Stability on a triangulated category and moduli

Inaba, Michi-aki

doi:10.1090/suga/415

Stability on a triangulated category and moduli
HTML articles powered by AMS MathViewer

by Michi-aki Inaba
Translated by: Michi-aki Inaba

Sugaku Expositions 30 (2017), 1-15

DOI: https://doi.org/10.1090/suga/415

Published electronically: March 16, 2017

PDF | Request permission

References

Paul S. Aspinwall and Michael R. Douglas, D-brane stability and monodromy, J. High Energy Phys. 5 (2002), no. 31, 35. MR 1915365, DOI 10.1088/1126-6708/2002/05/031
C. Bartocci, U. Bruzzo, and D. Hernández Ruipérez, A Fourier-Mukai transform for stable bundles on $K3$ surfaces, J. Reine Angew. Math. 486 (1997), 1–16. MR 1450748, DOI 10.1515/crll.1997.486.1
A. A. Beĭlinson, The derived category of coherent sheaves on $\textbf {P}^n$, Selecta Math. Soviet. 3 (1983/84), no. 3, 233–237. Selected translations. MR 863137
A. I. Bondal, Helices, representations of quivers and Koszul algebras, Helices and vector bundles, London Math. Soc. Lecture Note Ser., vol. 148, Cambridge Univ. Press, Cambridge, 1990, pp. 75–95. MR 1074784, DOI 10.1017/CBO9780511721526.008
A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 519–541. MR 1039961, DOI 10.1070/IM1990v035n03ABEH000716
Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), no. 3, 327–344. MR 1818984, DOI 10.1023/A:1002470302976
A. I. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, 1995, arXiv: alg-geom/9506012.
Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI 10.4007/annals.2007.166.317
Tom Bridgeland, Stability conditions on $K3$ surfaces, Duke Math. J. 141 (2008), no. 2, 241–291. MR 2376815, DOI 10.1215/S0012-7094-08-14122-5
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, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613–632. MR 1893007, DOI 10.1007/s002220100185
Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554. MR 1824990, DOI 10.1090/S0894-0347-01-00368-X
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
Ugo Bruzzo and Antony Maciocia, Hilbert schemes of points on some $K3$ surfaces and Gieseker stable bundles, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 2, 255–261. MR 1384468, DOI 10.1017/S0305004100074843
Alastair Craw and Akira Ishii, Flops of $G$-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124 (2004), no. 2, 259–307. MR 2078369, DOI 10.1215/S0012-7094-04-12422-4
S. I. Gelfand and Yu. I. Manin, Homological algebra, Springer-Verlag, Berlin, 1999. Translated from the 1989 Russian original by the authors; Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Algebra, V, Encyclopaedia Math. Sci., 38, Springer, Berlin, 1994; MR1309679 (95g:18007)]. MR 1698374
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original. MR 1438306, DOI 10.1007/978-3-662-03220-6
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1950475, DOI 10.1007/978-3-662-12492-5
A. L. Gorodentsev, Exceptional objects and mutations in derived categories, Helices and vector bundles, London Math. Soc. Lecture Note Ser., vol. 148, Cambridge Univ. Press, Cambridge, 1990, pp. 57–73. MR 1074783, DOI 10.1017/CBO9780511721526.007
A. L. Gorodentsev and A. N. Rudakov, Exceptional vector bundles on projective spaces, Duke Math. J. 54 (1987), no. 1, 115–130. MR 885779, DOI 10.1215/S0012-7094-87-05409-3
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
Daniel Huybrechts, Compact hyper-Kähler manifolds: basic results, Invent. Math. 135 (1999), no. 1, 63–113. MR 1664696, DOI 10.1007/s002220050280
Michi-aki Inaba, Toward a definition of moduli of complexes of coherent sheaves on a projective scheme, J. Math. Kyoto Univ. 42 (2002), no. 2, 317–329. MR 1966840, DOI 10.1215/kjm/1250283873
Michi-aki Inaba, Moduli of stable objects in a triangulated category, J. Math. Soc. Japan 62 (2010), no. 2, 395–429. MR 2662850
M. M. Kapranov, The derived category of coherent sheaves on Grassmann varieties, Funktsional. Anal. i Prilozhen. 17 (1983), no. 2, 78–79 (Russian). MR 705052
M. M. Kapranov, Derived category of coherent sheaves on Grassmann manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 1, 192–202 (Russian). MR 733363
M. M. Kapranov, On the derived category and $K$-functor of coherent sheaves on intersections of quadrics, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 1, 186–199 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 1, 191–204. MR 936529, DOI 10.1070/IM1989v032n01ABEH000752
Yujiro Kawamata, $D$-equivalence and $K$-equivalence, J. Differential Geom. 61 (2002), no. 1, 147–171. MR 1949787
Yujiro Kawamata, Francia’s flip and derived categories, Algebraic geometry, de Gruyter, Berlin, 2002, pp. 197–215. MR 1954065
Yujiro Kawamata, Derived categories of toric varieties, Michigan Math. J. 54 (2006), no. 3, 517–535. MR 2280493, DOI 10.1307/mmj/1163789913
Yujiro Kawamata, Derived equivalence for stratified Mukai flop on $G(2,4)$, Mirror symmetry. V, AMS/IP Stud. Adv. Math., vol. 38, Amer. Math. Soc., Providence, RI, 2006, pp. 285–294. MR 2282964, DOI 10.1090/amsip/038/13
A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
Max Lieblich, Moduli of complexes on a proper morphism, J. Algebraic Geom. 15 (2006), no. 1, 175–206. MR 2177199, DOI 10.1090/S1056-3911-05-00418-2
Antony Maciocia, The determinant line bundle over moduli spaces of instantons on abelian surfaces, Math. Z. 217 (1994), no. 2, 317–333. MR 1296399, DOI 10.1007/BF02571947
Antony Maciocia, Gieseker stability and the Fourier-Mukai transform for abelian surfaces, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 185, 87–100. MR 1380952, DOI 10.1093/qmath/47.1.87
Antony Maciocia, Generalized Fourier-Mukai transforms, J. Reine Angew. Math. 480 (1996), 197–211. MR 1420564, DOI 10.1515/crll.1996.480.197
Eyal Markman, Brill-Noether duality for moduli spaces of sheaves on $K3$ surfaces, J. Algebraic Geom. 10 (2001), no. 4, 623–694. MR 1838974
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
Shigeru Mukai, Fourier functor and its application to the moduli of bundles on an abelian variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 515–550. MR 946249, DOI 10.2969/aspm/01010515
Shigeru Mukai, Duality of polarized $K3$ surfaces, New trends in algebraic geometry (Warwick, 1996) London Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 311–326. MR 1714828, DOI 10.1017/CBO9780511721540.012
Yoshinori Namikawa, Mukai flops and derived categories, J. Reine Angew. Math. 560 (2003), 65–76. MR 1992802, DOI 10.1515/crll.2003.061
Yoshinori Namikawa, Mukai flops and derived categories. II, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, pp. 149–175. MR 2096144, DOI 10.1090/crmp/038/07
Kieran G. O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a $K3$ surface, J. Algebraic Geom. 6 (1997), no. 4, 599–644. MR 1487228
D. O. Orlov, Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk 58 (2003), no. 3(351), 89–172 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 3, 511–591. MR 1998775, DOI 10.1070/RM2003v058n03ABEH000629
Yukinobu Toda, Moduli stacks and invariants of semistable objects on $K3$ surfaces, Adv. Math. 217 (2008), no. 6, 2736–2781. MR 2397465, DOI 10.1016/j.aim.2007.11.010
Yukinobu Toda, Semistable objects in derived categories of $K3$ surfaces, Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu, B9, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 175–188. MR 2509700
Yukinobu Toda, Birational Calabi-Yau threefolds and BPS state counting, Commun. Number Theory Phys. 2 (2008), no. 1, 63–112. MR 2417847, DOI 10.4310/CNTP.2008.v2.n1.a2
Yukinobu Toda, Curve counting theories via stable objects I. DT/PT correspondence, J. Amer. Math. Soc. 23 (2010), no. 4, 1119–1157. MR 2669709, DOI 10.1090/S0894-0347-10-00670-3
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
K\B{o}ta Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), no. 4, 817–884. MR 1872531, DOI 10.1007/s002080100255
K\B{o}ta Yoshioka, Twisted stability and Fourier-Mukai transform. I, Compositio Math. 138 (2003), no. 3, 261–288. MR 2019443, DOI 10.1023/A:1027304215606
K\B{o}ta Yoshioka, Twisted stability and Fourier-Mukai transform. II, Manuscripta Math. 110 (2003), no. 4, 433–465. MR 1975096, DOI 10.1007/s00229-002-0340-6
K\B{o}ta Yoshioka, Stability and the Fourier-Mukai transform. I, Math. Z. 245 (2003), no. 4, 657–665. MR 2020704, DOI 10.1007/s00209-003-0562-7
K\B{o}ta Yoshioka, Some examples of Mukai’s reflections on $K3$ surfaces, J. Reine Angew. Math. 515 (1999), 97–123. MR 1717621, DOI 10.1515/crll.1999.080