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Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular cohomology $H^*_{sing}(X(\mathbb C)^{an},\mathbb C)$?

Such an example has to be singular (by a theorem of Grothendieck), but I am having a hard time finding one. The case $xy = 0$ doesn't work (both cohomology theories give the same answer).


EDIT: I would like a reduced example if possible...

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$\operatorname{Spec}\mathbb C[X]/(X^2)$ is an example, no? – Mariano Suárez-Alvarez Apr 26 '13 at 19:46
Right, I was hoping for a reduced one... – Nicolás Apr 26 '13 at 19:50
up vote 6 down vote accepted

A likely candidate for this would be a non-Du Bois singularity.

Du Bois, following Deligne's ideas, constructed a filtered complex of sheaves with coherent cohomology sheaves that gives a resolution of the constant sheaf $\mathbb C$ for any reduced finite type scheme over $\mathbb C$. This complex agrees with the de Rham complex for smooth varieties and in general its hypercohomology agrees with the singular cohomology of the (underlying topological space of the) scheme.

So, in some sense your question is to see an example when the Du Bois complex is different from the de Rham complex (I know that's not exactly what you are asking, but I think that's the important fact behind this issue).

There is a class of singularities, not surprisingly called Du Bois singularities, with the property that the $0^{\rm th}$ associated graded quotient of the Du Bois complex is quasi-isomorphic to the structure sheaf (this holds for smooth varieties).

So, a good start for finding an example like this is to look at non-Du Bois singularities. For curves being Du Bois is equivalent to semi-normal, so any curve with a cusp is non-Du Bois. I would start there.

In general, Du Bois singularities can be quite varied, but if your $X$ is normal and Gorenstein, then being Du Bois is equivalent to being log canonical. So, you have plenty of examples (For instance, take a normal hypersurface with worse than log canonical singularities). I am sure you will find an example easily.

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One might as well fill in the details. Take $A = k[x,y]/(y^2-x^3)$, $\mathrm{char}(k) \neq 2$, $3$. Make this a graded ring where $\deg x = 2$, $\deg y=3$ and grade $\Omega^1$ so that $d$ preserves grade. The degree $5$ part of $\Omega^1_A$ is two dimensional, spanned by $y dx$ and $x dy$. (One might naively think that $3 y dx = 2 x dy$, as this equality holds away from the cusp, but the derivation $A \to A/\langle x,y \rangle$ sending $x \to 1$ and $y \to 0$ shows otherwise.) The degree $5$ part of $A$ is only one dimensional, so $\Omega^1_A/d A$ is nontrivial in degree $5$. – David Speyer Aug 15 '14 at 20:45

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