I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the classical deformaion theory developed by Kodaira-Spencer. In particular, I have the following questions:

  1. In the last paragraph of page 11, Clemens write, If for an ideal $\mathfrak{A}\subseteq \mathfrak{m}=\{t_1,\cdots,t_s\}$, we let $$\Delta_{\mathfrak{A}}:= Spec\frac{\mathbb{C}[t]}{\mathfrak{A}}\subseteq \Delta.$$ I should add that $\Delta$ is a polydisc with coordinates $\{t_1,\cdots,t_s\}$. My question is that what is the meaning of the inclusion $Spec\frac{\mathbb{C}[t]}{\mathfrak{A}}\subseteq \Delta$? Is this inclusion in the category of schemes or complex sapces? I have the same question for $$M_{\mathfrak{A}}:=\pi^{-1}(\Delta_{\mathfrak{A}}),$$ where $\pi: M\to \Delta$ is a deformation of the central fiber $M_0$.

  2. In page 18. What is the meaning of the sentence "$M_{\mathfrak{A}}/\Delta_{\mathfrak{A}}$is an infinitesimal deformation of compact Kahler manifolds"? What is the definition of extending a family $M_{\mathfrak{A}}/\Delta_{\mathfrak{A}}$ to a family $M_{\mathfrak{A}'}/\Delta_{\mathfrak{A}'}$ for some ideal $\mathfrak{A}\supseteq \mathfrak{A}'\supseteq \mathfrak{m}\mathfrak{A}$.?

Edit: I find a good interpretation of the notion Spec is given in pp.36 of this note by Marco Manetti. See also pp.121 of H. Grauert, T, Peternell and R. Remmert. Several Complex Variables VII, Sheaf-Theoretical Methods in Complex Analysis, Springer-Verlag Berlin Heidelberg, 1994.


1 Answer 1

  1. It is an inclusion of analytic spaces — $\Delta $ is not a scheme. If $\mathfrak{A}=(f_1,\ldots ,f_p)$, $\Delta _{\mathfrak{A}}$ is the subspace of $\Delta $ defined by $f_1=\ldots =f_p=0$. I think $\mathfrak{A}\neq 0$ is implicitely assumed, so $\Delta _{\mathfrak{A}}$ is a fattening of the origin. Hence $M _{\mathfrak{A}}$ is what is called an infinitesimal deformation of $M_0$.

  2. If $\mathfrak{A}'\subset \mathfrak{A}$, $\Delta _{\mathfrak{A}}$ is a subspace of $\Delta _{\mathfrak{A}'}$, so you look for a family over $\Delta _{\mathfrak{A}'}$ which induces the given family over $\Delta _{\mathfrak{A}}$.

I think your questions would be more appropriate on MSE.

  • $\begingroup$ @abx You propose to intepret $\Delta _{\mathfrak{A}}$ as the compelx analytic space $(V(\mathfrak{A}), \mathscr{O}_{\Delta}/\mathscr{I})$, where $V(\mathfrak{A}) $ is the zero set of the ideal $\mathfrak{A}$ and $\mathscr{I}$ is the ideal sheaf of $V(\mathfrak{A}) $. But this has nothing to do with the notation $Spec\frac{\mathbb{C}[t]}{\mathfrak{A}}$. Besides, why the polynomial ring $\mathbb{C}[t]$ appear not the ring of holomorphic functions? $\endgroup$
    – Wei Xia
    Nov 5, 2018 at 2:07
  • $\begingroup$ I have seen three definitions of infinitesimal deformation in the literature:1. a cohomology class in $H^1(M, \Theta)$. 2.a deformation over the dual numbers $k[\epsilon]/\epsilon^2$. 3. a deformation over a zero dimensional compelx analytic space. Now if $M_{\mathfrak{A}}/\Delta_{\mathfrak{A}}$ is a infinitesimal deformation, then the ideal $\mathfrak{A}$ shoud not be arbitrary. Right? $\endgroup$
    – Wei Xia
    Nov 5, 2018 at 2:27
  • 1
    $\begingroup$ 1. "nothing to do": they are exactly the same object. 2. For any nonzero ideal $\mathfrak{A}$ the space $\Delta_{\mathfrak{A}}$ is zero-dimensional. $\endgroup$
    – abx
    Nov 5, 2018 at 5:05

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