Duke Mathematical Journal

The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic

Adrian Langer

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Abstract

We generalize Bogomolov’s inequality for Higgs sheaves and the Bogomolov– Miyaoka–Yau inequality in positive characteristic to the logarithmic case. We also generalize Shepherd-Barron’s results on Bogomolov’s inequality on surfaces of special type from rank $2$ to the higher-rank case. We use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo $p^{2}$. These examples contradict some claims by Xie.

Article information

Source
Duke Math. J. Volume 165, Number 14 (2016), 2737-2769.

Dates
Received: 29 August 2014
Revised: 12 October 2015
First available in Project Euclid: 2 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1464872424

Digital Object Identifier
doi:10.1215/00127094-3627203

Mathematical Reviews number (MathSciNet)
MR3551772

Zentralblatt MATH identifier
06653511

Subjects
Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
Secondary: 14G17: Positive characteristic ground fields 14J29: Surfaces of general type

Keywords
Bogomolov’s inequality logarithmic Higgs sheaves Bogomolov-Miyaoka-Yau inequality positive characteristic

Citation

Langer, Adrian. The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic. Duke Math. J. 165 (2016), no. 14, 2737--2769. doi:10.1215/00127094-3627203. https://projecteuclid.org/euclid.dmj/1464872424.


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References

  • [1] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties (in Russian), Izv. Akad. Nauk SSSR 42 (1978), no. 6, 1227–1287; English translation in Math. USSR-Izv. 13 (1979), no. 3, 499–555.
  • [2] F. R. Cossec and I. V. Dolgachev, Enriques Surfaces, I, Progr. Math. 76, Birkhäuser, Boston, 1989.
  • [3] T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 97–144.
  • [4] T. Ekedahl and N. I. Shepherd-Barron, On exceptional Enriques surfaces, preprint, arXiv:math/0405510v1 [math.AG].
  • [5] H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, DMV Seminar 20, Birkhäuser, Basel, 1992.
  • [6] F. Hirzebruch, “Arrangements of lines and algebraic surfaces” in Arithmetic and Geometry, Vol. II, Progr. Math. 36, Birkhäuser, Boston, 1983, 113–140.
  • [7] K. Kato, “Logarithmic structures of Fontaine–Illusie” in Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988), Johns Hopkins Univ. Press, Baltimore, 1989, 191–224.
  • [8] J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
  • [9] G. Lan, M. Sheng, Y. Yang, and K. Zuo, Semistable Higgs bundles of small ranks are strongly Higgs semistable, preprint, arXiv:1311.2405v1 [math.AG].
  • [10] G. Lan, M. Sheng, Y. Yang, and K. Zuo, Uniformizations of $p$-adic curves via Higgs–de Rham flows, preprint, arXiv:1404.0538.
  • [11] G. Lan, M. Sheng, and K. Zuo, Nonabelian Hodge theory in positive characteristic via exponential twisting, Math. Res. Lett. 22 (2015), 859–879.
  • [12] A. Langer, Chern classes of reflexive sheaves on normal surfaces, Math. Z. 235 (2000), 591–614.
  • [13] A. Langer, The Bogomolov–Miyaoka–Yau inequality for log canonical surfaces, J. Lond. Math. Soc. (2) 64 (2001), 327–343.
  • [14] A. Langer, Logarithmic orbifold Euler numbers of surfaces with applications, Proc. Lond. Math. Soc. (3) 86 (2003), 358–396.
  • [15] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251–276.
  • [16] A. Langer, On the S-fundamental group scheme, Ann. Inst. Fourier (Grenoble) 61 (2011), 2077–2119.
  • [17] A. Langer, Semistable modules over Lie algebroids in positive characteristic, Doc. Math. 19 (2014), 509–540.
  • [18] A. Langer, Bogomolov’s inequality for Higgs sheaves in positive characteristic, Invent. Math. 199 (2015), 889–920.
  • [19] A. Langer, Generic positivity and foliations in positive characteristic, Adv. Math. 277 (2015), 1–23.
  • [20] Ch. Liedtke and M. Satriano, On the birational nature of lifting, Adv. Math. 254 (2014), 118–137.
  • [21] V. B. Mehta and A. Ramanathan, “Homogeneous bundles in characteristic $p$” in Algebraic Geometry—Open Problems (Ravello, 1982), Lecture Notes in Math. 997, Springer, Berlin, 1983, 315–320.
  • [22] Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225–237.
  • [23] Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), 159–171.
  • [24] Y. Miyaoka, Theme and variations—inequalities between Chern numbers, Sugaku Expositions 4 (1991), 157–176.
  • [25] T. Mochizuki, Kobayashi–Hitchin correspondence for tame harmonic bundles and an application, Astérisque 309, Soc. Math. France, Paris, 2006.
  • [26] T. Mochizuki, Kobayashi–Hitchin correspondence for tame harmonic bundles, II, Geom. Topol. 13 (2009), 359–455.
  • [27] S. Mukai, Counterexamples to Kodaira’s vanishing and Yau’s inequality in positive characteristics, Kyoto J. Math. 53 (2013), 515–532.
  • [28] A. Ogus and V. Vologodsky, Nonabelian Hodge theory in characteristic $p$, Publ. Math. Inst. Hautes Études Sci. 106 (2007), 1–138.
  • [29] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. J. (2) 36 (1984), 269–291.
  • [30] M. Raynaud, “Contre-exemple au “vanishing theorem” en caractéristique $p>0$” in C. P. Ramanujam—A Tribute, Tata Inst. Fund. Res. Stud. Math. 8, Springer, Berlin, 1978, 273–278.
  • [31] A. N. Rudakov and I. R. Shafarevich, Quasi-elliptic K3 surfaces (in Russian), Uspekhi Mat. Nauk 33 (1978), no. 1, 227–228; English translation in Russian Math. Surveys 33 (1978), no. 1, 215–216.
  • [32] F. Sakai, Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980), 89–120.
  • [33] D. Schepler, Logarithmic nonabelian Hodge theory in characteristic $p$, preprint, arXiv:0802.1977.v1 [math.AG].
  • [34] J.-P. Serre, A Course in Arithmetic, Grad. Texts in Math. 7, Springer, New York, 1973.
  • [35] N. I. Shepherd-Barron, Unstable vector bundles and linear systems on surfaces in characteristic $p$, Invent. Math. 106 (1991), 243–262.
  • [36] C. Simpson, Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867–918.
  • [37] Q. Xie, Kawamata–Viehweg vanishing on rational surfaces in positive characteristic, Math. Z. 266 (2010), 561–570.
  • [38] Q. Xie, Strongly liftable schemes and the Kawamata–Viehweg vanishing in positive characteristic, Math. Res. Lett. 17 (2010), 563–572.
  • [39] Q. Xie, Strongly liftable schemes and the Kawamata–Viehweg vanishing in positive characteristic, II, Math. Res. Lett. 18 (2011), 315–328.
  • [40] Q. Xie and J. Wu, Strongly liftable schemes and the Kawamata–Viehweg vanishing in positive characteristic, III, J. Algebra 395 (2013), 12–23.
  • [41] S. T. Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA 74 (1977), 1798–1799.
  • [42] O. Zariski, The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615.
  • [43] X. Zheng, Counterexamples of Kodaira vanishing for smooth surfaces of general type in positive characteristic, preprint, arXiv:1509.04993v1 [math.AG].