Riemann-Roch Type Inequalities [antikvár]

RIEMANN ROCH TYPE INEQUALITIES By J. Kollár and T. Matsusaka* Dedicated to André Weil for his 77th birthday. Introduction. Our aim in this paper is to investigate the following classical problem in algebraic geometry: When X is an ample divisor on a projective variety, what can one say about l(rX), at leastfor large values ofr? One possible answer is provided by the Hilbert Polynomial, asserting that for somé large r, depending on the variety and X, from this value on, l(rX) is given by the Hilbert polynomial. Unfortunately the present proofs of this neither allow one to make the dependence explicit, nor provide information on / (rX) for small values of r. Therefore we try to give estimates of l(rX) which are not so sharp as the above one; but the dependence on various data is clear and valid for all positive r. Let us call X on a normál projective variety semi-ample, if somé multiple of it defines a complete linear system without base point and defines a birational map of the underlying variety. The main goal of this paper is the proof of the following: Theorem. For every n, there is a polynomial Q(x, y, z) such that its degree in z is at most n - 1, and if V" is any normál projective variety in characteristic 0, X a semi-ample Cartier divisor on V, then for all values of r the following estimate holds \l(rX)~{d/n\)r"\ < Q(d,H,r), where d = X{n\ L = I(KV, One of the interesting features of the theorem is that the estimate depends only on the two highest coefficients of the Hilbert polynomial of X. This is partially explained by the following Lemma, whose proof is simple but seems to have been unknown.