$\newcommand{\I}{\mathcal{I}}$ Let $X$ a variety smooth over the complex numbers. Then we know that $\Omega_{X/\mathbb{C}}$ is the (usual) pullback of the conormal sheaf $\I/\I^2$ where $\I$ the sheaf of ideals of the diagonal in $X\times_\mathbb{C} X$. You can of course prove this directly.
One could expect something of this nature (taking $\Omega_X$ to be defined only as the sheaf of Kahler differentials) from computations with the conormal exact sequence
$$ 0 \rightarrow \I /\I^2 \rightarrow \Omega_{X\times X} \otimes \mathcal{O}_\Delta \rightarrow \Omega_\Delta \rightarrow 0$$
$\textbf{Question:}$ Are there geometric ways to see this also? Is there a picture that geometers keep in mind making it obvious that say the tangent bundle to $X$ and the normal bundle of the diagonal $\Delta \subset X\times X$ are really the same thing? This is a very basic concept, yet I can't find such an explanation. For example, is this clear in the context of differential geometry or complex manifolds?
Best Answer
In the case of differential geometry everything reduces to vector spaces. Let $x \in X$. Then at any point $(x, y) \in X \times X$ $$ T_{(x, y)} X \times X = T_x X \oplus T_y X . $$ Using this identification the tangent to the diagonal at a point $(x, x)$ is the subspace of $T_{(x, x)} X \times X$ given by $$ T_{(x, x)} (\Delta) = \lbrace (\xi, \xi ) \mid \xi \in T_x X \rbrace \subset T_{(x, x)} X \times X. $$ On the other hand the normal is the quotient $$ (T_{(x, x)} X \times X) / T_{(x, x)} \Delta $$ and we can identify this with $T_x X$ in at least two slightly different ways. Either $$ \iota_1 \colon \xi \mapsto (\xi , - \xi) + T_{(x, x)} \Delta $$ or $$ \iota_2 \colon \xi \mapsto (-\xi , \xi) + T_{(x, x)} \Delta . $$
We can of course also identify $T_x X$ and $T_{(x, x)} \Delta $ by $ \xi \mapsto (\xi, \xi)$.
I guess one explanation for the two identifications of the normal bundle is that there is involution $\tau \colon X \times X \to X \times X $ given by $\tau(x, y) = (y, x)$ which fixes the diagonal pointwise and hence acts trivially on the tangent space to the diagonal. As a result it descends to an action on the normal bundle which interchanges the two identifications $\iota_1$ and $\iota_2$, that is $\tau \circ \iota_1 = \iota_2$