= Solution
Along the smooth complex curve, there is an exact sequence of complex <vector bundles>
$$
0\longrightarrow T\Sigma\longrightarrow TX|_\Sigma\longrightarrow N_{\Sigma/X}\longrightarrow0.
$$
The <Whitney sum formula for Chern classes> gives $c_1(TX)|_\Sigma=c_1(T\Sigma)+c_1(N_{\Sigma/X})$. The complex tangent line has first Chern number $\chi(\Sigma)=2-2g(\Sigma)$. The normal complex line is an oriented real rank-two bundle, whose <Euler number> is the <self-intersection> of the curve: a generic normal section has signed zeros equal to the intersections with a small pushed-off copy. Thus
$$
\langle c_1(TX),[\Sigma]\rangle=2-2g(\Sigma)+\Sigma\cdot\Sigma,
$$
and the <adjunction formula> is
$$
\boxed{2g(\Sigma)-2=\Sigma\cdot\Sigma-\langle c_1(TX),[\Sigma]\rangle.}
$$
Equivalently, since $c_1(K_X)=-c_1(TX)$, it is $2g-2=\Sigma\cdot(\Sigma+K_X)$. The derivation applies to a compact smooth complex curve; its complex orientation fixes all signs.
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