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A Witt Nadel vanishing theorem for threefolds

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  13 January 2020

Yusuke Nakamura  and
Hiromu Tanaka
Yusuke Nakamura
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan email nakamura@ms.u-tokyo.ac.jp
Hiromu Tanaka
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan email tanaka@ms.u-tokyo.ac.jp
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Abstract

In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_{q}$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$.

Keywords

rational pointsWitt vectorsNadel vanishing theorem

MSC classification

Primary: 14J45: Fano varieties
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 3 , March 2020 , pp. 435 - 475
Copyright
© The Authors 2020

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References

Ax, J., Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261.CrossRefGoogle Scholar
Berthelot, P., Bloch, S. and Esnault, H., On Witt vector cohomology for singular varieties, Compositio Math. 143 (2007), 363–392.CrossRefGoogle Scholar
Birkar, C., Existence of flips and minimal models for 3-folds in char p, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), 169–212 (in English, with English and French summaries).CrossRefGoogle Scholar
Chatzistamatiou, A. and Rülling, K., Hodge–Witt cohomology and Witt-rational singularities, Doc. Math. 17 (2012), 663–781.Google Scholar
Esnault, H., Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math. 151 (2003), 187–191.CrossRefGoogle Scholar
Fujino, O., Foundations of the minimal model program, MSJ Memoirs, vol. 35 (Mathematical Society of Japan, 2017).CrossRefGoogle Scholar
Gongyo, Y., Nakamura, Y. and Tanaka, H., Rational points on log Fano threefolds over a finite field, J. Eur. Math. Soc. (JEMS) 21 (2019), 3759–3795.CrossRefGoogle Scholar
Hacon, C. D. and Xu, C., On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), 711–744.CrossRefGoogle Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York–Heidelberg, 1977).CrossRefGoogle Scholar
Hashizume, K., Nakamura, Y. and Tanaka, H., Minimal model program for log canonical threefolds in positive characteristic, Math. Res. Lett., to appear. Preprint (2017), arXiv:1711.10706v2.Google Scholar
Illusie, L., Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501–661 (in French).CrossRefGoogle Scholar
Katz, N. M., On a theorem of Ax, Amer. J. Math. 93 (1971), 485–499.CrossRefGoogle Scholar
Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 283–360.CrossRefGoogle Scholar
Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge [Results in Mathematics and Related Areas, 3rd Series]. A Series of Modern Surveys in Mathematics, vol. 32 (Springer, Berlin, 1996).CrossRefGoogle Scholar
Kollár, J., Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200 (Cambridge University Press, Cambridge, 2013); with a collaboration of Sándor Kovács.CrossRefGoogle Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge [Results in Mathematics and Related Areas, 3rd Series]. A Series of Modern Surveys in Mathematics, vol. 49 (Springer, Berlin, 2004).CrossRefGoogle Scholar
Lipman, J., Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279; MR 0276239.CrossRefGoogle Scholar
Poonen, B., Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), 1099–1127.CrossRefGoogle Scholar
Raynaud, M., Contre-exemple au ‘vanishing theorem’ en caractéristique p > 0, Tata Institute of Fundamental Research Studies in Mathematics, vol. 8 (Springer, Berlin–New York, 1978), 273–278.Google Scholar
Tanaka, H., Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. 216 (2014), 1–70.CrossRefGoogle Scholar
Tanaka, H., The X-method for klt surfaces in positive characteristic, J. Algebraic Geom. 24 (2015), 605–628.CrossRefGoogle Scholar
Tanaka, H., Abundance theorem for semi log canonical surfaces in positive characteristic, Osaka J. Math. 53 (2016), 535–566; MR 3492812.Google Scholar
Tanaka, H., Behavior of canonical divisors under purely inseparable base changes, J. Reine Angew. Math. 744 (2018), 237–264; MR 3871445.Google Scholar
Tanaka, H., Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble) 68 (2018), 345–376, (in English, with English and French summaries); MR 3795482.CrossRefGoogle Scholar
Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994); MR 1269324.CrossRefGoogle Scholar