Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 20 / 4 / i / Solution

Along the smooth complex curve, there is an exact sequence of complex vector bundlesThe 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. Thusand the adjunction formula isEquivalently, 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.