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Nearby cycles and semipositivity in positive characteristic. (English) Zbl 1509.14089

Summary: We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D06 Fibrations, degenerations in algebraic geometry
14G17 Positive characteristic ground fields in algebraic geometry
13F35 Witt vectors and related rings
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[1] Arapura, D.: Kodaira-Saito vanishing via Higgs bundles in positive characteristic. J. Reine Angew. Math. 755, 293-312 (2019) Zbl 1468.14044 MR 4015235 · Zbl 1468.14044
[2] Bloch, S., Gieseker, D.: The positivity of the Chern classes of an ample vector bundle. Invent. Math. 12, 112-117 (1971) Zbl 0212.53502 MR 297773 · Zbl 0212.53502
[3] Brunebarbe, Y.: Symmetric differentials and variations of Hodge structures. J. Reine Angew. Math. 743, 133-161 (2018) Zbl 1403.14026 MR 3859271 · Zbl 1403.14026
[4] Brunebarbe, Y.: Semi-positivity from Higgs bundles. arXiv:1707.08495v1 (2017)
[5] Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52 137-252 (1980) Zbl 0456.14014 MR 601520 · Zbl 0456.14014
[6] Diaz, S., Harbater, D.: Strong Bertini theorems. Trans. Amer. Math. Soc. 324, 73-86 (1991) Zbl 0744.14004 MR 986689 · Zbl 0744.14004
[7] Ejiri, S.: Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers. J. Algebraic Geom. 26, 691-734 (2017) Zbl 1400.14026 MR 3683424 · Zbl 1400.14026
[8] Fujino, O., Fujisawa, T.: Variations of mixed Hodge structure and semipositivity theorems. Publ. RIMS Kyoto Univ. 50, 589-661 (2014) Zbl 1305.14004 MR 3273305 · Zbl 1305.14004
[9] Fujita, T.: On Kähler fiber spaces over curves. J. Math. Soc. Japan 30, 779-794 (1978) Zbl 0393.14006 MR 513085 · Zbl 0393.14006
[10] Griffiths, P. A.: Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping. Inst. Hautes Études Sci. Publ. Math. 38, 125-180 (1970) Zbl 0367.14001 MR 282990 · Zbl 0212.53503
[11] Hartshorne, R.: Algebraic Geometry. Grad. Texts in Math. 52, Springer, New York (1977) Zbl 0367.14001 MR 0463157 · Zbl 0367.14001
[12] Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. 2nd ed., Cambridge Math. Library, Cambridge Univ. Press, Cambridge (2010) Zbl 1206.14027 MR 2665168 · Zbl 1206.14027
[13] Illusie, L.: Réduction semi-stable et décomposition de complexes de de Rham à coefficients. Duke Math. J. 60, 139-185 (1990) Zbl 0708.14014 MR 1047120 · Zbl 0708.14014
[14] Katz, N. M.: Algebraic solutions of differential equations (p-curvature and the Hodge filtra-tion). Invent. Math. 18, 1-118 (1972) Zbl 0278.14004 MR 337959 · Zbl 0278.14004
[15] Kawamata, Y.: Characterization of abelian varieties. Compos. Math. 43, 253-276 (1981) Zbl 0471.14022 MR 622451 · Zbl 0471.14022
[16] Kleiman, S. L.: The Picard scheme. In: Fundamental Algebraic Geometry, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, RI, 235-321 (2005) Zbl 1085.14001 MR 2223410
[17] Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge (1998) Zbl 0926.14003 MR 1658959 · Zbl 0926.14003
[18] Lan, G., Sheng, M., Yang, Y., Zuo, K.: Uniformization of p-adic curves via Higgs-de Rham flows. J. Reine Angew. Math. 747, 63-108 (2019) Zbl 1439.14115 MR 3905130 · Zbl 1439.14115
[19] Lan, G., Sheng, M., Zuo, K.: Nonabelian Hodge theory in positive characteristic via exponen-tial twisting. Math. Res. Lett. 22, 859-879 (2015) Zbl 1326.14016 MR 3350108 · Zbl 1326.14016
[20] Lan, G., Sheng, M., Zuo, K.: Semistable Higgs bundles, periodic Higgs bundles and repres-entations of algebraic fundamental groups. J. Eur. Math. Soc. 21, 3053-3112 (2019) Zbl 1444.14048 MR 3994100 · Zbl 1444.14048
[21] Langer, A.: Moduli spaces of sheaves and principal G-bundles. In: Algebraic Geometry (Seattle, 2005), Part 1, Proc. Sympos. Pure Math. 80, Amer. Math. Soc., Providence, RI, 273-308 (2009) Zbl 1179.14010 MR 2483939 · Zbl 1179.14010
[22] Langer, A.: On positivity and semistability of vector bundles in finite and mixed characterist-ics. J. Ramanujan Math. Soc. 28A, 287-309 (2013) Zbl 1295.14032 MR 3115197 · Zbl 1295.14032
[23] Langer, A.: Semistable modules over Lie algebroids in positive characteristic. Doc. Math. 19, 509-540 (2014) Zbl 1330.14017 MR 3218782 · Zbl 1330.14017
[24] Langer, A.: Bogomolov’s inequality for Higgs sheaves in positive characteristic. Invent. Math. 199, 889-920 (2015) Zbl 1348.14048 MR 3314517 · Zbl 1348.14048
[25] Langer, A.: The Bogomolov-Miyaoka-Yau inequality for logarithmic surfaces in positive characteristic. Duke Math. J. 165, 2737-2769 (2016) Zbl 1386.14160 MR 3551772 · Zbl 1386.14160
[26] Ogus, A., Vologodsky, V.: Nonabelian Hodge theory in characteristic p. Publ. Math. Inst. Hautes Études Sci. 106, 1-138 (2007) Zbl 1140.14007 MR 2373230 · Zbl 1140.14007
[27] Patakfalvi, Z.: Semi-positivity in positive characteristics. Ann. Sci. École Norm. Sup. (4) 47, 991-1025 (2014) Zbl 1326.14015 MR 3294622 · Zbl 1326.14015
[28] Patakfalvi, Z.: On subadditivity of Kodaira dimension in positive characteristic over a general type base. J. Algebraic Geom. 27, 21-53 (2018) Zbl 1397.14030 MR 3722689 · Zbl 1397.14030
[29] Schepler, D. K.: Logarithmic nonabelian Hodge theory in characteristic p. arXiv:0802.1977 (2008)
[30] Sheng, M.: Twisted functoriality in nonabelian Hodge theory in positive characteristic. arXiv:2105.01385 (2021)
[31] Simpson, C. T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5-95 (1992) Zbl 0814.32003 MR 1179076 · Zbl 0814.32003
[32] Sun, R., Yang, J., Zuo, K.: Projective crystalline representations of étale fundamental groups and twisted periodic Higgs-de Rham flow. arXiv:1709.01485 (2017)
[33] Szpiro, L.: Sur le théorème de rigidité de Parsin et Arakelov. In: Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II, Astérisque 64, 169-202 (1979) Zbl 0425.14005 MR 563470 · Zbl 0425.14005
[34] Szpiro, L. (ed.): Séminaire sur les pinceaux de courbes de genre au moins deux. Astérisque 86 (1981) Zbl 0463.00009 MR 0642675 · Zbl 0463.00009
[35] Wahl, J. M.: Equisingular deformations of normal surface singularities. I. Ann. of Math. (2) 104, 325-356 (1976) Zbl 0358.14007 MR 422270 · Zbl 0358.14007
[36] Zuo, K.: On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applic-ations. Asian J. Math. 4, 279-301 (2000) Zbl 0983.32020 MR 1803724 · Zbl 0983.32020
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