Riemann–Roch theorem for algebraic surfaces ($\chi ({\mathcal{O}}_X(D))=\chi ({\mathcal{O}}_X)+\frac{1}{2}D\cdot (D-K_X)$)

For a Cartier divisor on a smooth projective surface, the displayed formula computes its Euler characteristic of a coherent sheaf. Combined with Serre duality, it turns intersection numbers into lower bounds for section spaces.

For a smooth compact complex surface, the degree-four part of its Todd class is $(c_1^2+c_2)/12$. Applying Riemann–Roch to its structure sheaf gives the formula. For a K3 surface, $c_1=0$ and $\chi ({\mathcal{O}}_X)=2$, so its topological Euler characteristic is $c_2=24$.