Difference between Kuranishi's existence theorem and Kodaira-Spencer's version.

Question

Conserning infinitesimal deformation of a complex compact manifold $M$, Kuranishi showed in his generalized existence theorem that a local moduli space exists and is unique up to isomorphism. I want to know if, with the addition of the condition $H^2(M,\Theta)=0$, we can actually prove the existence of a local moduli space using Kodaira-Spencer's existence theorem ? Also, what are in general the differences between the two versions of the theorem ?

Answer 1

Kodaira-Nirenberg-Spencer (1958) proved the existence of a versal deformation for a compact complex manifold $X$ satsifying the assumption $H^2(X, \Theta_X) = 0$. A deformation is named versal if is complete and if the pull-back morphism is uniquely determined up to 1-rst order.

The condition $H^2(X, \Theta_X) = 0$ assures that the base $S$ of the versal deformation is smooth: It is an open neightbourhood of zero in $H^1(X, \Theta_X)$.

Kuranishi (1962) generalized the result of Kodaira-Nirenberg-Spencer by dropping the condition on $H^2(X, \Theta_X)$. Now, the base of the versal deformation of $X$ is an analytic subset $S$ of a neighbourhood of zero in $H^1(X, \Theta_X)$. The tangent space of $S$ at zero is $H^1(X, \Theta_X)$. The obstructions against $S$ being smooth are located in $H^2(X, \Theta_X)$.

In both cases, $H^1(X, \Theta_X)$ parametrizes the 1-rst order deformations of $X$, i.e. the deformations over the double point. But in general, a deformation over the double point does not extend to a deformation over an open neighbourhood of zero in $H^1(X, \Theta_X)$.

Later, the theorem on the existence of a versal deformation has been generalized by different authors to deformations of an arbitrary compact complex space $X$ . E.g., different proofs have been given by Grauert (1974) and Forster-Knorr (1974).