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Suppose that I have a morphism of schemes of finite type over $\mathbb{C}$, which is a bijection on closed point (but in general I don't know if there exists an inverse for it).

I would like to know if there exist sufficient conditions on the source or the target such that the induced morphism of Hodge structures is an isomorphism.

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    $\begingroup$ The MHS is essentially topological for the non obvious reasons explained in ulrich's answer. In fact, you don't need a homeomorphism, a homotopy equivalence will do. E.g. (1) the MHS of a cuspidal curve and its normalization are the same, (2) the MHS of $GL_n(\mathcal{C}^*)$ and $(\mathcal{C}^*)^n$ the same because the inclusion is a homotopy equivalence. $\endgroup$ Jun 10, 2011 at 12:35
  • $\begingroup$ Silly question: MHS = mixed hodge structure? $\endgroup$ Jun 10, 2011 at 17:48
  • $\begingroup$ Yes, it is sorry. $\endgroup$ Jun 10, 2011 at 18:19

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The induced morphism of Hodge structures for any map of varieties will be an isomorphism if and only if the induced map on cohomology (forgetting the Hodge structure) is an isomorphism. This follows from the fact that morphisms of mixed Hode structures are strict with repsect to both the Hodge and weight filtration; see Deligne, Theorie de Hodge II.

In your setup, if the map of underlying toplogical spaces is a homeomorphism it induces an isomorphism on cohomology and so by the above also an isomorphism of mixed Hodge structures. This might not always happen: consider for example the map $\mathbb{A}^1 - \{0\} \ \sqcup \{0\} \to \mathbb{A}^1$. Or consider the normalisation of a nodal curve with one of the points lying above the node removed.

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  • $\begingroup$ Just to add: at least in the smooth case, the map on the complex analytic manifolds will be a holomorphic bijection, and hence the inverse will also be holomorphic, so the second paragraph applies. $\endgroup$ Jun 10, 2011 at 20:15
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I think the answer is always, assuming that the map is finite/proper/a homeomorphism in the euclidean topology (for example if the schemes are proper over $\mathbb{C}$).

In particular, I think this is effectively answered on page 176, entry 2), of the Peters-Steenbrink book. Even the ``local Hodge-structure'' of the Deligne-Du Bois complex coincides.

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