A trichotomy for the autoequivalence groups on smooth projective surfaces

Uehara, Hokuto

doi:10.1090/tran/7439

A trichotomy for the autoequivalence groups on smooth projective surfaces
HTML articles powered by AMS MathViewer

by Hokuto Uehara PDF

Trans. Amer. Math. Soc. 371 (2019), 3529-3547 Request permission

Abstract:

We study autoequivalence groups of the derived categories on smooth projective surfaces. We show a trichotomy of types according to the maximal dimension of Fourier–Mukai kernels for autoequivalences. This number is $2$, $3$, or $4$, and we also pose a conjecture on the description of autoequivalence groups if it is $2$, and prove it in some special cases.

References

Arend Bayer and Tom Bridgeland, Derived automorphism groups of K3 surfaces of Picard rank 1, Duke Math. J. 166 (2017), no. 1, 75–124. MR 3592689, DOI 10.1215/00127094-3674332
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
Tom Bridgeland, Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math. 498 (1998), 115–133. MR 1629929, DOI 10.1515/crll.1998.046
Tom Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), no. 1, 25–34. MR 1651025, DOI 10.1112/S0024609398004998
T. Bridgeland and A. Maciocia, Fourier–Mukai transforms for quotient varieties, math.AG/9811101.
Tom Bridgeland and Antony Maciocia, Complex surfaces with equivalent derived categories, Math. Z. 236 (2001), no. 4, 677–697. MR 1827500, DOI 10.1007/PL00004847
Tom Bridgeland and Antony Maciocia, Fourier-Mukai transforms for $K3$ and elliptic fibrations, J. Algebraic Geom. 11 (2002), no. 4, 629–657. MR 1910263, DOI 10.1090/S1056-3911-02-00317-X
Nathan Broomhead and David Ploog, Autoequivalences of toric surfaces, Proc. Amer. Math. Soc. 142 (2014), no. 4, 1133–1146. MR 3162236, DOI 10.1090/S0002-9939-2014-11530-9
William Fulton, Intersection theory, 2nd ed., 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. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
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
Akira Ishii and Hokuto Uehara, Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces, J. Differential Geom. 71 (2005), no. 3, 385–435. MR 2198807
Akira Ishii, Kazushi Ueda, and Hokuto Uehara, Stability conditions on $A_n$-singularities, J. Differential Geom. 84 (2010), no. 1, 87–126. MR 2629510
Yujiro Kawamata, $D$-equivalence and $K$-equivalence, J. Differential Geom. 61 (2002), no. 1, 147–171. MR 1949787
Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
S. A. Kuleshov and D. O. Orlov, Exceptional sheaves on Del Pezzo surfaces, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 3, 53–87 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 44 (1995), no. 3, 479–513. MR 1286839, DOI 10.1070/IM1995v044n03ABEH001609
D. O. Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 3, 131–158 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 3, 569–594. MR 1921811, DOI 10.1070/IM2002v066n03ABEH000389
R. Potter, Derived autoequivalences of bielliptic surfaces, arXiv:1701.01015.
Hokuto Uehara, An example of Fourier-Mukai partners of minimal elliptic surfaces, Math. Res. Lett. 11 (2004), no. 2-3, 371–375. MR 2067481, DOI 10.4310/MRL.2004.v11.n3.a9
Hokuto Uehara, Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension, Algebr. Geom. 3 (2016), no. 5, 543–577. MR 3568337, DOI 10.14231/AG-2016-025
Hokuto Uehara, Fourier-Mukai partners of elliptic ruled surfaces, Proc. Amer. Math. Soc. 145 (2017), no. 8, 3221–3232. MR 3652778, DOI 10.1090/proc/13471