Srivastava, Tanya Kaushal

On Derived Equivalences of K3 Surfaces in Positive Characteristic

Doc. Math. 24, 1135-1177 (2019)
DOI: 10.25537/dm.2019v24.1135-1177
Communicated by Takeshi Saito

Summary

For an ordinary K3 surface over an algebraically closed field of positive characteristic we show that every automorphism lifts to characteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characteristic as well. We show that the image of the derived autoequivalence group of a K3 surface of finite height in the group of isometries of its crystalline cohomology has index at least two. Moreover, we provide a conditional upper bound on the kernel of this natural cohomological descent map. Further, we give an extended remark in the appendix on the possibility of an F-crystal structure on the crystalline cohomology of a K3 surface over an algebraically closed field of positive characteristic and show that the naive F-crystal structure fails in being compatible with inner product.

Mathematics Subject Classification

14F05, 14F30, 14J50, 14J28, 14G17

Keywords/Phrases

derived equivalences, K3 surfaces, Automorphisms, positive characteristic

References

  • 1. de Jong A. et al., Stacks Project, http://stacks.math.columbia.edu, 2018.
  • 2. Artin M., Versal deformation and algebraic stacks, Inventiones Math. 27, 165-189, 1974. DOI 10.1007/BF01390174; zbl 0317.14001; MR0399094.
  • 3. Artin M., Mazur B., Formal groups arising from algebraic varieties, Ann. Sc. Éc. Norm. Sup. 4e série 10, 87-132, 1977. DOI 10.24033/asens.1322; zbl 0351.14023; MR0457458.
  • 4. Barth B., Hulek K., Peters C. A. M. , Van De Ven A., Compact complex surfaces, A Series of Modern Surveys in Mathematics 4, Springer, 2003. zbl 0718.14023; MR2030225.
  • 5. Bartocci C., Bruzzo U., Ruipérez D. H., Fourier-Mukai and Nahm transforms in geometry and mathematical physics, Progress in Mathematics 276, Birkhäuser, 2009. DOI 10.1007/b11801; zbl 1186.14001; MR2511017.
  • 6. Bayer A., Bridgeland T., Derived automorphism groups of $K3$ surfaces of Picard rank 1, Duke Math. J. 166, no. 1, 75-124, 2017. DOI 10.1215/00127094-3674332; zbl 1358.14019; MR3592689; arxiv 1310.8266.
  • 7. Beilinson A. A., Bernstein J., Deligne, P., Faisceaux pervers, Astérisque, Soc. Math. France 100, 5-171, 1982. zbl 0536.14011; MR0751966.
  • 8. Berthelot P., Cohomologie cristalline des schémas de caractéristique $p > 0$, Lecture Notes in Math. 407, Springer, 1974. zbl 0298.14012; MR0384804.
  • 9. Berthelot P., Ogus A., Notes on crystalline cohomology, Annals of Math. Lecture Notes, Princeton University Press, 1978. DOI 10.1515/9781400867318; zbl 0383.14010; MR0491705.
  • 10. Berthelot P., Ogus A., F-isocrystals and de Rham cohomology. I. Inventiones Mathematicae 72, 159-200, 1983. DOI 10.1007/BF01389319; zbl 0516.14017; MR0700767.
  • 11. Berthelot P., Illusie L., Classes de Chern en cohomologie cristalline, C.R. Acad Sci. Series A 270, 1695-1697, 1750-1752, 1970. zbl 0198.26201; MR0269660.
  • 12. Bondal A., Orlov D., Reconstruction of a variety from the derived category and groups of autoequivalences, Comp. Math. 125, 327-344, 2001. DOI 10.1023/A:1002470302976; zbl 0994.18007; MR1818984 ; arxiv alg-geom/9712029.
  • 13. Bridgeland T., Stability conditions on triangulated categories, Annals of Mathematics, Second Series 166, no. 2, 317-345, 2007. DOI 10.4007/annals.2007.166.317; zbl 1137.18008; MR2373143; arxiv math/0212237.
  • 14. Bridgeland T., Stability conditions on $K3$ surfaces, Duke Math. J. 141, no. 2, 241-291, 2008. DOI 10.1215/S0012-7094-08-14122-5; zbl 1138.14022; MR2376815; arxiv math/0307164.
  • 15. Bridgeland T., Spaces of stability conditions, Algebraic geometry, Seattle 2005. Part 1, 1-21, Proc. Sympos. Pure Math. 80, Amer. Math. Soc., Providence, RI, 2009. zbl 1169.14303; MR2483930; arxiv math/0611510.
  • 16. Deligne P., Relevement des surfaces K3 en characteristique nulle, In: Surfaces Algebriques, Seminar Orsay, 1976-78. Lecture Notes in Math. 868, 58-79, Springer, 1981. zbl 0495.14024; MR0638598.
  • 17. Deligne P., Illusie L., Cristaux ordinaires et coordonées canoniques, In: Surfaces Algebriques, Seminar Orsay, 1976-78. Lecture Notes in Math. 868, Springer, 1981. zbl 0537.14012; MR0638599.
  • 18. Dieudonné J., Lie groups and Lie hyperalgebras over a field of characteristic $p > 0$. IV, American Journal of Mathematics 77 (3), 429-452, 1955. DOI 10.2307/2372633; zbl 0064.25602; MR0071718.
  • 19. Esnault H., Oguiso K., Non-liftability of automorphism groups of K3 surface in positive characteristic, Math. Ann. 363, 1187-1206, 2015. DOI 10.1007/s00208-015-1197-9; zbl 1349.14131; MR3412356; arxiv 1406.2761.
  • 20. Gelfand S., Manin Y. Methods of homological algebra, Springer Monographs in Mathematics, 1997. zbl 0855.18001.
  • 21. Grothendieck A., Éléments de géométrie algébrique, III, Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math. 11, 1961. zbl 0122.16102; MR0217085.
  • 22. Grothendieck A., Théorie des topos et cohomologie étale des schémas, SGA 4, Tome III, Lecture Notes in Mathematics 305, 1972. DOI 10.1007/BFb0070714; zbl 0245.00002.
  • 23. Grothendieck A., Sur quelques points d'algèbre homologique, I. Tohoku Math. J., Second series 9, 119-221, 1957. zbl 0118.26104; MR0102537.
  • 24. Gómez T., Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci. 111 (1), 1-31, 2001. DOI 10.1007/BF02829538; zbl 0982.14005; MR1818418; arxiv math/9911199.
  • 25. Hartshorne R., On the De Rham cohomology of algebraic varieties, Publications Mathematiques de l'IHÉS 45, 5-99, 1975. DOI 10.1007/BF02684298; zbl 0326.14004; MR0432647.
  • 26. Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics 52, Springer, 1977. zbl 0367.14001; MR0463157.
  • 27. Hartshorne R., Deformation theory, Graduate Texts in Mathematics 257, Springer, 2010. DOI 10.1007/978-1-4419-1596-2; zbl 1186.14004; MR2583634.
  • 28. Hosono S., Lian B. H., Oguiso K., Yau S.-T., Autoequivalences of derived category of a K3 surface and monodromy transformations, J. Alg. Geom. 13, 513-545, 2004. DOI 10.1090/S1056-3911-04-00364-9; zbl 1070.14042; MR2047679; arxiv math/0201047.
  • 29. Hosono S., Lian B. H., Oguiso K., Yau S.-T., Fourier-Mukai number of a K3 surface, CRM Proc. Lecture Notes 38, 177-192, 2004. zbl 1076.14045; MR2096145; arxiv math/0202014.
  • 30. Huybrechts D., Fourier Mukai transforms in algebraic geometry, Oxford Science Publication, 2006. zbl 1095.14002; MR2244106.
  • 31. Huybrechts D., Lehn M., The geometry of moduli spaces of sheaves, Second edition, Cambridge Mathematical Library, Cambridge University Press, 2010. zbl 1206.14027; MR2665168.
  • 32. Huybrechts D., Marci E., Stellari P., Derived equivalences of K3 surfaces and orientation, Duke Math. J. 149, 461-507, 2009. DOI 10.1215/00127094-2009-043; zbl 1237.18008; MR2553878; arxiv 0710.1645.
  • 33. Huybrechts D., Marci E., Stellari P., Stability conditions for generic K3 categories, Comp. Math. 144, 134-162, 2008. DOI 10.1112/S0010437X07003065; zbl 1152.14037; MR2388559; arxiv math/0608430.
  • 34. Huybrechts D., Introduction to stability conditions. In: Moduli Spaces, Cambridge University Press, edited by Brambila-Paz, García-Prada, Newstead and Thomas. Lectures at the Newton Institute January 2011. London Mathematical Society Lecture Note Series 411, 179-229, 2014. zbl 1316.14003; arxiv 1111.1745.
  • 35. Huybrechts D., Richard T., Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Math. Ann. 346, 545-569, 2013. DOI 10.1007/s00208-009-0397-6; zbl 1186.14014; MR2578562; arxiv 0805.3527.
  • 36. Huybrechts D., Lectures on K3 surfaces, Cambridge University Press, 2016. MR3586372.
  • 37. Illusie L., Report on crystalline cohomology. In: Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), American Mathematical Society, Providence, R.I., 459-478, 1975. zbl 0326.14005; MR0393034.
  • 38. Katz N., Slope filtration of F-crystals, Astérisque 63, 113-164, 1979. zbl 0426.14007; MR0563463.
  • 39. Kashiwara M., Schapira P., Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften 292, Springer, 1990. zbl 0709.18001; MR1074006.
  • 40. Knutson D., Algebraic spaces, Lecture Notes in Mathematics 203, 1971. DOI 10.1007/BFb0059750; zbl 0221.14001; MR0302647.
  • 41. Kontsevich M., Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians (Z\``urich 1994), Birkh\''auser, 120-139, 1995. zbl 0846.53021; MR1403918; arxiv alg-geom/9411018.
  • 42. Langer A., Semistable sheaves in positive characteristic, Annals of Math. 159, 251-276, 2004. DOI 10.4007/annals.2004.159.251; zbl 1080.14014; MR2051393.
  • 43. Lieblich M., Moduli of complexes on a proper morphism, J. Algebraic Geometry 15, 175-206, 2006. DOI 10.1090/S1056-3911-05-00418-2; zbl 1085.14015; MR2177199; arxiv math/0502198.
  • 44. Lieblich M., Moduli of twisted sheaves, Duke Math. J. 138, 23-118, 2007. DOI 10.1215/S0012-7094-07-13812-2; zbl 1122.14012; MR2309155; arxiv math/0411337.
  • 45. Lieblich M., Olsson M., Fourier Mukai partners of K3 surfaces in positive characteristic, Annales Scientifiques de l'ENS 48, fascicule 5, 1001-1033, 2015. DOI 10.24033/asens.2264; zbl 1342.14038; MR3429474; arxiv 1112.5114.
  • 46. Lieblich M., Olsson M., A stronger derived Torelli theorem for K3 surfaces, In: Bogomolov F., Hassett B., Tschinkel Y. (eds), Geometry Over Nonclosed Fields. Simons Symposia. Springer, Cham, 2017. DOI 10.1007/978-3-319-49763-1_5; zbl 06999625; MR3644252; arxiv 1512.06451.
  • 47. Lieblich M., Maulik D., A note on the cone conjecture for K3 surfaces in positive characteristic, Math. Res. Lett. 25, no. 6, 1879-1891, 2018. DOI 10.4310/MRL.2018.v25.n6.a9; zbl 07072582; MR3934849; arxiv 1102.3377.
  • 48. Liedtke C., Matsumoto Y., Good reduction of K3 surfaces, Compos. Math. 154, 1-35, 2018. DOI 10.1112/S0010437X17007400; zbl 1386.14138; MR3699071; arxiv 1411.4797.
  • 49. Liedtke C., Lectures on supersingular K3 surfaces and the crystalline Torelli theorem, In: K3 surfaces and their moduli, Progress in Mathematics 315, Birkhäuser, 171-325, 2016. DOI 10.1007/978-3-319-29959-4_8; zbl 1349.14072; MR3524169; arxiv 1403.2538.
  • 50. Manin Y., Theory of commutative formal groups over fields of finite characteristic, Uspekhi Mat. Nauk SSSR 18(6 (114)), 3-90, 1963. DOI 10.1070/rm1963v018n06ABEH001142; zbl 0128.15603; MR0157972.
  • 51. Matsumoto Y., Good reduction criterion for K3 surfaces, Math. Z. 279, no. 1-2, 241-266, 2015. DOI 10.1007/s00209-014-1365-8; zbl 1317.14089; MR3299851; arxiv 1401.1261.
  • 52. Mukai S., Duality between $D(X)$ and $D(\hat{X})$ with its applications to Picard sheaves, Nagoya Math. J. 81, 153-175, 1981. DOI 10.1017/S002776300001922X; zbl 0417.14036; MR0607081.
  • 53. Mukai S., On the moduli space of bundles on K3 surfaces I., In: Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 341-413, 1984. zbl 0674.14023; MR0893604.
  • 54. Matsusaka T., Mumford D., Two theorems on deformations of polarized varities, Amer. J. Math. 86, 668-684, 1964. DOI 10.2307/2373030; zbl 0128.15505; MR0171778.
  • 55. Neukirch J., Algebraic number theory, Grundlehren der mathematischen Wissenschaften 322, Springer, 1999. zbl 0956.11021; MR1697859.
  • 56. Nitsure N., Construction of Hilbert and Quot schemes, Fundamental algebraic geometry: Grothendieck's FGA explained, Mathematical Surveys and Monographs 123, American Mathematical Society, 105-137, 2005. MR2223407.
  • 57. Nygaard N., Tate conjecture for ordinary K3 surfaces over finite fields, Inv. Math. 74, 213-237, 1983. DOI 10.1007/BF01394314; zbl 0557.14002; MR0723215.
  • 58. Olsson M., Algebraic spaces and stacks, Colloquium Publications 62, American Mathematical Society, 2016. zbl 1346.14001; MR3495343.
  • 59. Orlov D., On equivalences of derived categories and K3 surfaces. J. Math Sci. (New York) 84, 1361-1381, 1997. DOI 10.1007/BF02399195; zbl 0938.14019; MR1465519.
  • 60. Orlov D., Derived categories and coherent sheaves and equivalences between them, Russian Math. Surveys 58, 511-591, 2003. DOI 10.1070/RM2003v058n03ABEH000629; zbl 1118.14021; MR1998775; arxiv alg-geom/9712017.
  • 61. Ogus A., Supersingular K3 crystals, Journées de Géométrie Algébraique de Rennes Vol. II, Astérisque 64, 3-86, 1979. zbl 0435.14003; MR0563467.
  • 62. Ogus A., A crystalline Torelli theorem of supersingular K3 surfaces, Arithmetic and Geometry II, Progress in Mathematics 36, 361-394, Birkhäuser, 1983. zbl 0596.14013; MR0717616.
  • 63. Ploog D., Group of autoequivalences of derived categories of smooth projective varieties, PhD thesis, Freie Universität Berlin, 2005.
  • 64. Rudakov A. N., Shaferevich I. R., Inseparable morphisms of algebraic surfaces, Izv. Akad. Nauk SSSR 40, 1269-1307 (1976). MR0460344.
  • 65. Serre J.P., A course in arithmetic, Graduate Text in Mathematics 7, Springer, 1973. zbl 0256.12001; MR0344216.
  • 66. Sernesi E., Deformation of algebraic varieties, Grundlehren der mathematischen Wissenschaften 334, Springer, Berlin, Heidelberg, 2006. DOI 10.1007/978-3-540-30615-3; zbl 1102.14001; MR2247603.
  • 67. Srivastava T.K., On derived equivalences of K3 surfaces in positive characteristic, PhD Thesis, Freie Universität Berlin, 2018.
  • 68. Taelman L., Ordinary K3 surfaces over finite fields, preprint, 2017. DOI 10.1515/crelle-2018-0023; arxiv 1711.09225.
  • 69. Ward M., Arithmetic properties of the derived category for Calabi-Yau varieties, PhD thesis, University of Washington, 2014.

Affiliation

Srivastava, Tanya Kaushal
Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

Downloads