Let $X$ be a smooth projective algebraic variety over an algebraically closed field $k$. If $k=\mathbb{C}$, we know by work of Kuranishi that the base of the versal deformation of $X$ is the germ at $0$ of the fiber over $0$ of a holomorphic map $K:H^1(X, T\_X)\to H^2(X, T\_X)$ (defined in the neighborhood of 0), called the *Kuranishi map*. 


This means that, if $S\_i$ are the power series rings associated to $H^i(X, T\_X)$ (i.e., the completions of $Sym^\* H^i(X, T\_X)^\*$), there is a map $k: S\_2 \to S\_1$ such that $R = S\_1 \otimes\_{S\_2} k$ pro-represents the deformation functor of $X$.


**Question**. Can one construct the map $k$ using algebraic methods? 


Probably one should assume that $k$ is of characteristic zero (or replace power series rings by completed divided power algebras...).



Algebraic definition of the Kuranishi map
You can look at Manetti's paper [Deformation theory via differential graded Lie algebras](http://arxiv.org/abs/math/0507284), arXiv:math/0507284.


As the title suggest, it follows the philosophy that every deformation problem is governed by a DGLA, via solution of Maurer-cartan equations (module gauge action).


One of the main advantages with respect to the classical Kodaira-Spencer's approach is that the theory works over any field $\mathbb{K}$ of characteristic $0$.


The construction of the Kuranishi map is presented in Section $4$.


