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 $a_i$?
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1$\begingroup$ Compute in terms of what? E.g. do you know defining equations? $\endgroup$– Allen KnutsonFeb 12, 2015 at 21:48
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$\begingroup$ I don't know anyting in non-smooth case. But I think for instance $\omega^n$ make scence as "current" and not forms. In page 2 in definition 1 you can find the coeficients $a_i$ but in smooth case link.springer.com/article/10.1007/s00208-012-0788-y $\endgroup$– user21574Feb 13, 2015 at 0:20
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2$\begingroup$ If you mean that $\dim H^0(X, L^k)$ is a polynomial in $k$, this is false (even if $X$ is smooth). What is true is that there exists a polynomial $P$ such that $\dim H^0(X, L^k)=P(k)$ for $k$ large enough. $\endgroup$– abxFeb 13, 2015 at 16:32
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3 Answers
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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.
answered Feb 13, 2015 at 13:35
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I guess you are mostly interested in $a_0,a_1$ for applications to K-stability.
If $X$ is normal, then still $a_0 = \frac{L^n}{n!}$ and $a_1 = \frac{-K_X.L^{n-1}}{2(n-1)!}$. This follows from asymptotic Riemann-Roch for normal varieties, see for example Odaka's "A generalization of Ross-Thomas' slope theory" Lemma 3.5 (the published version, not the arXiv version). Note that since $X$ is normal, its canonical class exists as a Weil divisor and the intersection number still makes sense.
In fact this holds more generally in the case that $X$ is Gorenstein in codimension one.
Note that $a_0 = \frac{L^n}{n!}$ holds without any assumption on the singularities of $X$, see for example "Positivity in Algebraic Geometry Theorem" 1.2.22.
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The answer is contained in the book by W. Fulton "Intersection theory", chapter 18 "Riemann-Roch for singular varieties".
