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Let $U$ be the complement of a divisor with normal crossings in a smooth compact complex manifold $X$. If $X$ is algebraic, then Deligne describes in "Th\'eorie de Hodge 2" a procedure to equip the rational cohomology of $U$ with a mixed Hodge structure. Most of this procedure can be carried out if the $\partial\bar\partial$-lemma holds for $X$, in particular, if $X$ is bimeromorphic to a K\"ahler manifold. But there are also some things that break down, for example, the functoriality and the independence of the compactification (for both the resolution of singularities is used).

  1. Is it true that the rest of section 3 (of Th\'eorie de Hodge 2) works as soon as $X$ is, say, K\"ahler? For example, is it true that the rational cohomology of $U$ carries a mixed Hodge structure such that the weight filtration is the Leray filtration induced by the open embedding $U\to X$?

  2. If so, are there examples of K\"ahler compactifications that give different rational mixed Hodge strutures? Non-isomorphic Hodge structures?

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1 Answer 1

I may be misunderstanding question 2, but there is an example (due to Serre) of a complex surface U with 2 different K\"ahler compactifications with different mixed Hodge structures. By projectivizing a nontrivial extension of the trivial line bundle by itself over an elliptic curve E, you get a P^1 bundle over the curve, and the trivial subbundle gives a section, which you delete to get U, which is a C-bundle over E. U is complex-analytically isomorphic to $\mathbb{C}^\times \times \mathbb{C}^\times$ and can be compactified to $\mathbb{P}^1 \times \mathbb{P}^1$. H^1(U) in the P^1-bundle lies in W_1, while H^1(U) in the product of projective lines has pure weight 2.

(From Peters-Steenbrink Mixed Hodge Structures section 4.5)

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