} \cap W_{n} H^{n} (Y)$ , the target weight bound $W_{n} H^{n} (Y) = H^{n} (Y)$ of [F2] reduces the right-hand side to $im g^{*}$ ; by [F1] the image of $W_{n} H^{n} (U)$ under $g^{*}$ is $g^{*} (im j^{*}) = im (g^{*} j^{*}) = im ((j g)^{*})$ . Hence the two asserted images agree as actual subspaces of $H^{n} (Y, Q)$ , without choosing a splitting of the weight filtration on $H^{n} (U, Q)$ .

2.1F1F2F3step 1.1algebra

Put $I = im g^{*}$ . Strictness gives $W_{n - 1} I = g^{*} (W_{n - 1} H^{n} (U))$ , and by [F1] the source has weights at least $n$ , so $W_{n - 1} I = 0$ ; the target bound gives $W_{n} I = I$ . Hence the induced mixed Hodge structure on $I$ is pure of weight $n$ , and by step 1.1 the image of $(j g)^{*}$ is the same pure substructure.

3.1F1F2F4givencases: empty zero one degenerate endpoints∎

Boundaries and Choice accounting. If $U = \emptyset$ or $Y = \emptyset$ all groups vanish and the claim is vacuous; the degree $n = 0$ case gives weight-zero images by the same strictness argument; the case $U = X$ has $j = i d$ and $W_{n} H^{n} (U) = H^{n} (U)$ , so both images are the full image of $g^{*}$ , consistent with the statement; and $Y$ smooth or reduced is not required, the target bound being the only hypothesis on the target. The Axiom of Choice [F4] is inherited from [F1] and [F2]; no selection is made in this argument.

Source notes

Deligne III (8.2.6), proved from II (3.2.17) and III (8.2.4), is the source; the locally expanded argument avoids the compressed slogan that equality on associated graded pieces implies equality of images: strictness identifies the actual image with the intersection of the target weight filtration, and the target has weights at most $n$ in degrees handled here.

Depends on

Used by

Simultaneous projective Lefschetz, Hodge and perverse induction step Lemma

Dependency tree · two levels

53 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pierre Deligne, Theorie de Hodge II, Publ. Math. IHES 40 (1971) 5-57, (3.2.16)-(3.2.18) (standard reference, not scraped)
Pierre Deligne, Theorie de Hodge III, Publ. Math. IHES 44 (1974) 5-77, (8.2.6) (standard reference, not scraped)
M. A. A. de Cataldo and L. Migliorini, The Hodge theory of algebraic maps, arXiv:math/0306030v1 (standard reference, not scraped)