## Riemann-Roch type inequalities.(English)Zbl 0538.14006

The aim of the paper is to give estimates for $$\ell(rX)$$ depending only on the two highest coefficients of the Hilbert polynomial of X. Here X is a semi-ample Cartier divisor on a normal projective variety (semi-ample divisor means that some multiple of it has no base points and defines a birational map). The precise result is the following: For every n, there is a polynomial Q(x,y,z) with positive rational coefficients of degree in z at most n-1, satisfying the property: for any normal projective variety V of dimension n, any semi-ample Cartier divisor on V, and any integer r one has $$| \ell(rX)-(d/n!)r^ n| \leq Q(d,\xi,r),$$ where $$d=X^{(n)}$$ and $$\xi =I(K_ V,X^{(n-1)})$$. - The paper starts with the particular case of ample divisors on smooth projective varieties, which is easier to prove, and ends with a proof that the family of smooth polarized varieties (V,X) having bounded intersection numbers $$X^{(n)}$$, $$(K_ V\cdot X^{(n-1)})$$ is a bounded family.
Reviewer: M.Stoia

### MSC:

 14C40 Riemann-Roch theorems 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14C20 Divisors, linear systems, invertible sheaves
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