Generic vanishing and classification of irregular surfaces in positive characteristics

Wang, Yuan

doi:10.1090/tran/6914

Generic vanishing and classification of irregular surfaces in positive characteristics
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by Yuan Wang PDF

Trans. Amer. Math. Soc. 369 (2017), 8559-8585 Request permission

Abstract:

We establish a generic vanishing theorem for surfaces in characteristic $p$ that lift to $W_2(k)$ and use it for classification of surfaces of general type with Euler characteristic $1$ and large Albanese dimension.

References

A. Beauville, L’inegalité $p_g\ge 2q-4$ pour les surfaces de type général, Appendix to \cite{Debarre82}, 1982.
Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314, DOI 10.1017/CBO9780511623936
E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. $p$. II, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 23–42. MR 0491719
E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. $p$. III, Invent. Math. 35 (1976), 197–232. MR 491720, DOI 10.1007/BF01390138
Fabrizio Catanese, Ciro Ciliberto, and Margarida Mendes Lopes, On the classification of irregular surfaces of general type with nonbirational bicanonical map, Trans. Amer. Math. Soc. 350 (1998), no. 1, 275–308. MR 1422597, DOI 10.1090/S0002-9947-98-01948-5
O. Debarre, Inégalités numériques pour les surfaces de type général, Bull. Soc. Math. France 110 (1982), no. 3, 319–346 (French, with English summary). With an appendix by A. Beauville. MR 688038
Pierre Deligne and Luc Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270 (French). MR 894379, DOI 10.1007/BF01389078
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Renée Elkik, Rationalité des singularités canoniques, Invent. Math. 64 (1981), no. 1, 1–6 (French). MR 621766, DOI 10.1007/BF01393930
F. Enriques, Sulla classificazione delle superficie algebriche e particolarmente sulle superfichie di genere $p^1=1$, Atti. Acc. Lincei V Ser. 23 (1914).
Federigo Enriques, Le Superficie Algebriche, Nicola Zanichelli, Bologna, 1949 (Italian). MR 0031770
Hélène Esnault and Eckart Viehweg, Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkhäuser Verlag, Basel, 1992. MR 1193913, DOI 10.1007/978-3-0348-8600-0
Mark Green and Robert Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2, 389–407. MR 910207, DOI 10.1007/BF01388711
Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 308104, DOI 10.1007/BF01390094
Christopher D. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math. 575 (2004), 173–187. MR 2097552, DOI 10.1515/crll.2004.078
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
C. D. Hacon and S. J. Kovács, Generic vanishing fails for singular varieties and in characteristic $p>0$, Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser., vol. 417, Cambridge Univ. Press, Cambridge, 2015, pp. 240–253. MR 3380452
Christopher D. Hacon and Rita Pardini, Surfaces with $p_g=q=3$, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2631–2638. MR 1895196, DOI 10.1090/S0002-9947-02-02891-X
Christopher D. Hacon and Zsolt Patakfalvi, Generic vanishing in characteristic $p>0$ and the characterization of ordinary abelian varieties, Amer. J. Math. 138 (2016), no. 4, 963–998. MR 3538148, DOI 10.1353/ajm.2016.0031
D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2244106, DOI 10.1093/acprof:oso/9780199296866.001.0001
Jun-ichi Igusa, A fundamental inequality in the theory of Picard varieties, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 317–320. MR 71113, DOI 10.1073/pnas.41.5.317
J. Jang, Generic ordinarity for Semi-Stable Fibrations, arXiv: 0805.3982v1, 2008.
J. Kollár and S. Kovács, Birational geometry of log surfaces, preprint.
János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751–798. MR 187255, DOI 10.2307/2373157
K. Kodaira, On the structure of compact complex analytic surfaces. II, Amer. J. Math. 88 (1966), 682–721. MR 205280, DOI 10.2307/2373150
K. Kodaira, On the structure of compact complex analytic surfaces. III, Amer. J. Math. 90 (1968), 55–83. MR 228019, DOI 10.2307/2373426
János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
Adrian Langer, Bogomolov’s inequality for Higgs sheaves in positive characteristic, Invent. Math. 199 (2015), no. 3, 889–920. MR 3314517, DOI 10.1007/s00222-014-0534-z
Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
Christian Liedtke, Algebraic surfaces in positive characteristic, Birational geometry, rational curves, and arithmetic, Simons Symp., Springer, Cham, 2013, pp. 229–292. MR 3114931, DOI 10.1007/978-1-4614-6482-2_{1}1
V. B. Mehta and V. Srinivas, Varieties in positive characteristic with trivial tangent bundle, Compositio Math. 64 (1987), no. 2, 191–212. With an appendix by Srinivas and M. V. Nori. MR 916481
Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081
David Mumford, Enriques’ classification of surfaces in $\textrm {char}\ p$. I, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 325–339. MR 0254053
D. Mumford, Abelian Variety, American Mathematical Society, 2012. MR2514037
B. Moonen and van der Geer. Abelian Variety.
Gian Pietro Pirola, Surfaces with $p_g=q=3$, Manuscripta Math. 108 (2002), no. 2, 163–170. MR 1918584, DOI 10.1007/s002290200253
Giuseppe Pareschi and Mihnea Popa, Regularity on abelian varieties. I, J. Amer. Math. Soc. 16 (2003), no. 2, 285–302. MR 1949161, DOI 10.1090/S0894-0347-02-00414-9
Giuseppe Pareschi and Mihnea Popa, GV-sheaves, Fourier-Mukai transform, and generic vanishing, Amer. J. Math. 133 (2011), no. 1, 235–271. MR 2752940, DOI 10.1353/ajm.2011.0000
Richard Pink and Damian Roessler, A conjecture of Beauville and Catanese revisited, Math. Ann. 330 (2004), no. 2, 293–308. MR 2089427, DOI 10.1007/s00208-004-0549-7
Francesco Zucconi, Surfaces with $p_g=q=2$ and an irrational pencil, Canad. J. Math. 55 (2003), no. 3, 649–672. MR 1980618, DOI 10.4153/CJM-2003-027-8