Let $X$ be a complex projective manifold. Let $B$ be the closed subscheme of $H^1(X,T\_X)$ defined by $\mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal defining the origin. In other words, $B$ is a fat point with tangent space equal to $H^1(X,T\_X)$.


There should be a universal infinitesimal deformation $\mathcal{X}$ over $B$. This should correspond to the surjectivity of the Kodaira-Spencer map. Is there a canonical way to define it? Any reference?


ps: I know how to define a first order deformation out of an element $[\eta]\in H^1(X,T\_X)$ using local charts and transition functions. I would like a similar result on the full $B$, possibly without using local charts and without choosing a basis of $H^1(X,T\_X)$.



Surjectivity of the Kodaira-Spencer map
**Edit.** I corrected the second short exact sequence.


Here is a "global" description of the sheaf of algebras $\mathcal{O}\_{\mathcal{X}}$. For the affine space $\mathbb{A}$ associated to the vector space $\text{Ext}^1\_{\mathcal{O}\_X}(\Omega\_{X/k},\mathcal{O}\_X)$, on the product $\mathbb{A}\times\_{\text{Spec}(k)} X$, there is a universal short exact sequence of locally free sheaves, $$ 0 \to \mathcal{O} \to \mathcal{E} \to \text{pr}\_X^\* \Omega\_{X/k} \to 0.$$ If you restrict this to $B$, you get a short exact sequence of locally free $\mathcal{O}\_X$-modules on the underlying topological space $X$, $$ 0 \to \mathcal{O}\_X\otimes\_k \mathcal{O}\_B/\mathfrak{m}\_B^2 \to \mathcal{F} \xrightarrow{p} \Omega\_{X/k} \otimes\_k \mathcal{O}\_B/\mathfrak{m}\_B^2 \to 0. $$ The inclusion of constants $k\to \mathcal{O}\_B/\mathfrak{m}\_B^2$ gives both a $\mathcal{O}\_X$-homomorphism $\Omega\_{X/k}\otimes\_k k \to \Omega\_{X/k}\otimes\_k \mathcal{O}\_B/\mathfrak{m}\_B^2$ and a "quotient by constants" $\mathcal{O}\_X$-homomorphism $\mathcal{O}\_X\otimes\_k \mathcal{O}\_B/\mathfrak{m}\_B^2 \to \mathcal{O}\_X\otimes\_k \mathfrak{m}\_B/\mathfrak{m}\_B^2$. Taking the pullback of the short exact sequence by the first map and the pushout by the second gives a short exact sequence of $\mathcal{O}\_X$-modules, $$ 0 \to \mathcal{O}\_X\otimes\_k \mathfrak{m}\_B/\mathfrak{m}\_B^2 \to \mathcal{F} \xrightarrow{p} \Omega\_{X/k} \otimes\_k k \to 0. $$ Note that every local section $f$ of $\mathcal{F}$ comes from a local section $e$ of $\mathcal{E}$ such that $p(e)$ equals $\omega\otimes 1$ for some local section $\omega$ of $\Omega\_{X/k}$. Also, two such sections $e$ and $e'$ are considered equivalent if their difference is the image of some $b\otimes 1$ with $b$ a local section of $\mathcal{O}\_X$.


There is a natural $k$-linear derivation, $$d\_{X/k}: \mathcal{O}\_X \to \Omega\_{X/k}.$$ The pullback of the short exact sequence by this derivation is a short exact sequence, $$ 0 \to \mathcal{O}\_X\otimes\_k \mathfrak{m}\_B/\mathfrak{m}\_B^2 \to \mathcal{O}\_{\mathcal{X}} \xrightarrow{q} \mathcal{O}\_X \otimes\_k k,$$ where $\mathcal{O}\_{\mathcal{X}}$ consists of pairs $(a,f)$ of a section $a$ of $\mathcal{O}\_X\otimes\_k k$ and a section $f$ of $\mathcal{F}$ such that $d\_{X/k}(a)$ equals $p(f)$. Define the product of $(a,f)$ and $(a',f')$ by $(aa',af'+a'f)$. This makes $\mathcal{O}\_{\mathcal{X}}$ into a sheaf of algebras, and $q$ is a homomorphism of sheaves of algebras. For every constant $c\in k$, there is an element $(a,f)=(c,0)$. Similarly, for every element $t\in \mathfrak{m}\_B/\mathfrak{m}\_B^2$, there is an element $(a,f)=(0,1\otimes t)$. Altogether, this defines a ring homomorphism from $\mathcal{O}\_B/\mathfrak{m}\_B^2$ to $\Gamma(X,\mathcal{O}\_{\mathcal{X}})$ making $\mathcal{O}\_{\mathcal{X}}$ into a sheaf of $\mathcal{O}\_B/\mathfrak{m}\_B^2$-algebras. The homomorphism $q$ is the quotient by the ideal generated by $\mathfrak{m}\_B/\mathfrak{m}\_B^2$. Thus, the natural $\mathcal{O}\_B/\mathfrak{m}\_B^2$-module homomorphism, $$\mathcal{O}\_{\mathcal{X}}/\mathfrak{m}\_B \otimes\_k \mathfrak{m}\_B/\mathfrak{m}\_B^2 \to \mathfrak{m}\_B\cdot \mathcal{O}\_{\mathcal{X}},$$ is an isomorphism. From the local flatness criterion, $\mathcal{O}\_{\mathcal{X}}$ is flat as an $\mathcal{O}\_B/\mathfrak{m}\_B^2$-algebra. Thus the sheaf of $\mathcal{O}\_B/\mathfrak{m}\_B^2$-algebras $\mathcal{O}\_{\mathcal{X}}$ together with the algebra homomorphism $q$ defines a flat deformation $\mathcal{X}$ of $X$ over $B$.


