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On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic \(p\). (English) Zbl 0724.14008

This paper is about sheaf cohomology for varieties (schemes) in characteristic \(p>0.\) We assume the presence of a Frobenius splitting [see V. B. Mehta and A. Ramanathan, Ann. Math., II. Ser. 122, 27-40 (1985; Zbl 0601.14043)].
The main result is that a nonzero higher direct image under a proper map of the ideal sheaf of a compatibly Frobenius split subvariety can not have a support whose inverse image is contained in that subvariety. Earlier vanishing theorems for Frobenius split varieties were based on direct limits and Serre’s vanishing theorem, but our theorem is based on inverse limits and Grothendieck’s theorem on formal functions. The result implies a Grauert-Riemenschneider type theorem.

MSC:

14F17 Vanishing theorems in algebraic geometry
14G15 Finite ground fields in algebraic geometry

Citations:

Zbl 0601.14043
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References:

[1] Raynaud, M.: Contre-exemple au ”vanishing theorem” en caractéristiquep>0. In: Ramanathan, K.G. (ed.) C.P. Ramanujam – a tribute, pp. 273–278. (Tata Inst. Fund. Res. Stud. Math., vol. 8) Berlin Heidelberg New York: Springer 1978 · Zbl 0441.14006
[2] Boutot, J.-F.: Frobenius et Cohomologie locale. Séminaire Bourbaki, 1974/75, no. 453. (Lect. Notes Math., vol. 514) Berlin Heidelberg New York: Springer 1976
[3] Grauert, H., Riemenschneider, O.: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Invent. Math.11, 263–292 (1970) · Zbl 0202.07602
[4] Hartshorne, R., Speiser, R.: Local cohomological dimension in characteristicp. Ann. Math.105, 45–79 (1977) · Zbl 0362.14002
[5] Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. Math.122, 27–40 (1985) · Zbl 0601.14043
[6] Mumford, D.: Some footnotes to the work of C.P. Ramanujam. In: Ramanathan, K.G. (ed.) C.P. Ramanujam – a tribute, pp. 247–262 (Tata Inst. Fund. Res. Stud. Math., vol. 8) Berlin Heidelberg New York: Springer 1978 · Zbl 0444.14002
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