I am reading the paper [Cohomology and Obstructions I: Geometry of formal Kuranishi theory][1] 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:
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 $M_0$.
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 $\mathfrak{A}\supseteq \mathfrak{A}'\supseteq \mathfrak{m}\mathfrak{A}$.? [1]: http://cn.arxiv.org/abs/math/9901084v4