Suppose $U\subset\mathbb C^n$ is an open subset and $f_1,\ldots,f_k$ are analytic functions on it, generating the coherent ideal sheaf $\mathcal I$ which defines a closed complex subspace $Z\hookrightarrow U$ (analytic analogue of closed algebraic subscheme). I want to define the normal cone to $Z$ in $U$ (and the blow-up of $U$ along $Z$) in this setting as a complex space.

For normal cones, we can proceed in the algebraic category as follows: we can take the subscheme $\Gamma^\circ$ defined by $\lambda w_i - f_i(z_1,\ldots,z_n) = 0$ for $i=1,\ldots,k$ inside $U\times\mathbb C^k\times\mathbb C^\times$ (with $\lambda$ the coordinate on $\mathbb C^\times$, $z$'s the coordinates on $U$ and $w$'s the coordinates on $\mathbb C^k$) and take its (Zariski) closure in $U\times\mathbb C^k\times\mathbb C$ to obtain a scheme $\Gamma$. The scheme-theoretic fibre over $0\in\mathbb C$ of $\Gamma$ is then the normal cone (embedded in $U\times\mathbb C^k$). Alternately, avoiding the embedding, one defines $\text{Spec}_{/U}\oplus_{i\ge 0}\mathcal I^i/\mathcal I^{i+1}$ to be the normal cone.

Now, this construction doesn't apriori make sense (at least not to me) in analytic geometry as we don't have an analogue of Zariski closure here. The definition based on $\oplus_{i\ge 0}\mathcal I^i/\mathcal I^{i+1}$ also doesn't apriori seem to make sense in the analytic category as we need to realize this algebra as a quotient of a polynomial algebra over $\mathcal O_U$ (or $\mathcal O_Z$) by a finitely generated ideal and how to do this is not entirely obvious to me.

So, my question is: Can we show that the normal cone algebra $\oplus_{i\ge 0}\mathcal I^i/\mathcal I^{i+1}$ and the blow-up algebra $\oplus_{i\ge 0}\mathcal I^i$ are finitely presented over $\mathcal O_U$? (i.e., are locally isomorphic as graded $\mathcal O_U$-algebras to $\mathcal O_U[t_1,\ldots,t_m]/(F_1,\ldots,F_r)$ where $t_i$ are indeterminates and $F_j$ are homogeneous polynomials in the $t_i$'s with coefficients being sections of $\mathcal O_U$ and $m,r\ge 0$ are integers)

  • $\begingroup$ In the first para of sec 2 in the paper "Normal cones in analytic Whitney stratifications" (numdam.org/article/PMIHES_1969__36__127_0.pdf), the author says that the normal cone algebra is finitely presented over $\mathcal O_U$ and uses its analytic Spec to define the normal cone. A reference is provided to Seminaire Henri Cartan (presumably the articles of Houzel on local analytic geometry). It doesn't seem like the finite presentation statement is proved there. $\endgroup$ May 6, 2019 at 21:38

1 Answer 1


Proposition 1.17 in the paper Complex Analytic Cones by Axelsson and Magnusson in Math Ann 273 pages 601-628 answers your question.


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