Is there a proof of Noether's formula without using general Riemann-Roch theorem?

Question

Is there an algebraic proof of Noether's formula $\chi(O_S)=\frac{1}{12}(K^2+c_2(T_S))$ without directly applying general Riemann-Roch theorem?

Answer 1

Yes, see Griffith and Harris, "Principles of algebraic geometry", last section in the chapter on surfaces.