# Is the weight filtration a topological invariant?

This question is somehow related to (but different from) the following MO question and the one linked from there

Diffeomorphic Kähler manifolds with different Hodge numbers

Let $X$ and $X'$ be two complex algebraic varieties that are diffeomorphic to each other. One may or may not assume they are irreducible. Then:

1. (Weak form) For each pair of integers $(i,n),$ do we always have $$\dim Gr^W_iH^n(X,\mathbb Q)=\dim Gr^W_iH^n(X',\mathbb Q),$$ where $W$ denotes the weight filtration on the mixed Hodge structures?

2. (Strong form) Let $f:X\to X'$ be a diffeomorphism. Does $f^*:H^n(X')\to H^n(X)$ respect the $W$-filtrations?

Variants: One can ask similar questions with "diffeomorphic" replaced by "homeomorphic" or "complex analytically isomorphic". For instance, does the weight filtration see the difference between tangential intersections and transverse intersections? One can also add coefficients.

At the first sight, it's not clear to me how to approach the question, since there seems to be no relation between the resolutions of $X$ and $X'.$

Edit: As algori pointed out to me, this is false without the compactness assumption, even for "complex analytic isomorphism". So I will assume $X$ and $X'$ are compact.

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I second algori's pessimism (optimism?) that it's false for the homeomorphisms for singular compact varieties. For diffeomorphisms, you need to be clear on what you mean in the singular case (often this entails fixing a stratification, but this involves more structure). –  Donu Arapura Apr 21 '11 at 0:32
Dear Donu: Can you give me a counter-example for homeomorphisms for compact singular varieties? And by diffeomorphisms, I mean isomorphisms as real analytic varieties. I take the following for the definition of the structural sheaf $\mathscr O_X$ of analytic functions on a real analytic variety: locally the space can be embedded in an open ball, and an analytic function on the space is simply the restriction of an analytic function on a small neighborhood. –  shenghao Apr 21 '11 at 12:19
Dear Shenghao, unfortunately my comment came out stronger than I intended; I meant to say it felt unlikely to me. Actually, I do know many examples of homeomorphic (also real analytically equivalent) noncompact varieties where this fails. What's interesting is that these examples arise naturally (from nonabelian Hodge theory). –  Donu Arapura Apr 21 '11 at 12:58

If one does not assume $X$ and $X'$ compact, then there are relatively easy counter-examples: there is a vector bundle over an elliptic curve with total space analytically equivalent to $\mathbb{C}^\*\times\mathbb{C}^\*$, see e.g., Peters-Steenbrink, Mixed Hodge structures, p. 102.

There are positive results as well, but the only one I can think of is this: if $X$ and $X'$ are (possibly singular) compact algebraic surfaces, the answer to question 2 is positive, see Steenbrink-Stevens, Indag. Math., 46, 1984, no. 1, p. 63-76.

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Thanks, algori. I added the compactness assumption. In fact before posting the question, I went to look up at the same example of Serre in Hartshorne's AG, Appen. B, but didn't realize anything strange about it... –  shenghao Apr 20 '11 at 14:33
shenghao -- I think this should be false for threefolds. –  algori Apr 20 '11 at 15:11
Dear algori: could you give me a counter-example or direct me to a reference? –  shenghao Apr 21 '11 at 12:25
Dear shengao -- I think there may be an example somewhere in the paper by Steenbrink and Stevens (I don't have that paper, so can't tell where exactly). –  algori Apr 21 '11 at 12:37
It seems that Steenbrink and Stevens' paper provides a 3-dim. counter example, for a variant of the original question. –  shenghao Apr 28 '11 at 23:49

Not an answer, but related somewhat is an important topological property of the weight filtration. Namely if $f: X \to S$ is morphism of complex algebraic varieties such that $R^if_*Q$ are local systems (e.g. all fibers are homeomorphic) on $S$, then the weight filtration assembles to give sub-local systems. So the weight filtration is locally constant, in families.

An arithmetic version (over Spec $Z$) is also available.

See Deligne's ICM 1974 address, especially Theorems 2 and 14.

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SGP -- just a small point: it often happens that the fact that fibres are not homeomorphic is reflected in the fact that $R^i f_\* \mathbb{Q}$ is non-constant, but this needn't always be the case: a typical example would be the quotient map $X\to X/G$ where $X$ is an affine variety on which a reductive group $G$ acts with finite stabilizers. And conversely, it can happen that the fibres are all homeomorphic, but some of the derived images of the constant sheaf are non-constant. –  algori Apr 21 '11 at 11:24
@algori: Good point! Thanks! –  SGP Apr 21 '11 at 18:16