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On the Kawamata-Viehweg vanishing theorem for a surface in positive characteristic. (English) Zbl 0940.14010

From the introduction: We prove the Kawamata-Viehweg vanishing theorem for a nonsingular projective surface defined over an algebraically closed field of characteristic \(p>0\). In particular, we give an explicit condition that the cohomology groups vanish.
Theorem. Let \(S\) be a nonsingular projective surface defined over an algebraically closed field of characteristic \(p>0\) and \(M\) a nef and big \(\mathbb{Q}\)-divisor on \(S\). For the fractional part \(\{M\}\) of \(M\), write \(\{M\}=\sum_i{a_i\over b_i}D_i\) with \(0\leq a_i<b_i\) and \(a_i,b_i\in\mathbb{Z}\) for all \(i\). With this notation, we further suppose that \(p\nmid b_i\) for all \(i\). Assume that one of the following situations holds:
(1) \(S\) is not of general type and further not a quasi-elliptic surface of Kodaira dimension 1;
(2) \(S\) is of general type with minimal model \(S'\), \(p\geq 3\), and \(M^2>K^2_{S'}\);
(3) \(S\) is of general type with minimal model \(S'\), \(p=2\), and \(M^2>\max \{K^2_{S'},K^2_{S'} -3\chi({\mathcal O}_S)+2\}\).
Then we have \(H^i(S,{\mathcal O}_S(K_S+ \lceil M\rceil))=0\) for \(i> 0\).
We first prove the Kodaira vanishing theorem for a surface in positive characteristic, which is based on N. I. Shepherd-Barron’s result [Invent. Math. 106, No. 2, 243-262 (1991; Zbl 0769.14006)] on the instability of rank 2 locally free sheaves. And we give a slight improvement of the covering lemma. Finally, from these results and Sakai’s lemma, we obtain the theorem.

MSC:

14F17 Vanishing theorems in algebraic geometry
14J25 Special surfaces
14C20 Divisors, linear systems, invertible sheaves
14G15 Finite ground fields in algebraic geometry

Citations:

Zbl 0769.14006
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