# Surjectivity of the Kodaira-Spencer map

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 $H^1(X,T_X)$.

• For the scheme $\mathcal{X}$, there may not exist any ample invertible sheaf on $\mathcal{X}$. So it is unlikely you will find some "extrinsic" construction of $\mathcal{X}$. You can describe the sheaf of algebras $\mathcal{O}_{\mathcal{X}}$ on the underlying topological space $X$. Vistoli's notes on deformations of local complete intersection schemes are a great resource. – Jason Starr Jul 15 '16 at 18:26

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 $B$.