Abstract
We study the relationship between derived categories of factorizations on gauged Landau–Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and we completely describe all components appearing in these semi-orthogonal decompositions. We show how this general framework encompasses many well-known semi-orthogonal decompositions. We then proceed to give applications of this complete description.
In this setting, we verify a question posed by Kawamata: we show that D-equivalence and K-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne–Mumford stacks. This recovers Kawamata’s theorem that all projective toric Deligne–Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover and extend Orlov’s σ-model/Landau–Ginzburg model correspondence.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS 0636606 RTG
Award Identifier / Grant number: DMS 0838210 RTG
Award Identifier / Grant number: DMS 0854977 FRG
Award Identifier / Grant number: DMS 0600800
Award Identifier / Grant number: DMS 0652633 FRG
Award Identifier / Grant number: DMS 0854977
Award Identifier / Grant number: MS 0901330
Funding source: Austrian Science Fund
Award Identifier / Grant number: P24572 N25
Award Identifier / Grant number: P20778
Funding statement: The first named author was funded by NSF DMS 0636606 RTG, NSF DMS 0838210 RTG, and NSF DMS 0854977 FRG. The second and third named authors were funded by NSF DMS 0854977 FRG, NSF DMS 0600800, NSF DMS 0652633 FRG, NSF DMS 0854977, NSF DMS 0901330, FWF P24572 N25, by FWF P20778 and by an ERC Grant.
Acknowledgements
The authors have benefited immensely from conversations and correspondence with Yujiro Kawamata, Manfred Herbst, Colin Diemer, Gabriel Kerr, Alastair Craw, Sukhendu Mehrotra, Andrei Căldăraru, Paolo Stellari, Ed Segal, Michael Thaddeus, Dmitri Orlov, Alexei Bondal, Kentaro Hori, Will Donovan, R. Paul Horja, Dragos Deliu, M. Umut Isik, Pawel Sosna, Emanuele Macrì, Alexander Kuznetsov, Daniel Halpern-Leistner, and Maxim Kontsevich and would like to thank them all for their time, patience, and insight.
References
[1] J. Alper, Good moduli spaces for Artin stacks, Ph.D. thesis, Stanford University, 2008, http://arxiv.org/abs/0804.2242. Search in Google Scholar
[2] M. Ballard, Derived categories of singular schemes with an application to reconstruction, preprint (2011), http://arxiv.org/abs/0801.2599. 10.1016/j.aim.2011.02.016Search in Google Scholar
[3] M. Ballard, D. Deliu, D. Favero, M. U. Isik and L. Katzarkov, Resolutions in factorization categories, preprint (2014), http://arxiv.org/abs/1212.3264. 10.1016/j.aim.2016.02.008Search in Google Scholar
[4] M. Ballard and D. Favero, Hochschild dimensions of tilting objects, Int. Math. Res. Not. IMRN 2012 (2012), no. 11, 2607–2645. 10.1093/imrn/rnr124Search in Google Scholar
[5] M. Ballard, D. Favero and L. Katzarkov, A category of kernels for graded matrix factorizations and its implications for Hodge theory, preprint (2011), http://arxiv.org/abs/1105.3177v3. Search in Google Scholar
[6] M. Ballard, D. Favero and L. Katzarkov, Orlov spectra: Bounds and gaps, Invent. Math. 189 (2012), no. 2, 359–430. 10.1007/s00222-011-0367-ySearch in Google Scholar
[7] M. Bernardara and M. Bolognesi, Categorical representability and intermediate Jacobians of Fano threefolds, preprint (2011), http:arxiv.org/abs/1103.3591. 10.4171/115-1/1Search in Google Scholar
[8] M. Bernardara, E. Macrì, S. Mehrotra and P. Stellari, A categorical invariant for cubic threefolds, Adv. Math. 229 (2012), no. 2, 770–803. 10.1016/j.aim.2011.10.007Search in Google Scholar
[9] A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497. 10.2307/1970915Search in Google Scholar
[10] A. Bondal, Representations of associative algebras and coherent sheaves (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44; translation in Math. USSR-Izv. 34 (1990), no. 1, 23–42. 10.1070/IM1990v034n01ABEH000583Search in Google Scholar
[11] A. Bondal and M. Kapranov, Representable functors, Serre functors, and reconstructions (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337; translation in Math. USSR-Izv. 35 (1990), no. 3, 519–541. Search in Google Scholar
[12] A. Bondal and D. Orlov, Semi-orthogonal decompositions for algebraic varieties, preprint (1995), http://arxiv.org/abs/alg-geom/9506012. Search in Google Scholar
[13] A. Bondal and D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math. 125 (2001), no. 3, 327–344. 10.1023/A:1002470302976Search in Google Scholar
[14] L. Borisov, L. Chen and G. Smith, The orbifold Chow ring of toric Deligne–Mumford stacks, J. Amer. Math. Soc. 18 (2005), no. 1, 193–215. 10.1090/S0894-0347-04-00471-0Search in Google Scholar
[15] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613–632. 10.1007/s002220100185Search in Google Scholar
[16] M. Brion and C. Procesi, Action d’un tore dans une variété projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris 1989), Progr. Math. 92, Birkhäuser, Boston (1990), 509–539. Search in Google Scholar
[17] R.-O. Buchweitz, Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, preprint (1986). Search in Google Scholar
[18]
A. Căldăraru and J. Tu,
Curved
[19] P. Clarke and J. Guffin, On the existence of affine Landau–Ginzburg phases in gauged linear sigma models, preprint (2010), http://arxiv.org/abs/1004.2937. 10.4310/ATMP.2015.v19.n4.a1Search in Google Scholar
[20] D. Cox, J. Little and H. Schenck, Toric varieties, Grad. Stud. Math. 124, American Mathematical Society, Providence 2011. 10.1090/gsm/124Search in Google Scholar
[21] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 75–109. 10.1007/BF02684599Search in Google Scholar
[22] I. Dolgachev and Y. Hu, Variation of geometric invariant theory quotients. With an appendix by N. Ressayre, Publ. Math. Inst. Hautes Études Sci. 87 (1998), 5–56. 10.1007/BF02698859Search in Google Scholar
[23] W. Donovan, Grassmannian twists on the derived category via spherical functors, preprint (2011), http://arxiv.org/abs/1111.3774. 10.1112/plms/pdt008Search in Google Scholar
[24] W. Donovan and E. Segal, Window shifts, flop equivalences, and Grassmannian twists, preprint (2012), http://arxiv.org/abs/1206.0219. 10.1112/S0010437X13007641Search in Google Scholar
[25] D. Edidin, Notes on the construction of the moduli space of curves, Recent progress in intersection theory (Bologna 1997), Trends Math., Birkhäuser, Boston (2000), 85–113. 10.1007/978-1-4612-1316-1_3Search in Google Scholar
[26] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64. 10.1090/S0002-9947-1980-0570778-7Search in Google Scholar
[27] D. Favero, Reconstruction and finiteness results for Fourier–Mukai partners, Adv. Math. 230 (2012), no. 4–6, 1955–1971. 10.1016/j.aim.2012.03.025Search in Google Scholar
[28] W. Fulton and J. Harris, Representation theory. A first course, Grad. Texts in Math. 129, Springer, New York 1991. Search in Google Scholar
[29] W. Fulton and R. MacPherson, A compactification of configuration spaces, Ann. of Math. (2) 139 (1994), no. 1, 183–225. 10.2307/2946631Search in Google Scholar
[30] I. Gel’fand, M. Kapranov and A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Mod. Birkhäuser Class., Birkhäuser, Boston 2008. Search in Google Scholar
[31] P. Griffiths, On the periods of certain rational integrals, Ann. of Math. (2) 90 (1969), 460–541. 10.2307/1970746Search in Google Scholar
[32] M. Halic, Quotients of affine spaces for actions of reductive groups, preprint (2004), http://arxiv.org/abs/math/0412278. Search in Google Scholar
[33] D. Halpern-Leistner, The derived category of a GIT quotient, preprint (2014), http://arxiv.org/abs/1203.0276. 10.1090/S0894-0347-2014-00815-8Search in Google Scholar
[34] B. Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316–352. 10.1016/S0001-8708(02)00058-0Search in Google Scholar
[35]
M. Herbst, K. Hori and D. Page,
Phases of
[36] M. Herbst and J. Walcher, On the unipotence of autoequivalences of toric complete intersection Calabi–Yau categories, Math. Ann. 353 (2012), no. 3, 783–802. 10.1007/s00208-011-0704-xSearch in Google Scholar
[37] W. Hesselink, Desingularizations of varieties of nullforms, Invent. Math. 55 (1979), no. 2, 141–163. 10.1007/BF01390087Search in Google Scholar
[38] Y. Hu and S. Keel, A GIT proof of Włodarczyk’s weighted factorization theorem, preprint (1999), http://arxiv.org/abs/math/9904146. Search in Google Scholar
[39] M. U. Isik, Equivalence of the derived category of a variety with a singularity category, preprint (2010), http://arxiv.org/abs/1011.1484. 10.1093/imrn/rns125Search in Google Scholar
[40] M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479–508. 10.1007/BF01393744Search in Google Scholar
[41]
M. Kapranov,
Veronese curves and Grothendieck–Knudsen moduli space
[42] Y. Kawamata, D-equivalence and K-equivalence, J. Differential Geom. 61 (2002), no. 1, 147–171. 10.4310/jdg/1090351323Search in Google Scholar
[43] Y. Kawamata, Francia’s flip and derived categories, Algebraic geometry, De Gruyter, Berlin (2002), 197–215. 10.1515/9783110198072.197Search in Google Scholar
[44] Y. Kawamata, Log crepant birational maps and derived categories, J. Math. Sci. Univ. Tokyo 12 (2005), no. 2, 211–231. Search in Google Scholar
[45] Y. Kawamata, Derived categories of toric varieties, Michigan Math. J. 54 (2006), no. 3, 517–535. 10.1307/mmj/1163789913Search in Google Scholar
[46] Y. Kawamata, Derived categories of toric varieties. II, preprint (2012), http://arxiv.org/abs/1201.3460. 10.1307/mmj/1370870376Search in Google Scholar
[47] S. Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. Search in Google Scholar
[48] G. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316. 10.2307/1971168Search in Google Scholar
[49] Y.-H. Kiem and H.-B. Moon, Moduli spaces of weighted pointed stable rational curves via GIT, Osaka J. Math. 48 (2011), no. 4, 1115–1140. Search in Google Scholar
[50] A. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. 10.1093/qmath/45.4.515Search in Google Scholar
[51] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes 31, Princeton University Press, Princeton 1984. 10.1515/9780691214566Search in Google Scholar
[52] A. Kuznetsov, Hochschild homology and semiorthogonal decompositions, preprint (2009), http://arxiv.org/abs/0904.4330. Search in Google Scholar
[53] A. Kuznetsov, Derived categories of cubic fourfolds, Cohomological and geometric approaches to rationality problems, Progr. Math. 282, Birkhäuser, Boston (2010), 219–243. 10.1007/978-0-8176-4934-0_9Search in Google Scholar
[54] L. Li, Wonderful compactification of an arrangement of subvarieties, Michigan Math. J. 58 (2009), no. 2, 535–563. 10.1307/mmj/1250169076Search in Google Scholar
[55]
Y. Manin and M. Smirnov,
On the derived category of
[56] M. Marcolli and G. Tabuaba, From exceptional collections to motivic decompositions, preprint (2013), http://arxiv.org/abs/1202.6297v2. 10.1515/crelle-2013-0027Search in Google Scholar
[57] E. Miller and B. Sturmfels, Combinatorial commutative algebra, Grad. Texts in Math. 227, Springer, New York 2005. Search in Google Scholar
[58] I. Mirkovíc and S. Riche, Linear Koszul duality, Compos. Math. 146 (2010), no. 1, 233–258. 10.1112/S0010437X09004357Search in Google Scholar
[59] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 3, Springer, Berlin 1994. 10.1007/978-3-642-57916-5Search in Google Scholar
[60] A. Mustaţă and A. Mustaţă, Intermediate moduli spaces of stable maps, Invent. Math. 167 (2007), no. 1, 47–90. 10.1007/s00222-006-0006-1Search in Google Scholar
[61] L. Ness, A stratification of the null cone via the moment map. With an appendix by D. Mumford, Amer. J. Math. 106 (1984), no. 6, 1281–1329. 10.2307/2374395Search in Google Scholar
[62] D. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852–862. 10.1070/IM1993v041n01ABEH002182Search in Google Scholar
[63] D. 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. 270, Birkhäuser, Boston (2009), 503–531. 10.1007/978-0-8176-4747-6_16Search in Google Scholar
[64] D. Orlov, Remarks on generators and dimensions of triangulated categories, Mosc. Math. J. 9 (2009), no. 1, 153–159. 10.17323/1609-4514-2009-9-1-143-149Search in Google Scholar
[65] D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226 (2011), no. 1, 206–217. 10.1016/j.aim.2010.06.016Search in Google Scholar
[66] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, preprint (2014), http://arxiv.org/abs/0905.2621. 10.1090/S0065-9266-2010-00631-8Search in Google Scholar
[67] L. Positselski, Coherent analogues of matrix factorizations and relative singularity categories, preprint (2015), http://arxiv.org/abs/1102.0261. Search in Google Scholar
[68] N. Ressayre, The GIT-equivalence for G-line bundles, Geom. Dedicata 81 (2000), no. 1–3, 295–324. 10.1023/A:1005275524522Search in Google Scholar
[69] E. Segal, Equivalences between GIT quotients of Landau–Ginzburg B-models, Comm. Math. Phys. 304 (2011), no. 2, 411–432. 10.1007/s00220-011-1232-ySearch in Google Scholar
[70] J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278. 10.1007/978-3-642-39816-2_29Search in Google Scholar
[71] I. Shipman, A geometric approach to Orlov’s theorem, Compos. Math. 148 (2012), no. 5, 1365–1389. 10.1112/S0010437X12000255Search in Google Scholar
[72] H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28. 10.1215/kjm/1250523277Search in Google Scholar
[73] C. Teleman, The quantization conjecture revisited, Ann. of Math. (2) 152 (2000), no. 1, 1–43. 10.2307/2661378Search in Google Scholar
[74] M. Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. 10.1090/S0894-0347-96-00204-4Search in Google Scholar
[75] R. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math. 65 (1987), no. 1, 16–34.10.1016/0001-8708(87)90016-8Search in Google Scholar
[76] H. Uehara, An example of Fourier–Mukai partners of minimal elliptic surfaces, Math. Res. Lett. 11 (2004), 371–375. 10.4310/MRL.2004.v11.n3.a9Search in Google Scholar
[77] M. Van den Bergh, Non-commutative crepant resolutions, The legacy of Niels Henrik Abel, Springer, Berlin (2004), 749–770. 10.1007/978-3-642-18908-1_26Search in Google Scholar
[78] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. 10.1007/BF01388892Search in Google Scholar
[79]
E. Witten,
Phases of
[80] J. Włodarczyk, Birational cobordisms and factorization of birational maps, J. Algebraic Geom. 9 (2000), no. 3, 425–449. Search in Google Scholar
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