If you want to look at Deformation Theory from a complex-analytic point of view ( i.e. in the spirit of Kodaira-Spencer "deformations of complex structures" ), you need to solve the Maurer-Cartan equation

$\partial \bar \bar partial \varphi + \frac{1}{2}[\varphi, \varphi]=0$,

were $\varphi \in H^1(X, T_X)$, identified with the space of harmonic 1-forms. \mathcal{E}^{0,1}(T^{1,0})$. In order to do this, one first look at a solution which is a formal power series$\varphi(t)= \varphi(t)=\varphi_1 t \varphi_1 + t^2 \varphi_2 t^2 + t^3 \varphi_3+...$varphi_3 t^3 +...$

Collecting powers of $t$ we obtain equations

$\partial \bar \bar partial \varphi_1=0$

$\partial \bar \bar partial \varphi_2 + \frac{1}{2}[\varphi_1, \varphi_1]=0$

...

The first equation states that $\varphi_1$ is an harmonic form, according to the fact that is an element of $\mathcal{H}^1(T^{1,0})$. By Hodge Theorem, this space can be identified with $H^1(X, T_X)$, which is exactly the space parametrizing "first-order" deformationsare parametrized by $H^1(X, T_X)$.

The second equation states that you can extend the first order deformation to a second-order one (i.e., you can solve the Maurer-Cartan equation modulo $(t^3)$ ) if and only if the 2-cocycle $[\varphi_1, \varphi_1] \in H^2(X, T_X)$ varphi_1]$is a coboundary. So the class of$[\varphi_1, \varphi_1]$in$H^2(X, T_X)$is the "primary obstruction" to your deformation problem. In this way, you can try to solve modulo higher and higher powers of$t$. If all the higher order obstructions vanish and the series defining$\varphi(t)$converges, you obtain a "genuine" deformation, namely a deformation over a small disk. Now it should be clear that, in order to generalize this in the algebraic framework, you need a substitute for the step "solve the Maurer-Cartan equation modulo$(t^k)$". This substitute is roughly speaking obtained by considering deformations over Spec$k[\epsilon]/(\epsilon^k)$. 4 deleted 4 characters in body If you want to look at Deformation Theory from a complex-analytic point of view ( i.e. in the spirit of Kodaira-Spencer "deformations of complex structures" ), you need to solve the Maurer-Cartan equation$\partial \bar \varphi + \frac{1}{2}[\varphi, \varphi]=0$, were$\varphi \in H^1(X, T_X)$, identified with the space of harmonic 1-forms. In order to do this, one first look at a solution which is a formal power series$\varphi(t)= t \varphi_1 + t^2 \varphi_2 + t^3 \varphi_3+...$Collecting powers of$t$we obtain equations$\partial \bar \varphi_1=0\partial \bar \varphi_2 + \frac{1}{2}[\varphi_1, \varphi_1]=0$... The first equation states that$\varphi_1$is an harmonic form, according to the fact that the "first-order" deformations are parametrized by$H^1(X, T_X)$. The second equation states that you can extend the first order deformation to a second-order one (i.e., you can solve the Maurer-Cartan equation modulo$(t^3)$) if and only if the 2-cocycle$[\varphi_1, \varphi_1] \in H^2(X, T_X)$is a coboundary. So$[\varphi_1, \varphi_1]$is the "primary obstruction" to your deformation problem. In this way, you can try to solve modulo higher and higher powers of$t$. If all the higher order obstructions vanish and the series defining$\varphi(t)$converges, you obtain a "genuine" deformation, namely a deformation over a small disk. Now it should be clear that, in order to generalize this in the algebraic framework, you need a substitute for the step "solve the Maurer-Cartan equation modulo$(t^k)$". This substitute is roughly speaking obtained by considering deformations over Spec$\mathbb{C}[\epsilon]/(\epsilon^k)$.k[\epsilon]/(\epsilon^k)$.

3 edited body; added 1 characters in body

If you want to look at Deformation Theory from a complex-analytic point of view ( i.e. in the spirit of Kodaira-Spencer "deformations of complex structures" ), you need to solve the Maurer-Cartan equation

$\partial \bar \varphi + \frac{1}{2}[\varphi, \varphi]=0$,

were $\varphi \in H^1(X, T_X)$, identified with the space of harmonic 1-forms. In order to do this, one first look at a solution which is a formal power series

$\varphi(t)= t \varphi_1 + t^2 \varphi_2 + t^3 \varphi_3+...$

Collecting powers of $t$ we obtain equations

$\partial \bar \varphi_1=0$

$\partial \bar \varphi_2 + \frac{1}{2}[\varphi_1, \varphi_1]=0$

...

The first equation states that $\varphi_1$ is an harmonic form, according to the fact that the "first-order" deformations are parametrized by $H^1(X, T_X)$.

The second equation states that you can extend the first order deformation to a second-order one (i.e., you can solve the Maurer-Cartan equation modulo $(t^2)$ (t^3)$) if and only if the 2-cocycle$[\varphi_1, \varphi_1] \in H^2(X, T_X)$is a coboundary. So$[\varphi_1, \varphi_1]$is the "primary obstruction" to your deformation problem. In this way, you can try to solve modulo higher and higher powers of$t$. If all the higher order obstructions vanish and the series defining$\varphi(t)$converges, you obtain a "genuine" deformation, namely a deformation over a small disk. Now it should be clear that, in order to generalize this in the algebraic framework, you need a substitute to for the step "solve the Maurer-Cartan equation modulo$(t^k)$". This substitute is roughly speaking obtained by considering deformations over$\mathbb{C}[\epsilon]/(\epsilon^k)\$.

2 added 2 characters in body
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