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...).