Table of contents for this issue | Next article
Alexander Polishchuk; Arkady Vaintrob
Matrix factorizations and singularity categories for stacks
(Factorisations matricielles et catégories des singularités pour les champs algébriques)
Annales de l'institut Fourier, 61 no. 7 (2011), p. 2609-2642, doi: 10.5802/aif.2788
Article PDF | Reviews MR 3112502 | Zbl 1278.13014
Class. Math.: 14A20, 14J17, 18E30
Keywords: matrix factorizations, singularity category, algebraic stack
[1] D. Abramovich, A. Corti & A. Vistoli, “Twisted bundles and admissible covers”, Comm. Algebra 31 (2003) no. 8, p. 3547-3648 MR 2007376 | Zbl 1077.14034
[2] D. Arinkin & R. Bezrukavnikov, “Perverse coherent sheaves”, Moscow Math. J. 10 (2010) no. 1, p. 3-29 MR 2668828 | Zbl 1205.18010
[3] H. Bass, “Big projective modules are free”, Illinois J. Math. 7 (1963), p. 24-31 MR 143789 | Zbl 0115.26003
[4] I. Brunner, M. Herbst, W. Lerche & J. Walcher, Matrix Factorizations And Mirror Symmetry: The Cubic Curve, J. High Energy Phys., University of Chicago Press, 2006, 006 MR 2270440
[5] I. Brunner & D. Roggenkamp, B-type defects in Landau-Ginzburg models, J. High Energy Phys., University of Chicago Press, 2007, 093 MR 2342020
[6] R. Buchweitz, G. Greuel & F. Schreyer, “Cohen-Macaulay modules on hypersurface singularities II”, Invent. Math. 88 (1987), p. 165-182 MR 877011 | Zbl 0617.14034
[7] M. Bökstedt & A. Neeman, “Homotopy limits in triangulated categories”, Compositio Math. 86 (1993) no. 2, p. 209-234 Numdam | MR 1214458 | Zbl 0802.18008
[8] X.-W. Chen, “Unifying two results of D. Orlov”, Abhandlungen Mathem. Seminar Univ. Hamburg 80 (2010), p. 207-212 MR 2734686 | Zbl 1214.18013
[9] A. I. Efimov, “Homological mirror symmetry for curves of higher genus”, Adv. Math. 230 (2012) no. 2, p. 493-530 MR 2914956 | Zbl 1242.14039
[10] D. Eisenbud, “Homological algebra on a complete intersection, with an application to group representations”, Trans. Amer. Math. Soc. 260 (1980), p. 35-64 MR 570778 | Zbl 0444.13006
[11] H. Fan, T. Jarvis & Y. Ruan, “The Witten equation and its virtual fundamental cycle”, preprint math.AG/0712.4025
[12] H. Fan, T. Jarvis & Y. Ruan, “The Witten equation, mirror symmetry and quantum singularity theory”, Ann. Math., to appear [math.AG/0712.4021]
[13] L. Gruson & M. Raynaud, “Critères de platitude et de projectivité. Techniques de “platification” d’un module”, Invent. Math. 13 (1971), p. 1-89 MR 308104 | Zbl 0227.14010
[14] M. Herbst, K. Hori & D. Page, “Phases Of $N=2$ Theories In $1+1$ Dimensions With Boundary”, preprint arXiv:0803.2045
[15] L. Illusie, Conditions de finitude relative, SGA6, Théorie des intersections et théorème de Riemann-Roch, SGA6, exp. III, Lecture Notes in Math. 225, Springer-Verlag, 1971 Zbl 0229.14009
[16] L. Illusie, Existence de résolutions globales, Théorie des intersections et théorème de Riemann-Roch, SGA6, exp. II, Lecture Notes in Math. 225, Springer-Verlag, 1971 Zbl 0241.14002
[17] L. Illusie, Géneralités sur les conditions de finitude dans les catégories dérivées, Théorie des intersections et théorème de Riemann-Roch, SGA6, exp. I, Lecture Notes in Math. 225, Springer-Verlag, 1971 Zbl 0229.14010
[18] H. Kajiura, K. Saito & A. Takahashi, “Matrix factorization and representations of quivers. II. Type ADE case”, Adv. Math. 211 (2007) no. 1, p. 327-362 MR 2313537 | Zbl 1167.16011
[19] A. Kapustin & Y. Li, “D-branes in Landau-Ginzburg models and algebraic geometry”, J. High Energy Phys. (2003) no. 12, 005, 44 pp MR 2041170
[20] A. Kapustin & Y. Li, “Topological correlators in Landau-Ginzburg models with boundaries”, Adv. Theor. Math. Phys. 7 (2003) no. 4, p. 727-749 MR 2039036 | Zbl 1058.81061
[21] L. Katzarkov, M. Kontsevich & T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT $tt^*$-geometry, Amer. Math. Soc., 2008, p. 87-174 MR 2483750 | Zbl 1206.14009
[22] M. Khovanov & L. Rozansky, “Matrix factorizations and Link homology”, Fund. Math. 199 (2008), p. 1-91 MR 2391017 | Zbl 1145.57009
[23] M. Kontsevich, Hodge structures in non-commutative geometry, Non-commutative geometry in mathematics and physics, Amer. Math. Soc., 2008, p. 1-21 MR 2444365
[24] A. Kresch, On the geometry of Deligne-Mumford stacks, Algebraic geometry (Seattle 2005). Part 1, Amer. Math. Soc., 2009, p. 259-271 MR 2483938 | Zbl 1169.14001
[25] D. Orlov, “Triangulated categories of singularities and D-branes in Landau-Ginzburg models”, Proc. Steklov Inst. Math. (2004) no. 3 (246), p. 227-248 MR 2101296 | Zbl 1101.81093
[26] D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Birkhäuser, 2009, p. 503-531 MR 2641200 | Zbl 1200.18007
[27] D. Orlov, “Formal completions and idempotent completions of triangulated categories of singularities”, Adv. Math. 226 (2011) no. 1, p. 206-217 MR 2735755 | Zbl 1216.18012
[28] A. Polishchuk & A. Vaintrob, “Matrix factorizations and cohomological field theories”, preprint math.AG/1105.2903
[29] A. Polishchuk & A. Vaintrob, “Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations”, Duke Math. J. 161 (2012) no. 10, p. 1863-1926 MR 2954619 | Zbl 1249.14001
[30] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Mem. Amer. Math. Soc. 212, 2011, 133 pp., [math.AG/0905.2621] MR 2830562 | Zbl pre05947402
[31] A. Quintero Vélez, “McKay correspondence for Landau-Ginzburg models”, Commun. Number Theory Phys. 3 (2009) no. 1, p. 173-208 MR 2504756 | Zbl 1169.14011
[32] M. Romagny, “Group actions on stacks and applications”, Michigan Math. J. 53 (2005), p. 209-236 MR 2125542 | Zbl 1100.14001
[33] E. Segal, “Equivalences between GIT quotients of Landau-Ginzburg B-models”, Commun. Math. Phys. 304 (2011), p. 411-432 MR 2795327 | Zbl 1216.81122
[34] P. Seidel, “Homological mirror symmetry for the genus two curve”, preprint J. Algebraic Geom. 20 (2011) no. 4, p. 727-769 MR 2819674 | Zbl 1226.14028
[35] R. W. Thomason, Algebraic $K$-theory of group scheme actions, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983), Princeton Univ. Press, 1987, p. 539-563 MR 921490 | Zbl 0701.19002
[36] J. Walcher, “Stability of Landau-Ginzburg branes”, J. Math. Phys. 46 (2005) no. 8, 082305, 29 pp MR 2165838 | Zbl 1110.81152
[37] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1990 MR 1079937 | Zbl 0745.13003
Alexander Polishchuk; Arkady Vaintrob
Matrix factorizations and singularity categories for stacks
(Factorisations matricielles et catégories des singularités pour les champs algébriques)
Annales de l'institut Fourier, 61 no. 7 (2011), p. 2609-2642, doi: 10.5802/aif.2788
Article PDF | Reviews MR 3112502 | Zbl 1278.13014
Class. Math.: 14A20, 14J17, 18E30
Keywords: matrix factorizations, singularity category, algebraic stack
Résumé - Abstract
We study matrix factorizations of a potential W which is a section of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.
Bibliography
[2] D. Arinkin & R. Bezrukavnikov, “Perverse coherent sheaves”, Moscow Math. J. 10 (2010) no. 1, p. 3-29 MR 2668828 | Zbl 1205.18010
[3] H. Bass, “Big projective modules are free”, Illinois J. Math. 7 (1963), p. 24-31 MR 143789 | Zbl 0115.26003
[4] I. Brunner, M. Herbst, W. Lerche & J. Walcher, Matrix Factorizations And Mirror Symmetry: The Cubic Curve, J. High Energy Phys., University of Chicago Press, 2006, 006 MR 2270440
[5] I. Brunner & D. Roggenkamp, B-type defects in Landau-Ginzburg models, J. High Energy Phys., University of Chicago Press, 2007, 093 MR 2342020
[6] R. Buchweitz, G. Greuel & F. Schreyer, “Cohen-Macaulay modules on hypersurface singularities II”, Invent. Math. 88 (1987), p. 165-182 MR 877011 | Zbl 0617.14034
[7] M. Bökstedt & A. Neeman, “Homotopy limits in triangulated categories”, Compositio Math. 86 (1993) no. 2, p. 209-234 Numdam | MR 1214458 | Zbl 0802.18008
[8] X.-W. Chen, “Unifying two results of D. Orlov”, Abhandlungen Mathem. Seminar Univ. Hamburg 80 (2010), p. 207-212 MR 2734686 | Zbl 1214.18013
[9] A. I. Efimov, “Homological mirror symmetry for curves of higher genus”, Adv. Math. 230 (2012) no. 2, p. 493-530 MR 2914956 | Zbl 1242.14039
[10] D. Eisenbud, “Homological algebra on a complete intersection, with an application to group representations”, Trans. Amer. Math. Soc. 260 (1980), p. 35-64 MR 570778 | Zbl 0444.13006
[11] H. Fan, T. Jarvis & Y. Ruan, “The Witten equation and its virtual fundamental cycle”, preprint math.AG/0712.4025
[12] H. Fan, T. Jarvis & Y. Ruan, “The Witten equation, mirror symmetry and quantum singularity theory”, Ann. Math., to appear [math.AG/0712.4021]
[13] L. Gruson & M. Raynaud, “Critères de platitude et de projectivité. Techniques de “platification” d’un module”, Invent. Math. 13 (1971), p. 1-89 MR 308104 | Zbl 0227.14010
[14] M. Herbst, K. Hori & D. Page, “Phases Of $N=2$ Theories In $1+1$ Dimensions With Boundary”, preprint arXiv:0803.2045
[15] L. Illusie, Conditions de finitude relative, SGA6, Théorie des intersections et théorème de Riemann-Roch, SGA6, exp. III, Lecture Notes in Math. 225, Springer-Verlag, 1971 Zbl 0229.14009
[16] L. Illusie, Existence de résolutions globales, Théorie des intersections et théorème de Riemann-Roch, SGA6, exp. II, Lecture Notes in Math. 225, Springer-Verlag, 1971 Zbl 0241.14002
[17] L. Illusie, Géneralités sur les conditions de finitude dans les catégories dérivées, Théorie des intersections et théorème de Riemann-Roch, SGA6, exp. I, Lecture Notes in Math. 225, Springer-Verlag, 1971 Zbl 0229.14010
[18] H. Kajiura, K. Saito & A. Takahashi, “Matrix factorization and representations of quivers. II. Type ADE case”, Adv. Math. 211 (2007) no. 1, p. 327-362 MR 2313537 | Zbl 1167.16011
[19] A. Kapustin & Y. Li, “D-branes in Landau-Ginzburg models and algebraic geometry”, J. High Energy Phys. (2003) no. 12, 005, 44 pp MR 2041170
[20] A. Kapustin & Y. Li, “Topological correlators in Landau-Ginzburg models with boundaries”, Adv. Theor. Math. Phys. 7 (2003) no. 4, p. 727-749 MR 2039036 | Zbl 1058.81061
[21] L. Katzarkov, M. Kontsevich & T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT $tt^*$-geometry, Amer. Math. Soc., 2008, p. 87-174 MR 2483750 | Zbl 1206.14009
[22] M. Khovanov & L. Rozansky, “Matrix factorizations and Link homology”, Fund. Math. 199 (2008), p. 1-91 MR 2391017 | Zbl 1145.57009
[23] M. Kontsevich, Hodge structures in non-commutative geometry, Non-commutative geometry in mathematics and physics, Amer. Math. Soc., 2008, p. 1-21 MR 2444365
[24] A. Kresch, On the geometry of Deligne-Mumford stacks, Algebraic geometry (Seattle 2005). Part 1, Amer. Math. Soc., 2009, p. 259-271 MR 2483938 | Zbl 1169.14001
[25] D. Orlov, “Triangulated categories of singularities and D-branes in Landau-Ginzburg models”, Proc. Steklov Inst. Math. (2004) no. 3 (246), p. 227-248 MR 2101296 | Zbl 1101.81093
[26] D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Birkhäuser, 2009, p. 503-531 MR 2641200 | Zbl 1200.18007
[27] D. Orlov, “Formal completions and idempotent completions of triangulated categories of singularities”, Adv. Math. 226 (2011) no. 1, p. 206-217 MR 2735755 | Zbl 1216.18012
[28] A. Polishchuk & A. Vaintrob, “Matrix factorizations and cohomological field theories”, preprint math.AG/1105.2903
[29] A. Polishchuk & A. Vaintrob, “Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations”, Duke Math. J. 161 (2012) no. 10, p. 1863-1926 MR 2954619 | Zbl 1249.14001
[30] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Mem. Amer. Math. Soc. 212, 2011, 133 pp., [math.AG/0905.2621] MR 2830562 | Zbl pre05947402
[31] A. Quintero Vélez, “McKay correspondence for Landau-Ginzburg models”, Commun. Number Theory Phys. 3 (2009) no. 1, p. 173-208 MR 2504756 | Zbl 1169.14011
[32] M. Romagny, “Group actions on stacks and applications”, Michigan Math. J. 53 (2005), p. 209-236 MR 2125542 | Zbl 1100.14001
[33] E. Segal, “Equivalences between GIT quotients of Landau-Ginzburg B-models”, Commun. Math. Phys. 304 (2011), p. 411-432 MR 2795327 | Zbl 1216.81122
[34] P. Seidel, “Homological mirror symmetry for the genus two curve”, preprint J. Algebraic Geom. 20 (2011) no. 4, p. 727-769 MR 2819674 | Zbl 1226.14028
[35] R. W. Thomason, Algebraic $K$-theory of group scheme actions, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983), Princeton Univ. Press, 1987, p. 539-563 MR 921490 | Zbl 0701.19002
[36] J. Walcher, “Stability of Landau-Ginzburg branes”, J. Math. Phys. 46 (2005) no. 8, 082305, 29 pp MR 2165838 | Zbl 1110.81152
[37] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1990 MR 1079937 | Zbl 0745.13003
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