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:

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 $\mathfrak{A}\supseteq \mathfrak{A}'\supseteq \mathfrak{m}\mathfrak{A}$.?

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.