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Hyperplane sections and stable derived categories

Author: Kazushi Ueda
Journal: Proc. Amer. Math. Soc. 142 (2014), 3019-3028
MSC (2010): Primary 13C14, 13D09; Secondary 14J33
Published electronically: June 2, 2014
MathSciNet review: 3223358
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Abstract: We discuss the relation between the graded stable derived category of a hypersurface and that of its hyperplane section. The motivation comes from the compatibility between homological mirror symmetry for the Calabi-Yau manifold defined by an invertible polynomial and that for the singularity defined by the same polynomial.

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  • [AR87] Maurice Auslander and Idun Reiten, Almost split sequences for $ {\bf Z}$-graded rings, (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 232-243. MR 915178 (89b:13031),
  • [BH93] Per Berglund and Tristan Hübsch, A generalized construction of mirror manifolds, Nuclear Phys. B 393 (1993), no. 1-2, 377-391. MR 1214325 (94k:14031),
  • [Bor] Lev A. Borisov, Berglund-Hübsch mirror symmetry via vertex algebras, Comm. Math. Phys. 320 (2013), no. 1, 73-99. MR 3046990
  • [Buc87] Ragnar-Olaf Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, 1987. Available at
  • [CR11] Alessandro Chiodo and Yongbin Ruan, LG/CY correspondence: the state space isomorphism, Adv. Math. 227 (2011), no. 6, 2157-2188. MR 2807086 (2012g:14069),
  • [Eis80] David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35-64. MR 570778 (82d:13013),
  • [FU] Masahiro Futaki and Kazushi Ueda, Homological mirror symmetry for singularities of type D, Math. Z. 273 (2013), no. 3-4, 633-652. MR 3030671,
  • [FU11] Masahiro Futaki and Kazushi Ueda, Homological mirror symmetry for Brieskorn-Pham singularities, Selecta Math. (N.S.) 17 (2011), no. 2, 435-452. MR 2803848 (2012e:14083),
  • [Hap91] Dieter Happel, On Gorenstein algebras (Bielefeld, 1991), Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 389-404. MR 1112170 (92k:16022)
  • [IT] Osamu Iyama and Ryo Takahashi, Tilting and cluster tilting for quotient singularities, Math. Ann. 356 (2013), no. 3, 1065-1105. MR 3063907
  • [KMU12] Masanori Kobayashi, Makiko Mase, and Kazushi Ueda, A note on exceptional unimodal singularities and K3 surfaces, Internat. Math. Res. Notices 2013, no. 7. MR 3044454
  • [Kon95] Maxim Kontsevich, Homological algebra of mirror symmetry, Proc. ICM, Vol. 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120-139. MR 1403918 (97f:32040)
  • [Kra] Marc Krawitz, FJRW rings and Landau-Ginzburg mirror symmetry, University of Michigan, 2010, 67 pp., ProQuest LLC. MR 2801653
  • [Kra05] Henning Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128-1162. MR 2157133 (2006e:18019),
  • [NU12] Yuichi Nohara and Kazushi Ueda, Homological mirror symmetry for the quintic 3-fold, Geom. Topol. 16 (2012), no. 4, 1967-2001. MR 2975297
  • [Orl04] D. O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models (Russian, with Russian summary), Tr. Mat. Inst. Steklova 246 (2004, Algebr. Geom. Metody, Svyazi i Prilozh.), 240-262; English transl., Proc. Steklov Inst. Math. 3 (246) (2004), 227-248. MR 2101296 (2006i:81173)
  • [Orl09] Dmitri Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, arithmetic, and geometry: In honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270, Birkhäuser Boston Inc., Boston, MA, 2009, pp. 503-531. MR 2641200 (2011c:14050),
  • [Sei02] Paul Seidel, Fukaya categories and deformations (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 351-360. MR 1957046 (2004a:53110)
  • [Sei08] Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR 2441780
  • [Sei11] Paul Seidel, Homological mirror symmetry for the quartic surface, math.AG/0310414, 2011.
  • [She11] Nick Sheridan, On the homological mirror symmetry conjecture for pairs of pants, J. Differential Geom. 89 (2011), no. 2, 271-367. MR 2863919 (2012m:53196)
  • [Tak10] Atsushi Takahashi, Weighted projective lines associated to regular systems of weights of dual type, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) Adv. Stud. Pure Math., vol. 59, Math. Soc. Japan, Tokyo, 2010, pp. 371-388. MR 2683215 (2012b:14080)

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Additional Information

Kazushi Ueda
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan

Received by editor(s): July 18, 2012
Received by editor(s) in revised form: October 2, 2012
Published electronically: June 2, 2014
Communicated by: Lev Borisov
Article copyright: © Copyright 2014 American Mathematical Society

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