Xu, Chenyang; Zhang, Lei Nonvanishing for 3-folds in characteristic \(p>5\). (English) Zbl 1468.14032 Duke Math. J. 168, No. 7, 1269-1301 (2019). Summary: We prove the nonvanishing theorem for 3-folds over an algebraically closed field \(k\) of characteristic \(p>5\). Cited in 6 Documents MSC: 14E30 Minimal model program (Mori theory, extremal rays) 14G17 Positive characteristic ground fields in algebraic geometry Keywords:positive characteristic; nonvanishing; abundance; minimal model × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] V. Alexeev, Boundedness and \(K^2\) for log surfaces, Internat. J. Math. 5 (1994), no. 6, 779-810. · Zbl 0838.14028 [2] T. Bauer, F. Campana, T. Eckl, S. Kebekus, T. Peternell, S. Rams, T. Szemberg, and L. Wotzlaw, “A reduction map for nef line bundles” in Complex Geometry (Göttingen, 2000), Springer, Berlin, 2002, 27-36. · Zbl 1054.14019 [3] B. Bhatt, O. Gabber, and M. Olsson, Finiteness of étale fundamental groups by reduction modulo \(p\), preprint, arXiv:1705.07303v1 [math.AG]. [4] C. Birkar, Existence of flips and minimal models for 3-folds in char \(p\), Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 1, 169-212. · Zbl 1346.14040 [5] C. Birkar and J. Waldron, Existence of Mori fibre spaces for 3-folds in \(\operatorname{char}p\), Adv. Math. 313 (2017), 62-101. · Zbl 1373.14019 [6] J. Carvajal-Rojas, K. Schwede, and K. Tucker, Fundamental groups of \(F\)-regular singularities via \(F\)-signature, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 4, 993-1016. · Zbl 1408.13015 · doi:10.24033/asens.2370 [7] P. Cascini, H. Tanaka, and C. Xu, On base point freeness in positive characteristic, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 5, 1239-1272. · Zbl 1408.14020 · doi:10.24033/asens.2269 [8] O. Das and J. Waldron, On the abundance problem for \(3\)-folds in characteristic \(p>5\), Math. Z., published online 13 August 2018. · Zbl 1462.14017 [9] T. Ekedahl, Diagonal complexes and \(F\)-gauge structures, Travaux en Cours, Hermann, Paris, 1986. · Zbl 0593.14016 [10] T. Ekedahl, “Foliations and inseparable morphisms” in Algebraic Geometry, Bowdoin, 1985, Part 2 (Bowdoin, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, 1987, 139-149. · Zbl 0659.14018 [11] T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 97-144. · Zbl 0674.14028 · doi:10.1007/BF02699128 [12] H. Esnault and V. Mehta, Simply connected projective manifolds in characteristic \(p>0\) have no nontrivial stratified bundles, Invent. Math. 181 (2010), no. 3, 449-465. · Zbl 1203.14029 · doi:10.1007/s00222-010-0250-2 [13] B. Fantechi, L. Göttsche, L. Illusie, S. Kleiman, N. Nitsure, and A. Vistoli, Fundamental Algebraic Geometry, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, 2005. · Zbl 1085.14001 [14] A. Grothendieck and J. Murre, The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme, Lecture Notes in Math. 208, Springer, Berlin, 1971. · Zbl 0216.33001 [15] A. Grothendieck and M. Raynaud, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois-Marie (SGA1), Doc. Math. (Paris) 3, Soc. Math. France, Paris, 2003. [16] A. Grothendieck and M. Raynaud, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, Séminaire de Géométrie Algébrique du Bois Marie (SGA2), Doc. Math. (Paris) 4, Soc. Math. France, Paris, 2005. [17] C. Hacon, Z. Patakfalvi, and L. Zhang, Birational characterization of abelian varieties and ordinary abelian varieties in characteristic \(p>0\), to appear in Duke Math. J., preprint, arXiv:1703.06631v1 [math.AG]. · Zbl 1436.14033 [18] C. Hacon and J. Witaszek, On the rationality of kawamata log terminal singularities in positive characteristic, preprint, arXiv:1706.03204v2 [math.AG] · Zbl 1439.14056 [19] C. Hacon and C. Xu, On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), no. 3, 711-744. · Zbl 1326.14032 · doi:10.1090/S0894-0347-2014-00809-2 [20] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. · Zbl 0367.14001 [21] K. Hashizume, Y. Nakamura, and H. Tanaka, Minimal model program for log canonical threefolds in positive characteristic, to appear in Math. Res. Lett., preprint, arXiv:1711.10706v2 [math.AG]. · Zbl 1472.14018 [22] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Cambridge Univ. Press, Cambridge, 2010. · Zbl 1206.14027 [23] L. Illusie, “Finiteness, duality, and Künneth theorems in the cohomology of the de Rham-Witt complex” in Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer, Berlin, 1983, 20-72. · Zbl 0538.14013 [24] N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math. Inst. Hautes Études Sci. 39 (1970), 175-232. · Zbl 0221.14007 · doi:10.1007/BF02684688 [25] D. Keeler, Ample filters of invertible sheaves, J. Algebra 259 (2003), no. 1, 243-283. · Zbl 1082.14004 · doi:10.1016/S0021-8693(02)00557-4 [26] D. Keeler, Fujita’s conjecture and Frobenius amplitude, Amer. J. Math. 130 (2008), no. 5, 1327-1336. · Zbl 1159.14003 · doi:10.1353/ajm.0.0015 [27] M. Kerz and A. Schmidt, On different notions of tameness in arithmetic geometry, Math. Ann. 346 (2010), no. 3, 641-668. · Zbl 1185.14019 · doi:10.1007/s00208-009-0409-6 [28] J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996. · Zbl 0877.14012 [29] J. Kollár, Singularities of the Minimal Model Program, Cambridge Tracts in Math. 200, Cambridge Univ. Press, Cambridge, 2013. · Zbl 1282.14028 [30] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998. · Zbl 0926.14003 [31] J. Kollár et al., Flips and Abundance for Algebraic Threefolds (Salt Lake City, 1991) Astérisque 211, Soc. Math. France, Paris, 1992. · Zbl 0782.00075 [32] H. Lange and U. Stuhler, Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1977), no. 1, 73-83. · Zbl 0349.14018 · doi:10.1007/BF01215129 [33] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), no. 1, 251-276. · Zbl 1080.14014 · doi:10.4007/annals.2004.159.251 [34] A. Langer, “Moduli spaces of sheaves and principal \(G\)-bundles” in Algebraic Geometry (Seattle, 2005), Part 1, Proc. Sympos. Pure Math. 80, Amer. Math. Soc, Providence, 2009, 273-308. · Zbl 1179.14010 [35] A. Langer, On the S-fundamental group scheme, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 5, 2077-2119. · Zbl 1247.14019 · doi:10.5802/aif.2667 [36] A. Langer, On the S-fundamental group scheme, II, J. Inst. Math. Jussieu 11 (2012), no. 4, 835-854. · Zbl 1252.14028 · doi:10.1017/S1474748012000011 [37] A. Langer, Generic positivity and foliations in positive characteristic, Adv. Math. 277 (2015), 1-23. · Zbl 1348.14070 · doi:10.1016/j.aim.2015.02.015 [38] R. Lazarsfeld, Positivity in Algebraic Geometry, I, Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin, 2004. · Zbl 1066.14021 [39] Y. Miyaoka, On the Kodaira dimension of minimal threefolds, Math. Ann. 281 (1988), no. 2, 325-332. · Zbl 0625.14023 · doi:10.1007/BF01458437 [40] Z. Patakfalvi and J. Waldron, Singularities of general fibers and the LMMP, preprint, arXiv:1708.04268v2 [math.AG]. · Zbl 1498.14036 [41] J. Serre, Sur la topologie des variétés algébriques en caractéristique \(p\), Symposium internacional de topología algebraica, Universidad Nacional Autónoma de México/UNESCO, Mexico City, 1958, 24-53. · Zbl 0098.13103 [42] L. Szpiro, “Sur le théorème de rigidité de Parsin et Arakelov” in Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II, Astérisque 64, Soc. Math. France, Paris, 1979, 169-202. · Zbl 0425.14005 [43] H. Tanaka, The X-method for klt surfaces in positive characteristic, J. Algebraic Geom. 24 (2015), no. 4, 605-628. · Zbl 1338.14017 · doi:10.1090/S1056-3911-2014-00627-5 [44] H. Tanaka, Abundance theorem for surfaces over imperfect fields, preprint, arXiv:1502.01383v5 [math.AG]. · Zbl 1445.14029 [45] J. Witaszek, On the canonical bundle formula and fibrations of relative dimension one in positive characteristic, preprint, arXiv:1711.04380v2 [math.AG]. · Zbl 1492.14026 [46] C. Xu, Finiteness of algebraic fundamental groups, Compos. Math. 150 (2014), no. 3, 409-414. · Zbl 1291.14057 · doi:10.1112/S0010437X13007562 [47] C. Xu, On the base-point-free theorem of 3-folds in positive characteristic, J. Inst. Math. Jussieu 14 (2015), no. 3, 577-588. · Zbl 1346.14020 · doi:10.1017/S1474748014000097 [48] L. Zhang, Abundance for non-uniruled 3-folds with non-trivial Albanese maps in positive characteristics, J. Lond. Math. Soc., published online 13 September 2018. · Zbl 1410.14013 [49] L. Zhang, Abundance for 3-folds with non-trivial Albanese maps in positive characteristic, preprint, arXiv:1705.00847v2 [math.AG]. · Zbl 1455.14032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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