Using the laguage of derived category, the Kodaira-Spencer map

$\kappa(x) : Ext^1_X(k(x), k(x)) \rightarrow Ext^1_X(\mathcal F_x, \mathcal F_x)$

can be described as a Fourier-Mukai transformation $\Phi_{\mathcal F}$.

$\Phi_{\mathcal F} : Hom_{D^b(X)} (k(x), k(x)[1]) \rightarrow Hom_{D^b(X)}(\mathcal F_x, \mathcal F_x[1]) $

Here, $x \in X$ is a point of a smooth projective variety over an algebraically closed field.

$\mathcal F$ is a coherent sheaf on $X \times X$ which is flat over the first factor, and $\mathcal F_x$ means a $\mathcal {i}^*_{x\times X} \mathcal F$.

(which is equal to $\Phi_{\mathcal F} (k(x))$)

Of course, KS map is not defined as a differential of some morphism, but please consider the following argument ;

Suppose $\mathcal F_x$ is concentrated at $x$ for all $x \in X$. It means the map $f:x \mapsto \mathcal F_x$ is injective. Hence $\kappa(x) := df(x)$ is injective at generic point.

(see D.Huybrechts, Fourier-Mukai transform in algebraic geometry, Ch7, Prop7.1)

Here, the definition of $f$ is a nonsense. It is not even clear where the image of $f$ lives. But the argument is quite persuasive. And it is even **geometrically intuitive**, because it describes a KS map as a differential of some "function".

I think this must be a shadow of rigorous mathematical contents(probably a deformation theory), but I failed to make it so. Could someone explain to me what's going on? Thanks in advance.