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Symmetries and stabilization for sheaves of vanishing cycles (with an appendix by Jörg Schürmann). (English) Zbl 1325.14057

This paper studies symmetries and stabilization properties of perverse sheaves of vanishing cycles of a regular function \(f: U\to \mathbb C\), where \(U\) is a smooth \(\mathbb C\)-scheme. The results can also be generalized to replace \(U\) by a complex analytic space and perverse sheaves of vanishing cycles by \(\mathsf D\)-modules or mixed Hodge modules. These results have applications in the categorification of Donaldson-Thomas invariants of Calabi-Yau three-folds and in defining a new kind of ‘Fukaya category’ of Lagrangians in a complex symplectic manifold using perverse sheaves.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
32S30 Deformations of complex singularities; vanishing cycles
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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[1] V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differentiable maps, Volume 1, Monographs in Math. 82, Birkh\"{}auser, 1985. · Zbl 1297.32001
[2] K. Behrend, Donaldson–Thomas type invariants via microlocal geometry, Ann. of Math. 170 (2009), 1307– 1338.DOI: 10.4007/annals.2009.170.1307 · Zbl 1191.14050
[3] K. Behrend, J. Bryan and B. Szendr˝oi, Motivic degree zero Donaldson–Thomas invariants, Invent. Math. 192 (2013), 111–160.DOI: 10.1007/s00222-012-0408-1 · Zbl 1267.14008
[4] K. Behrend and B. Fantechi, Gerstenhaber and Batalin–Vilkovisky structures on Lagrangian intersections, pages 1–47 in Algebra, arithmetic, and geometry, Progr. Math. 269, Birkh\"{}auser, Boston, MA, 2009. · Zbl 1198.53090
[5] A.A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Ast\'{}erisque 100, 1982. · Zbl 0536.14011
[6] O. Ben-Bassat, C. Brav, V. Bussi, and D. Joyce, A ’Darboux Theorem’ for shifted symplectic structures on derived Artin stacks, with applications, Geom. & Top. 19 (2015), 1287–1359.DOI: 10.2140/gt.2015.19.1287 · Zbl 1349.14003
[7] J.-E. Bj\"{}ork, AnalyticD-modules and applications, Kluwer, Dordrecht, 1993.
[8] A. Borel et al., AlgebraicD-modules, Perspectives in Mathematics 2, Academic Press, 1987.
[9] C. Brav, V. Bussi and D. Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, 2013.arXiv: 1305.6302 · Zbl 1349.14003
[10] V. Bussi, Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles, 2014.arXiv: 1404.1329
[11] V. Bussi, D. Joyce and S. Meinhardt, On motivic vanishing cycles of critical loci, 2013.arXiv: 1305.6428
[12] S.C. Coutinho, A primer of algebraicD-modules, L.M.S. Student texts 33, Cambridge University Press, 1995.DOI: 10.1017/CBO9780511623653
[13] A. D’Agnolo and P. Schapira, Quantization of complex Lagrangian submanifolds, Advances in Math. 213 (2007), 358–379.DOI: 10.1016/j.aim.2006.12.009 · Zbl 1122.46051
[14] A. Dimca, Sheaves in Topology, Universitext, Springer-Verlag, Berlin, 2004.DOI: 10.1007/978-3-642-18868-8 · Zbl 1043.14003
[15] A. Dimca and B. Szendr˝oi, The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on C3, Math. Res. Lett. 16 (2009), 1037–1055.DOI: 10.4310/MRL.2009.v16.n6.a12 · Zbl 1191.14013
[16] S.K. Donaldson and R.P. Thomas, Gauge Theory in Higher Dimensions, Chapter 3 in S.A. Huggett, L.J. Mason, K.P. Tod, S.T. Tsou and N.M.J. Woodhouse, editors, The Geometric Universe, Oxford University Press, Oxford, 1998. · Zbl 0926.58003
[17] E. Freitag and R. Kiehl, Etale cohomology and the Weil Conjecture, Ergeb. der Math. und ihrer Grenzgebiete 13, Springer-Verlag, 1988.DOI: 10.1007/978-3-662-02541-3 · Zbl 0643.14012
[18] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory – anomaly and obstruction, Parts I & II. AMS/IP Studies in Advanced Mathematics, 46.1 & 46.2, A.M.S./International Press, 2009. · Zbl 1181.53003
[19] T. Gaffney and H. Hauser, Characterizing singularities of varieties and of mappings, Invent. math. 81 (1985), 427–447.DOI: 10.1007/BF01388580 · Zbl 0627.14004
[20] G. Guibert, F. Loeser and M. Merle, Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink, Duke Math. J. 132 (2006), 409–457.DOI: 10.1215/S0012-7094-06-13232-5 · Zbl 1173.14301
[21] R. Hotta, T. Tanisaki and K. Takeuchi,D-modules, perverse sheaves, and representation theory, Progr. Math. 236, Birkh\"{}auser, Boston, MA, 2008. · Zbl 1136.14009
[22] D. Huybrechts and R.P. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira– Spencer classes, Math. Ann. 346 (2010), 545–569.DOI: 10.1007/s00208-009-0397-6 150C. BRAV, V. BUSSI, D. DUPONT, D. JOYCE, AND B. SZENDR ˝OI
[23] D. Joyce, A classical model for derived critical loci, to appear in Journal of Differential Geometry, 2015. arXiv: 1304.4508
[24] D. Joyce and Y. Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), no. 1020. · Zbl 1259.14054
[25] M. Kashiwara, The Riemann–Hilbert problem for holonomic systems, Publ. RIMS 20 (1984), 319–365. DOI: 10.2977/prims/1195181610 · Zbl 0566.32023
[26] M. Kashiwara,D-modules and microlocal calculus, Trans. Math. Mono. 217, A.M.S., 2003. · Zbl 1017.32012
[27] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Math. Wiss. 292, Springer-Verlag, Berlin, 1990.DOI: 10.1007/978-3-662-02661-8 · Zbl 0709.18001
[28] M. Kashiwara and P. Schapira, Constructibility and duality for simple modules on symplectic manifolds, Amer. J. Math. 130 (2008), 207–237.DOI: 10.1353/ajm.2008.0007 · Zbl 1142.53071
[29] M. Kashiwara and P. Schapira, Deformation quantization modules, Ast\'{}erisque 345, 2012. · Zbl 1260.32001
[30] R. Kiehl and R. Weissauer, Weil Conjectures, perverse sheaves and l’adic Fourier transform, Springer-Verlag, 2001.DOI: 10.1007/978-3-662-04576-3 · Zbl 0988.14009
[31] Y.-H. Kiem and J. Li, Categorification of Donaldson–Thomas invariants via perverse sheaves, 2012. arXiv: 1212.6444
[32] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, 2008.arXiv: 0811.2435 · Zbl 1248.14060
[33] M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Commun. Number Theory Phys. 5 (2011), 231–352. DOI: 10.4310/CNTP.2011.v5.n2.a1 · Zbl 1248.14060
[34] P. Maisonobe and Z. Mebkhout, Le th\'{}eor‘eme de comparaison pour les cycles \'{}evanescents, pages 311–389 in \'{}El\'{}ements de la th\'{}eorie des syst‘emes diff\'{}erentiels g\'{}eom\'{}etriques, S\'{}emin. Congr., 8, Soc. Math. France, Paris, 2004.
[35] D. Massey, The Sebastiani–Thom isomorphism in the derived category, Compositio Math. 125 (2001), 353– 362.DOI: 10.1023/A:1002608716514 · Zbl 0986.32004
[36] D. Massey, Notes on perverse sheaves and vanishing cycles, 1999. arXiv: math.AG/9908107
[37] D. Massey, Natural commuting of vanishing cycles and the Verdier dual, 2009.arXiv: 0908.2799 · Zbl 1348.32004
[38] J. Mather and S.S. Yau, Classification of isolated hypersurface singularities by their moduli algebra, Invent. Math. 69 (1982), 243–251.DOI: 10.1007/BF01399504 · Zbl 0499.32008
[39] L. Maxim, M. Saito and J. Sch\"{}urmann, Symmetric products of mixed Hodge modules, J. Math. Pures Appl. 96 (2011), 462–483.DOI: 10.1016/j.matpur.2011.04.003 · Zbl 1238.32021
[40] D. Nadler, Microlocal branes are constructible sheaves, Selecta Math. 15 (2009), 563–619. DOI: 10.1007/s00029-009-0008-0 · Zbl 1197.53116
[41] D. Nadler and E. Zaslow, Constructible sheaves and the Fukaya category, J. Amer. Math. Soc. 22 (2009), 233–286.DOI: 10.1090/S0894-0347-08-00612-7 · Zbl 1227.32019
[42] T. Pantev, B. To\"{}en, M. Vaqui\'{}e and G. Vezzosi, Shifted symplectic structures, Publ. Math. I.H.E.S. 117 (2013), 271–328.DOI: 10.1007/s10240-013-0054-1
[43] K. Rietsch, An introduction to perverse sheaves, pages 391–429 in V. Dlab and C.M. Ringel, editors, Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun. 40, A.M.S., Providence, RI, 2004. · Zbl 1058.55002
[44] C. Sabbah, On a twisted de Rham complex, II, 2010.arXiv: 1012.3818
[45] M. Saito, Modules de Hodge polarisables, Publ. RIMS 24 (1988), 849–995.DOI: 10.2977/prims/1195173930 · Zbl 0691.14007
[46] M. Saito, Duality for vanishing cycle functors, Publ. RIMS 25 (1989), 889–921. DOI: 10.2977/prims/1195172510 · Zbl 0712.32011
[47] M. Saito, Mixed Hodge Modules, Publ. RIMS 26 (1990), 221–333.DOI: 10.2977/prims/1195171082 · Zbl 0727.14004
[48] M. Saito,D-modules on analytic spaces, Publ. RIMS 27 (1991), 291–332.DOI: 10.2977/prims/1195169840 · Zbl 0742.32009
[49] M. Saito, Thom–Sebastiani Theorem for Hodge Modules, preprint, 2010.
[50] J. Sch\"{}urmann, Topology of singular spaces and constructible sheaves, Monografie Matematyczne 63, Birkh\"{}auser, Basel, 2003. · Zbl 1041.55001
[51] B. Szendr˝oi, Nekrasov’s partition function and refined Donaldson–Thomas theory: the rank one case, SIGMA (2012) 088, 16pp. · Zbl 1284.14079
[52] R.P. Thomas, A holomorphic Casson invariant for Calabi–Yau 3–folds, and bundles on K3 fibrations, J. Diff. Geom. 54 (2000), 367–438. · Zbl 1034.14015
[53] J.-L. Verdier, Sp\'{}ecialization de faisceaux et monodromie mod\'{}er\'{}ee, Ast\'{}erisque 101-102 (1983), 332–364. SYMMETRIES AND STABILIZATION FOR SHEAVES OF VANISHING CYCLES151 Address for Christopher Brav: Faculty of Mathematics, Higher School of Economics, 7 Vavilova Str., Moscow, Russia E-mail: chris.i.brav@gmail.com. Address for Vittoria Bussi: ICTP, Strada Costiera 11, Trieste, Italy E-mail: vbussi@ictp.it. Address for Delphine Dupont, Dominic Joyce and Bal\'{}azs Szendr˝oi: The Mathematical Institute, Woodstock Road, Oxford, UK E-mails: joyce@maths.ox.ac.uk, szendroi@maths.ox.ac.uk Address for J\"{}org Sch\"{}urmann: Mathematische Institut, Universit\"{}at M\"{}unster, Einsteinstrasse 62, 48149 M\"{}unster, Germany. E-mail: jschuerm@math.uni-muenster.de.
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