no. 7
Matrix factorizations and singularity categories for stacks
[Factorisations matricielles et catégories des singularités pour les champs algébriques]
Polishchuk, Alexander1 ; Vaintrob, Arkady1
1 Department of Mathematics, University of Oregon, Eugene, OR 97405
Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2609-2642.

On étudie les factorisations matricielles d’un potentiel W qui est une section d’un fibré en droites sur un champ algébrique. On établit une relation entre la catégorie dérivée correspondante (la catégorie des D-branes de type B dans le modèle de Landau-Ginzburg avec potentiel W) et la catégorie des singularités du lieu des zéros de W généralisant un théorème d’Orlov. On utilise ce résultat pour construire des foncteurs image directe pour les factorisations matricielles à supports relativement propres.

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.

DOI : 10.5802/aif.2788
Classification : 14A20, 14J17, 18E30
Keywords: matrix factorizations, singularity category, algebraic stack
Mot clés : factorisations matricielles, catégorie des singularités, champ algébrique

Polishchuk, Alexander 1 ; Vaintrob, Arkady 1

1 Department of Mathematics, University of Oregon, Eugene, OR 97405 
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Polishchuk, Alexander; Vaintrob, Arkady. Matrix factorizations and singularity categories for stacks. Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2609-2642. doi : 10.5802/aif.2788. https://aif.centre-mersenne.org/articles/10.5802/aif.2788/

[1] Abramovich, D.; Corti, A.; Vistoli, A. Twisted bundles and admissible covers, Comm. Algebra, Volume 31 (2003) no. 8, pp. 3547-3648 | DOI | MR | Zbl

[2] Arinkin, D.; Bezrukavnikov, R. Perverse coherent sheaves, Moscow Math. J., Volume 10 (2010) no. 1, pp. 3-29 | DOI | MR | Zbl

[3] Bass, H. Big projective modules are free, Illinois J. Math., Volume 7 (1963), pp. 24-31 | DOI | MR | Zbl

[4] Brunner, I.; Herbst, M.; Lerche, W.; Walcher, J. Matrix Factorizations And Mirror Symmetry: The Cubic Curve, J. High Energy Phys., University of Chicago Press, 2006 no. 11 (006) | MR

[5] Brunner, I.; Roggenkamp, D. B-type defects in Landau-Ginzburg models, J. High Energy Phys., University of Chicago Press, 2007 no. 8 (093) | MR | Zbl

[6] Buchweitz, R.; Greuel, G.; Schreyer, F. Cohen-Macaulay modules on hypersurface singularities II, Invent. Math., Volume 88 (1987), pp. 165-182 | DOI | MR | Zbl

[7] Bökstedt, M.; Neeman, A. Homotopy limits in triangulated categories, Compositio Math., Volume 86 (1993) no. 2, pp. 209-234 | Numdam | MR | Numdam | Zbl

[8] Chen, X.-W. Unifying two results of D. Orlov, Abhandlungen Mathem. Seminar Univ. Hamburg, Volume 80 (2010), pp. 207-212 | DOI | MR | Zbl

[9] Efimov, A. I. Homological mirror symmetry for curves of higher genus, Adv. Math., Volume 230 (2012) no. 2, pp. 493-530 | DOI | MR | Zbl

[10] Eisenbud, D. Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc., Volume 260 (1980), pp. 35-64 | DOI | MR | Zbl

[11] Fan, H.; Jarvis, T.; Ruan, Y. The Witten equation and its virtual fundamental cycle (preprint math.AG/0712.4025)

[12] Fan, H.; Jarvis, T.; Ruan, Y. The Witten equation, mirror symmetry and quantum singularity theory (Ann. Math., to appear [math.AG/0712.4021]) | Zbl

[13] Gruson, L.; Raynaud, M. Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | MR | Zbl

[14] Herbst, M.; Hori, K.; Page, D. Phases Of N=2 Theories In 1+1 Dimensions With Boundary (preprint arXiv:0803.2045) | Zbl

[15] Illusie, L. Conditions de finitude relative, SGA6, Théorie des intersections et théorème de Riemann-Roch, SGA6, exp. III (Lecture Notes in Math.), Volume 225, Springer-Verlag, Berlin-New York, 1971 | DOI | Zbl

[16] Illusie, L. Existence de résolutions globales, Théorie des intersections et théorème de Riemann-Roch, SGA6, exp. II (Lecture Notes in Math.), Volume 225, Springer-Verlag, Berlin-New York, 1971 | DOI | Zbl

[17] Illusie, L. 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.), Volume 225, Springer-Verlag, Berlin-New York, 1971 | DOI | Zbl

[18] Kajiura, H.; Saito, K.; Takahashi, A. Matrix factorization and representations of quivers. II. Type ADE case, Adv. Math., Volume 211 (2007) no. 1, pp. 327-362 | DOI | MR | Zbl

[19] Kapustin, A.; Li, Y. D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Phys. (2003) no. 12 (005, 44 pp) | DOI | MR

[20] Kapustin, A.; Li, Y. Topological correlators in Landau-Ginzburg models with boundaries, Adv. Theor. Math. Phys., Volume 7 (2003) no. 4, pp. 727-749 | DOI | MR | Zbl

[21] Katzarkov, L.; Kontsevich, M.; Pantev, T. Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT t t * -geometry, Amer. Math. Soc., Providence, RI, 2008, pp. 87-174 | DOI | MR | Zbl

[22] Khovanov, M.; Rozansky, L. Matrix factorizations and Link homology, Fund. Math., Volume 199 (2008), pp. 1-91 | DOI | MR | Zbl

[23] Kontsevich, M. Hodge structures in non-commutative geometry, Non-commutative geometry in mathematics and physics, Amer. Math. Soc., Providence, RI, 2008, pp. 1-21 | MR | Zbl

[24] Kresch, A. On the geometry of Deligne-Mumford stacks, Algebraic geometry (Seattle 2005). Part 1, Amer. Math. Soc., Providence, RI, 2009, pp. 259-271 | DOI | MR | Zbl

[25] Orlov, D. Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. (2004) no. 3 (246), pp. 227-248 | MR | Zbl

[26] Orlov, D. Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Volume II, Birkhäuser, Boston, 2009, pp. 503-531 | DOI | MR | Zbl

[27] Orlov, D. Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math., Volume 226 (2011) no. 1, pp. 206-217 | DOI | MR | Zbl

[28] Polishchuk, A.; Vaintrob, A. Matrix factorizations and cohomological field theories (preprint math.AG/1105.2903) | DOI | Numdam | Zbl

[29] Polishchuk, A.; Vaintrob, A. Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations, Duke Math. J., Volume 161 (2012) no. 10, pp. 1863-1926 | DOI | MR | Zbl

[30] Positselski, L. Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Mem. Amer. Math. Soc., 212, 2011 no. 996 (133 pp., [math.AG/0905.2621]) | MR

[31] Quintero Vélez, A. McKay correspondence for Landau-Ginzburg models, Commun. Number Theory Phys., Volume 3 (2009) no. 1, pp. 173-208 | MR | Zbl

[32] Romagny, M. Group actions on stacks and applications, Michigan Math. J., Volume 53 (2005), pp. 209-236 | DOI | MR | Zbl

[33] Segal, E. Equivalences between GIT quotients of Landau-Ginzburg B-models, Commun. Math. Phys., Volume 304 (2011), pp. 411-432 | DOI | MR | Zbl

[34] Seidel, P. Homological mirror symmetry for the genus two curve, preprint J. Algebraic Geom., Volume 20 (2011) no. 4, pp. 727-769 | DOI | MR | Zbl

[35] Thomason, R. W. Algebraic K-theory of group scheme actions, Algebraic topology and algebraic K -theory (Princeton, N.J., 1983), Princeton Univ. Press, 1987, pp. 539-563 | MR | Zbl

[36] Walcher, J. Stability of Landau-Ginzburg branes, J. Math. Phys., Volume 46 (2005) no. 8 (082305, 29 pp) | DOI | MR | Zbl

[37] Yoshino, Y. Cohen-Macaulay modules over Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1990 | DOI | MR | Zbl

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