Suppose that $X$ is a complex algebraic (or complex analytic) variety, and $x \in X$ is a singular point. I am interested in two types of local differential forms at $x$: analytic and formal.

First, let $\mathcal{O}_{X,x}^{\text{an}}$ be the ring of analytic germs of functions at $x$. I am interested in the complex $\Omega_{X,x}^{\text{an}}$ of analytic germs of differential forms, i.e., linear combinations of elements $h_0 dh_1 \wedge \cdots \wedge dh_k$ for $h_0, \ldots, h_k \in \mathcal{O}_{X,x}^{\text{an}}$, and the corresponding de Rham cohomology $H^\bullet(\Omega_{X,x}^{\text{an}})$. Explicitly, if $X \subseteq \mathbf{A}^n$ is a subvariety of affine space cut out by equations $f_1, \ldots, f_m$, then this complex is defined as $\Omega_{\mathbf{A}^n,x}^{\text{an}} / (f_1, \ldots, f_m, df_1, \ldots, df_m)$, where we quotient by the ideal in the de Rham differential graded algebra generated by the $f_i$ and $df_i$.

Next, let $\hat {\mathcal{O}}_{X,x}$ be the completion of the local ring of algebraic functions at $x$, i.e., the ring of (not-necessarily convergent) formal power series of functions at $x$. Let $\hat {\Omega}_{X,x}$ be the complex of formal differential forms, i.e., linear combinations of elements $h_0 dh_1 \wedge \cdots \wedge dh_k$ for $h_0, \ldots, h_k \in \hat {\mathcal{O}}_{X,x}$. For $X \subseteq \mathbf{A}^n$, this is defined as a quotient of $\hat \Omega_{\mathbf{A}^n,x}$ in the same manner as above.

Since one has a canonical inclusion $\mathcal{O}_{X,x}^{\text{an}} \hookrightarrow \hat {\mathcal{O}}_{X,x}$, one obtains a canonical comparison map

$H^\bullet(\Omega_{X,x}^{\text{an}}) \to H^\bullet(\hat \Omega_{X,x}).$

My question is: When is this map an isomorphism?

I am particularly interested in the case that the LHS is finite-dimensional, e.g., when $X$ has an isolated singularity at $x$ (finite-dimensionality of the LHS then follows from the Theorem of Section 3.17 of Bloom and Herrera's paper ``De Rham Cohomology of an Analytic Space,'' (Invent. Math. 7, 275--296 (1969)).

More details and reformulations:

Under the finite-dimensionality hypothesis, the comparison map is definitely surjective: the RHS is the inverse limit of $H^\bullet(\Omega_{X,x} / \mathfrak{m}_{X,x}^N \cdot \Omega_{X,x})$, where $\mathfrak{m}_{X,x} \subseteq \mathcal{O}_{X,x}$ is the maximal ideal, and the LHS surjects to each of these (by lifting closed or exact forms modulo $\mathfrak{m}_{X,x}^N$ to closed or exact analytic forms). So both sides are finite-dimensional and the comparison map is surjective.

Thus, under this hypothesis, the question reduces to: When it is true that, if a closed analytic form $\alpha \in \Omega_{X,x}^{\text{an}}$ is the differential of a formal form in $\hat \Omega_{X,x}$, then it is also the differential of an analytic form in $\Omega_{X,x}^{\text{an}}$? (Perhaps, an analytic approximation theorem could be applied to answer this.)

Next, I will restrict this question to the special case that interests me: isolated singularities which are locally complete intersections. In this case, by results of Sections 4 and 5 of Greuel's paper ``Der Gauss-Manin-Zusammenhang isolierter Singularitaeten von vollstaendigen Durchschnitten,'' (Math. Ann. 214, 235--266 (1975)), one has the formula

$H^\bullet(\Omega_{X,x}^{\text{an}}) \cong \mathbf{C}^{\mu_x-\tau_x}[-\operatorname{dim} X],$

where $\mu_x$ is the Milnor number of the singularity at $x$, and the notation above indicates that the de Rham cohomology of the analytic neighborhood of $x$ is concentrated in degree equal to the dimension of $X$. Also, $\tau_x$ is the Tjurina number, which is the dimension of the singularity ring at $x$: explicitly, if $X$ is locally a complete intersection of dimension $n-m$ cut out at $x \in \mathbf{A}^n$ by functions $f_1, \ldots, f_m$, then the singularity ring is the quotient of $\mathcal{O}_{X,x}^{\text{an}}$ by the ideal generated by the $f_i$ together with the determinants of the $(n-m) \times (n-m)$ minors of the Jacobian matrix $(\frac{\partial f_i}{\partial x_j})$. In other words, the Tjurina number here is the dimension of the torsion of the germs of differential forms $\Omega_{X,x}^{\operatorname{dim}(X),\text{an}}$ of degree $\operatorname{dim}(X)$.

In this case, I would only want to know whether the same formula holds for the de Rham cohomology of the formal neighborhood, i.e., that the dimension of $H^\bullet(\hat{\Omega}_{X,x})$ is equal to the Milnor number, and not less.

[Readers who are tired of reading can stop here---I will give one more alternative formulation:]

Alternatively, one can work with the de Rham complex modulo torsion, $\tilde{\Omega}_{X,x}^{\text{an}}$, obtained from $\Omega^{\text{an}}_{X,x}$ by modding by the torsion submodule over $\mathcal{O}_{X,x}^{\text{an}}$. This is equivalent to working with germs of forms modulo those forms that become zero when restricted to the smooth locus, i.e., whose representatives on open neighborhoods of $x$ have zero restriction to smooth open subsets. In this case, Greuel's formula (still for an isolated singularity at $x$ which is locally a complete intersection) remains the same,

$H^\bullet(\tilde{\Omega}_{X,x}^{\text{an}}) \cong \mathbf{C}^{\mu_x-\tau_x}[-\operatorname{dim} X].$

In the alternative formulation, I would like to know again if the same formula holds replacing analytic germs of forms mod torsion, $\tilde{\Omega}_{X,x}^{\text{an}}$, by formal forms mod torsion. It follows from Greuel's paper that, still assuming $x$ is an isolated singularity which is locally a complete intersection, the two questions are equivalent.


1 Answer 1


For the case where x is an isolated singularity (which is probably a necessary hypothesis), we positively answered this question in arXiv:1211.1883, Theorem 4.45.

  • $\begingroup$ This result at least dates back to Deligne. See Hartshorne, On the de Rham cohomology of algebraic varieties, §IV.3. $\endgroup$
    – Z. M
    Commented Jul 3, 2022 at 20:38

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