3 removed some words

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

http://mathoverflow.net/questions/42744/diffeomorphic-kahler-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 use $H^n_c,$ or add local systems $L,L'$ (say with the same monodromy and underlie VHS) as 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.

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

http://mathoverflow.net/questions/42744/diffeomorphic-kahler-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 use $H^n_c,$ or add local systems $L,L'$ (say with the same monodromy and underlie VHS) as 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.

1

# 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

http://mathoverflow.net/questions/42744/diffeomorphic-kahler-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 use $H^n_c,$ or add local systems $L,L'$ (say with the same monodromy and underlie VHS) as 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 $X'.$