Let $(X,L,\omega)$ be a projective variety with polarization $L$. then we can write


$$\dim H^0(X,L^k)=a\_0k^n+a\_1k^{n-1}+...$$


If $X$ is smooth then $a\_0=Vol(X)$ and we can compute $a\_i$.


If $X$ is non-smooth then how can we compute $a\_i$?



Computing the coefficients of the polynomial $\dim H^0(X,L^k)$ in non-smooth case
Sometimes one can at least compute $\chi(\mathcal{O}\_X(X, \, L^k))$. 


The formula that one expects will be of the form "ordinary Riemann-Roch formula plus correction terms depending on the singularities of $X$".
This is of course a bit vague, nevertheless the correction terms can be explicitly computed
in some particular cases. 


For instance, surfaces with rational double points and threefolds with canonical singularities are treated in detail in [M. Reid, *Young person guide to canonical singularities*], Chapter III "Contribution of $\mathbb{Q}$-divisors to RR". In particular, the correction terms are expressed in terms of Dedekind sums.


