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Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. (English) Zbl 1476.14060

Let \(f: X \rightarrow X\) be a surjective endomorphism of a projective variety over a fixed algebraically closed field k of arbitrary characteristic \( p \geq 0\). \(f\) is said to be \( q\)-polarized (resp. numerically \(q\)-polarized) by \(H\), if there is an ample Cartier divisor \(H\) such that \(f^*H \sim q H\), a linear equivalence (resp. \(f^*H \equiv q H\), a numerical equivalence) for some integer \( q > 1\). In this paper, the authors prove the following results.
1. If \(\mathrm{char} k = 0\), in 2010, N. Nakayama and D.-Q. Zhang [Math. Ann. 346, No. 4, 991–1018 (2010; Zbl 1189.14043)] showed that the numerically polarized condition is equivalent to the polarized condition after replacing \(H\). If \( X\) is normal and \(f\) is separable, the authors generalize this result to arbitrary characteristic.
2. The authors generalize the result of S. Boucksom et al. [Duke Math. J. 161, No. 8, 1455–1520 (2012; Zbl 1251.14026)] for \(\mathbb{Q}\)-Gorenstein normal projective variety \(X\) and prove:
Let \(f: X \rightarrow X\) be a polarized separable endomorphism of a \(\mathbb{Q}\)-Gorenstein normal projective variety \(X\) of dimension \(n \geq 0\) over the field \(k\) of arbitrary characteristic. Then \(-K_X\) is numerically equivalent to an effective \(\mathbb{Q}\)-Cartier divisor. In particular, if \(\mathrm{Alb}(X)\) is trivial, then the Iitaka \(D\)-dimension \(\kappa (X, -K_X ) \geq 0\).
3. They give an affirmative answer to a question of H. Krieger and P. Reschke [Bull. Soc. Math. Fr. 145, No. 3, 449–468 (2017; Zbl 1387.32022)] and with a very different proof, generalize a theorem of S. Meng and D.-Q. Zhang [Adv. Math. 325, 243–273 (2018; Zbl 1387.14057)] to arbitrary characteristics.
4. They prove:
If \(f: X \rightarrow X\) is a numerically polarized separable endomorphism of a normal projective variety \(X\) with \( K_X\) being pseudo-effective and \(\mathbb{Q}\)-Cartier, then \(f\) is quasi-étale and \(K_X\sim_\mathbb{Q} 0 \).
5. Let \(f^{\mathrm{Gal}}: \bar{X} \rightarrow X\) be the Galois closure of \( f\). Nakayama’s result on a normal projective surface \(X\) defined over a field of characteristic 0 is generalized to char \(p>5\) and \(p\nmid\) deg \(f^{\mathrm{Gal}}\):
If \(f: X \rightarrow X\) is a polarized endomorphism of a normal projective surface \(X\) over the field \(k\) of characteristic \(p > 5\). Suppose \(p\nmid\) deg \( f^{\mathrm{Gal}}\) and \(K_X\) is pseudo-effective, then \(X\) is a \(\mathbb{Q}\)-abelian surface. In particular, \(X\) is \(\mathbb{Q}\)-factorial and klt.
6. They show:
If \( p > 5\) and co-prime to degree \(f^{\mathrm{Gal}}\), then one can run the minimal model program (MMP) \(f\)-equivariantly, after replacing \(f\) by a positive power, for a mildly singular threefold \(X\) and reach a variety \(Y\) with torsion canonical divisor (and also with \( Y\) being a quasi-étale quotient of an abelian variety when dim\((Y ) \leq 2\)). They also show that a power of \(f\) acts as a scalar multiplication on the Néron-Severi group of \( X\) (modulo torsion) when \(X\) is a smooth and rationally chain connected projective variety of dimension at most three.

MSC:

14H30 Coverings of curves, fundamental group
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
14E30 Minimal model program (Mori theory, extremal rays)
11G10 Abelian varieties of dimension \(> 1\)
08A35 Automorphisms and endomorphisms of algebraic structures
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References:

[1] Beauville, A., Variété Kählerinnes dont la premiere classe de Chern est nulle, J. Diff. Geom., 18, 755-782 (1983) · Zbl 0537.53056 · doi:10.4310/jdg/1214438181
[2] Bhatt, B.; Carvajal-Rojas, J.; Graf, P.; Schwede, K.; Tucker, K., Étale fundamental groups of strongly \(F\)-regular schemes, Int. Math. Res. Not. IMRN, 2019, 14, 4325-4339 (2019) · Zbl 1457.14045 · doi:10.1093/imrn/rnx253
[3] Birkar, C.; Waldron, J., Existence of Mori fibre spaces for \(3\)-folds in \({{\rm char}}\, p\), Adv. Math., 313, 62-101 (2017) · Zbl 1373.14019 · doi:10.1016/j.aim.2017.03.032
[4] Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char.\(p\), II, In: Complex analysis and algebraic geometry. Cambridge University Press, pp. 23-42 (1977) · Zbl 0348.14021
[5] Boucksom, S.; de Fernex, T.; Favre, C., The volume of an isolated singularity, Duke Math. J., 161, 8, 1455-1520 (2012) · Zbl 1251.14026 · doi:10.1215/00127094-1593317
[6] Danilov, VI, Algebraic varieties and schemes. Algebraic geometry, I, 167-297, Encyclopaedia Math. Sci. 23 (1994), Berlin: Springer, Berlin · Zbl 0787.00008
[7] Debarre, O., Higher-dimensional algebraic geometry, Universitext (2001), New York: Springer, New York · Zbl 0978.14001
[8] Deligne, P.; Lusztig, G., Representations of reductive groups over finite fields, Ann. Math. (2), 103, 1, 103-161 (1976) · Zbl 0336.20029 · doi:10.2307/1971021
[9] Fantechi, B.; Göttsche, L.; Illusie, L.; Kleiman, SL; Nitsure, N.; Vistoli, A., Fundamental algebraic geometry, Grothendieck’s FGA explained, mathematical surveys and monographs, 123 (2005), Providence: American Mathematical Society, Providence · Zbl 1085.14001
[10] Fujino, O.; Tanaka, H., On log surfaces, Proc. Jpn. Acad. Ser. A Math. Sci., 88, 8, 109-114 (2012) · Zbl 1268.14012 · doi:10.3792/pjaa.88.109
[11] Gongyo, Y., Nakamura, Y., Tanaka, H.: Rational points on log Fano threefolds over a finite field, J. Eur. Math. Soc. (to appear). arXiv:1512.05003 · Zbl 1462.14045
[12] Greb, D.; Kebekus, S.; Peternell, T., Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties, Duke Math. J., 165, 10, 1965-2004 (2016) · Zbl 1360.14094 · doi:10.1215/00127094-3450859
[13] Hacon, C.; Patakfalvi, Z.; Zhang, L., Birational characterization of abelian varieties and ordinary abelian varieties in characteristic \(p>0\), Duke Math. J., 168, 9, 1723-1736 (2019) · Zbl 1436.14033 · doi:10.1215/00127094-2019-0008
[14] Hacon, C.; Xu, C., On the three dimensional minimal model program in positive characteristic, J. Am. Math. Soc., 28, 3, 711-744 (2015) · Zbl 1326.14032 · doi:10.1090/S0894-0347-2014-00809-2
[15] Hara, N., Classification of two-dimensional \(F\)-regular and \(F\)-pure singularities, Adv. Math., 133, 1, 33-53 (1998) · Zbl 0905.13002 · doi:10.1006/aima.1997.1682
[16] Hartshorne, R., Algebraic geometry, Grad. Texts in Mathematics (1977), New York: Springer, New York · Zbl 0367.14001
[17] Hu, W., The Lawson-Yau formula and its generalization, J. Pure Appl. Algebra, 217, 1, 45-53 (2013) · Zbl 1312.14028 · doi:10.1016/j.jpaa.2012.04.009
[18] Kollár, J., Shafarevich maps and plurigenera of algebraic varieties, Invent. Math., 113, 177-215 (1993) · Zbl 0819.14006 · doi:10.1007/BF01244307
[19] Kollár, J.; Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics (1998), Cambridge: Cambridge University Press, Cambridge · Zbl 0926.14003
[20] Krieger, H.; Reschke, P., Cohomological conditions on endomorphisms of projective varieties, Bull. Soc. Math. France, 145, 3, 449-468 (2017) · Zbl 1387.32022 · doi:10.24033/bsmf.2744
[21] Lang, S., Abelian varieties (1983), New York: Springer, New York · Zbl 0516.14031
[22] Langer, A., Generic positivity and foliations in positive characteristic, Adv. Math., 277, 1-23 (2015) · Zbl 1348.14070 · doi:10.1016/j.aim.2015.02.015
[23] Laumon, G., Comparaison de caractéristiques d’Euler-Poincaré en cohomologie \(\ell \)-adique, C. R. Acad. Sci. Paris Sér. I Math., 292, 3, 209-212 (1981) · Zbl 0468.14005
[24] Meng, S.; Zhang, D-Q, Building blocks of polarized endomorphisms of normal projective varieties, Adv. Math., 325, 243-273 (2018) · Zbl 1387.14057 · doi:10.1016/j.aim.2017.11.026
[25] Mumford, D.: Abelian varieties, with appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition, Tata Institute of Fundamental Research Studies in Mathematics, 5, Published for the Tata Institute of Fundamental Research, Bombay · Zbl 0223.14022
[26] Nakayama, N.: On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms, Preprint 2 September (2008)
[27] Nakayama, N.; Zhang, D-Q, Polarized endomorphisms of complex normal varieties, Math. Ann., 346, 4, 991-1018 (2010) · Zbl 1189.14043 · doi:10.1007/s00208-009-0420-y
[28] Okawa, S., Extensions of two Chow stability criteria to positive characteristics, Michigan Math. J., 60, 3, 687-703 (2011) · Zbl 1230.14068 · doi:10.1307/mmj/1320763055
[29] Pink, R.; Roessler, D., On \(\psi \)-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture, J. Algebraic Geom., 13, 4, 771-798 (2004) · Zbl 1072.14054 · doi:10.1090/S1056-3911-04-00368-6
[30] Tanaka, H., Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J., 216, 1-70 (2014) · Zbl 1311.14020 · doi:10.1215/00277630-2801646
[31] Wahl, J., A characteristic number for links of surface singularities, J. Am. Math. Soc., 3, 3, 625-637 (1990) · Zbl 0743.14026 · doi:10.1090/S0894-0347-1990-1044058-X
[32] Waldron, J., The LMMP for log canonical 3-folds in characteristic \(p>5\), Nagoya Math. J., 230, 48-71 (2018) · Zbl 1423.14113 · doi:10.1017/nmj.2017.2
[33] Yau, ST, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equations, I, Commun. Pure Appl. Math., 31, 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
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