×

Variation of geometric invariant theory quotients and derived categories. (English) Zbl 1432.14015

The bounded derived category of coherent sheaves \(\mathsf{D}^{\text b}(\text{coh}X)\) for a smooth, projective variety \(X\) is an important, and sometimes classifying, invariant of the variety. In this work, \(\mathsf{D}^{\text b}(\text{coh}X)\) is broken down into simpler pieces of building blocks that can be analysed to give information about \(X\). The tool used in this article for this procedure is called semi-orthogonal decomposition. There exist many examples of such decompositions, but no general algorithm to determine what semi-orthogonal decompositions that exist of a given bounded derived category \(\mathsf{D}^{\text b}(\text{coh}X).\) In this article, the authors give a method for finding semi-orthogonal decompositions of bounded derived categories of coherent sheaves together with a complete description of all the involved components.
The approach in this article is based on birational methods as one expects that the derived categories of sheaves on birational varieties should be related. There is no known semi-orthogonal decomposition for a sufficiently general class of birational transformations, where by sufficiently general means that the birational class should at least include blow-ups at smooth centres. In this article, the class of birational transformations comes from Geometric Invariant Theory (GIT).
The article makes the link between GIT and birational geometry clear. There is no canonical choice of linearisation of a group action on a variety, and this establishes a feature for constructing new birational models of a GIT quotient. The meaning of this sentence is explained by treating GIT thoroughly through a separate preliminary section: Changing the linearisation leads to birational transformations between the GIT quotients. This is what is called variations of GIT structures (VGIT). Conversely, any birational map between smooth projective varieties can be obtained through such GIT variations, and one then says that two different GIT quotients are related by wall-crossing. The last is because there is a natural fan structure on the set of linearisations.
The present methods focused on semi-orthogonal decompositions coming from wall-crossing in VGIT give a new perspective on the relationship between birational geometry and derived categories, leading to new and important results.
The main results on derived categories of sheaves are stated as follows in the article: Let \(X\) be a smooth, projective variety with an action of a reductive linear algebraic group \(G\). Assume that \(X\) has two \(G\)-equivariant ample line bundles \(\mathcal L_-\) and \(\mathcal L_+\) satisfying:
i)
For \(t\in[-1,1],\) let \(\mathcal L_t=\mathcal L_-^{\frac{1-t}{2}}\otimes\mathcal L_+^{\frac{1+t}{2}}.\)
Then the semi-stable locus should be constant for \(-1\leq t<0\) and for \(0<t\leq 1.\) Now name \(X^{\text{ss}}(-):=X^{\text{ss}}(\mathcal L_t)\) for \(-1\leq t<0,\) \(X^{\text{ss}}(0):=X^{\text{ss}}(\mathcal L_0),\) \(X^{\text{ss}}(+):=X^{\text{ss}}(\mathcal L_t)\) for \(0<t\leq 1.\) Then:
ii)
The set \(X^{\text{ss}}(0)\setminus(X^{\text{ss}}(-)\cup V(+))\) is connected,
iii)
For any point \(x\in X^{\text{ss}}(0)\setminus(X^{\text{ss}}(-)\cup X^{\text{ss}}(+)),\) the stabilizer \(G_x\) is isomorphic to \(\mathbb G_m.\)

When these conditions satisfied, work of M. Thaddeus [J. Am. Math. Soc. 9, No. 3, 691–723 (1996; Zbl 0874.14042)] and I. V. Dolgachev and Y. Hu [Publ. Math., Inst. Hautes Étud. Sci. 87, 5–56 (1998; Zbl 1001.14018)] show that there is a one-parameter subgroup \(\lambda:\mathbb G_m\rightarrow G,\) a connected component \(Z^0_\lambda\) on the fixed locus of \(\lambda\) in \(X^{\text{ss}}(0),\) and disjoint decompositions \(X^{\text{ss}}(0)=X^{\text{ss}}(+)\sqcup S_\lambda\text{ and }X^{\text{ss}}(0)=X^{\text{ss}}(-)\sqcup S_{-\lambda},\) where \(S_\lambda\) is the \(G\)-orbit of all points in \(X\) that flow to \(Z^0_\lambda\) as \(\alpha\rightarrow 0\) in \(\mathbb G_m\) and \(S_{-\lambda}\) is the \(G\)-orbit of all points in \(X\) that flow to \(Z^0_\lambda\) as \(\alpha\rightarrow\infty\) in \(\mathbb G_m.\)
To state the article’s first main statement: Let \(C(\lambda)\) be the centralizer of \(\lambda\) and \(G_\lambda=C(\lambda)/\lambda.\) Let \(X/\!/ +:=[X^{\text{ss}}(+)/G]\text{ and }X/\!/ - :=[X^{\text{ss}}(-)/G]\) be the global quotient stacks of the \((+)\) and \((-)\) semi-stable loci by \(G\). Let \(\mu\) be the weight of \(\lambda\) on the anti-canonical bundle of \(X\) along \(Z^0_\lambda.\) The authors assume for simplicity that there exists a splitting \(C(\lambda)\cong\lambda\times G_\lambda,\) and they put \(X^\lambda/\!/_0 G_\lambda\) the GIT quotient stack \([(X^\lambda)^{\text{ss}}(\mathcal L_0)/G_\lambda]\) of the fixed locus \(X_\lambda\) by \(G_\lambda\) using the equivariant bundle \(\mathcal L_0.\) We state theorem more or less verbatim:
Theorem 1. Fix \(d\in\mathbb Z.\) (a) If \(\mu>0,\) then there are fully-faithful functors \(\Phi_d^+:\mathsf{D}^{\text b}(\text{coh}X/\!/-)\rightarrow \mathsf{D}^{\text b}(\text{coh}X/\!/ +),\) and, for \(d\leq j\leq\mu+d-1,\Upsilon_j^+:\mathsf{D}^{\text b}(\text{coh}X^\lambda/\!/_0 G_\lambda)\rightarrow \mathsf{D}^{\text b}(\text{coh}X/\!/ +)\) and a semi-orthogonal decomposition \(\mathsf{D}^{\text b}(\text{coh}X/\!/ +)=\langle \Upsilon^+_d,\dots,\Upsilon^+_{\mu+d-1},\Phi^+_d\rangle.\) (b) If \(\mu=0,\) then there is an exact equivalence \(\Phi^+_d:\mathsf{D}^{\text b}(\text{coh}X/\!/ -)\rightarrow\mathsf{D}^{\text b}(\text{coh}X/\!/ +).\) (c) If \(\mu<0,\) then there are fully-faithful functors \(\Phi_d^-:\mathsf{D}^{\text b}(\text{coh}X/\!/ +)\rightarrow\mathsf{D}^{\text b}(\text{coh}X/\!/-)\) and, for \(\mu+d+1\leq j\leq d,\;\Upsilon_j^-:\mathsf{D}^{\text b}(\text{coh}X^\lambda/\!/_0 G_\lambda)\rightarrow\mathsf{D}^{\text b}(\text{coh}X/\!/ -)\) and a semi-orthogonal decomposition \(\mathsf{D}^{\text b}(\text{coh}X/\!/-)=\langle\Upsilon^-_{\mu+d+1},\dots,\Upsilon^-_d,\Phi^-_d\rangle.\)
The above theorem provides a framework to view some exsting results. For a particular choice of wall-crossing, D. O. Orlov’s description [Russ. Acad. Sci., Izv., Math. 41, No. 1, 1 (1992; Zbl 0798.14007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 4, 852–862 (1992)] of the derived category of a blow up with smooth center, can be recovered. Also, it can be used to prove the \(D\)-domination and \(K\)-domination for such variations, and to give a streamlined proof of a result of Y. Kawamata [J. Differ. Geom. 61, No. 1, 147–171 (2002; Zbl 1056.14021)] stating that for \(X\) a smooth projective toric variety, the derived category \(\mathsf{D}^{\text b}(\text{coh}X)\) possesses a full exceptional collection.
Now, M. M. Kapranov [J. Algebr. Geom. 2, No. 2, 239–262 (1993; Zbl 0790.14020)] presented \(\overline{M}_{0,n}\) as an iterated blow up of \(\mathbb P^{n-3}\) along strict transforms of linear spaces and so the existence of a full exceptional collection was known. The article generalizes this result by establishing the corresponding result for B. Hassett’s moduli spaces [Adv. Math. 173, No. 2, 316–352 (2003; Zbl 1072.14014)] of stable symmetrically-weighted rational curves \(\overline{M}_{0,n\times\epsilon}.\)
D. Orlov [Prog. Math. 270, 503–531 (2009; Zbl 1200.18007)] has given a result relating the derived categories of projective complete intersections and singularity categories of affine cones. As a final main result of the present work, it is proved how to recover this result using VGIT.
In addition to give the results mentioned above, the article recall the GIT in a relative elementary way. Thus the ideas and the extracted definitions from GIT are as important as the results themselves.
The ideas in the paper appeared first by Y. Kawamata [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 197–215 (2002; Zbl 1092.14023)] in his work on derived categories treating \(\mathbb G_m\) actions.
Independently, M. van den Bergh [in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer. 749–770 (2004; Zbl 1082.14005)] also studied these actions on affine space via windows, giving the fully-faithful functors and the criterion for equivalence. The present article makes manifest that windows and VGIT are an essential framework for Orlov’s work. The authors mention Segal’s, Orlov’s and others important work, using the framework developed by Orlov and highlighted in this article.
I would like to end the review with the author’s own words verbatim: “Neither of these works provides descriptions of the full semi-orthogonal decompositions arising from wall-crossing. Consequently, applications, outside of those to construction of equivalences, are more limited in these works than here. This includes all applications mentioned.”

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14L24 Geometric invariant theory
14D07 Variation of Hodge structures (algebro-geometric aspects)
14D22 Fine and coarse moduli spaces
18G80 Derived categories, triangulated categories
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Alper, Good moduli spaces for Artin stacks, Ph.D. thesis, Stanford University, 2008, .
[2] M. Ballard, Derived categories of singular schemes with an application to reconstruction, preprint (2011), . · Zbl 1213.14031
[3] M. Ballard, D. Deliu, D. Favero, M. U. Isik and L. Katzarkov, Resolutions in factorization categories, preprint (2014), .
[4] M. Ballard and D. Favero, Hochschild dimensions of tilting objects, Int. Math. Res. Not. IMRN 2012 (2012), no. 11, 2607-2645. · Zbl 1250.18011
[5] M. Ballard, D. Favero and L. Katzarkov, A category of kernels for graded matrix factorizations and its implications for Hodge theory, preprint (2011), .
[6] M. Ballard, D. Favero and L. Katzarkov, Orlov spectra: Bounds and gaps, Invent. Math. 189 (2012), no. 2, 359-430. · Zbl 1266.14013
[7] M. Bernardara and M. Bolognesi, Categorical representability and intermediate Jacobians of Fano threefolds, preprint (2011), . · Zbl 1287.18010
[8] M. Bernardara, E. Macrì, S. Mehrotra and P. Stellari, A categorical invariant for cubic threefolds, Adv. Math. 229 (2012), no. 2, 770-803. · Zbl 1242.14012
[9] A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480-497. · Zbl 0275.14007
[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.
[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.
[12] A. Bondal and D. Orlov, Semi-orthogonal decompositions for algebraic varieties, preprint (1995), .
[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. · Zbl 0994.18007
[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. · Zbl 1178.14057
[15] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613-632. · Zbl 1085.14017
[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. · Zbl 0741.14028
[17] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, preprint (1986).
[18] A. Căldăraru and J. Tu, Curved {A_{∞}} algebras and Landau-Ginzburg models, preprint (2013), .
[19] P. Clarke and J. Guffin, On the existence of affine Landau-Ginzburg phases in gauged linear sigma models, preprint (2010), .
[20] D. Cox, J. Little and H. Schenck, Toric varieties, Grad. Stud. Math. 124, American Mathematical Society, Providence 2011. · Zbl 1223.14001
[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. · Zbl 0181.48803
[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. · Zbl 1001.14018
[23] W. Donovan, Grassmannian twists on the derived category via spherical functors, preprint (2011), . · Zbl 1283.14009
[24] W. Donovan and E. Segal, Window shifts, flop equivalences, and Grassmannian twists, preprint (2012), . · Zbl 1354.14028
[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. · Zbl 0990.14008
[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. · Zbl 0444.13006
[27] D. Favero, Reconstruction and finiteness results for Fourier-Mukai partners, Adv. Math. 230 (2012), no. 4-6, 1955-1971. · Zbl 1253.14016
[28] W. Fulton and J. Harris, Representation theory. A first course, Grad. Texts in Math. 129, Springer, New York 1991.
[29] W. Fulton and R. MacPherson, A compactification of configuration spaces, Ann. of Math. (2) 139 (1994), no. 1, 183-225. · Zbl 0820.14037
[30] I. Gel’fand, M. Kapranov and A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Mod. Birkhäuser Class., Birkhäuser, Boston 2008.
[31] P. Griffiths, On the periods of certain rational integrals, Ann. of Math. (2) 90 (1969), 460-541. · Zbl 0215.08103
[32] M. Halic, Quotients of affine spaces for actions of reductive groups, preprint (2004), .
[33] D. Halpern-Leistner, The derived category of a GIT quotient, preprint (2014), . · Zbl 1354.14029
[34] B. Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), no. 2, 316-352. · Zbl 1072.14014
[35] M. Herbst, K. Hori and D. Page, Phases of {N=2} theories in 1+1 dimensions with boundary, preprint (2008), .
[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. · Zbl 1248.14022
[37] W. Hesselink, Desingularizations of varieties of nullforms, Invent. Math. 55 (1979), no. 2, 141-163. · Zbl 0401.14006
[38] Y. Hu and S. Keel, A GIT proof of Włodarczyk’s weighted factorization theorem, preprint (1999), .
[39] M. U. Isik, Equivalence of the derived category of a variety with a singularity category, preprint (2010), .
[40] M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479-508. · Zbl 0651.18008
[41] M. Kapranov, Veronese curves and Grothendieck-Knudsen moduli space {M_{0,n}}, J. Algebraic Geom. 2 (1993), no. 2, 239-262. · Zbl 0790.14020
[42] Y. Kawamata, D-equivalence and K-equivalence, J. Differential Geom. 61 (2002), no. 1, 147-171. · Zbl 1056.14021
[43] Y. Kawamata, Francia’s flip and derived categories, Algebraic geometry, De Gruyter, Berlin (2002), 197-215. · Zbl 1092.14023
[44] Y. Kawamata, Log crepant birational maps and derived categories, J. Math. Sci. Univ. Tokyo 12 (2005), no. 2, 211-231. · Zbl 1095.14014
[45] Y. Kawamata, Derived categories of toric varieties, Michigan Math. J. 54 (2006), no. 3, 517-535. · Zbl 1159.14026
[46] Y. Kawamata, Derived categories of toric varieties. II, preprint (2012), .
[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. · Zbl 0768.14002
[48] G. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299-316. · Zbl 0406.14031
[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. · Zbl 1253.14029
[50] A. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515-530. · Zbl 0837.16005
[51] F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes 31, Princeton University Press, Princeton 1984. · Zbl 0553.14020
[52] A. Kuznetsov, Hochschild homology and semiorthogonal decompositions, preprint (2009), .
[53] A. Kuznetsov, Derived categories of cubic fourfolds, Cohomological and geometric approaches to rationality problems, Progr. Math. 282, Birkhäuser, Boston (2010), 219-243. · Zbl 1202.14012
[54] L. Li, Wonderful compactification of an arrangement of subvarieties, Michigan Math. J. 58 (2009), no. 2, 535-563. · Zbl 1187.14060
[55] Y. Manin and M. Smirnov, On the derived category of {M_{0,n}}, preprint (2011), .
[56] M. Marcolli and G. Tabuaba, From exceptional collections to motivic decompositions, preprint (2013), .
[57] E. Miller and B. Sturmfels, Combinatorial commutative algebra, Grad. Texts in Math. 227, Springer, New York 2005.
[58] I. Mirkovíc and S. Riche, Linear Koszul duality, Compos. Math. 146 (2010), no. 1, 233-258. · Zbl 1275.14014
[59] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 3, Springer, Berlin 1994. · Zbl 0797.14004
[60] A. Mustaţă and A. Mustaţă, Intermediate moduli spaces of stable maps, Invent. Math. 167 (2007), no. 1, 47-90. · Zbl 1111.14018
[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. · Zbl 0604.14006
[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. · Zbl 0798.14007
[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. · Zbl 1200.18007
[64] D. Orlov, Remarks on generators and dimensions of triangulated categories, Mosc. Math. J. 9 (2009), no. 1, 153-159.
[65] D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226 (2011), no. 1, 206-217. · Zbl 1216.18012
[66] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, preprint (2014), . · Zbl 1275.18002
[67] L. Positselski, Coherent analogues of matrix factorizations and relative singularity categories, preprint (2015), . · Zbl 1333.14018
[68] N. Ressayre, The GIT-equivalence for G-line bundles, Geom. Dedicata 81 (2000), no. 1-3, 295-324. · Zbl 0955.14035
[69] E. Segal, Equivalences between GIT quotients of Landau-Ginzburg B-models, Comm. Math. Phys. 304 (2011), no. 2, 411-432. · Zbl 1216.81122
[70] J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197-278. · Zbl 0067.16201
[71] I. Shipman, A geometric approach to Orlov’s theorem, Compos. Math. 148 (2012), no. 5, 1365-1389. · Zbl 1253.14019
[72] H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1-28. · Zbl 0277.14008
[73] C. Teleman, The quantization conjecture revisited, Ann. of Math. (2) 152 (2000), no. 1, 1-43. · Zbl 0980.53102
[74] M. Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691-723. · Zbl 0874.14042
[75] R. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math. 65 (1987), no. 1, 16-34. · Zbl 0624.14025
[76] H. Uehara, An example of Fourier-Mukai partners of minimal elliptic surfaces, Math. Res. Lett. 11 (2004), 371-375. · Zbl 1060.14055
[77] M. Van den Bergh, Non-commutative crepant resolutions, The legacy of Niels Henrik Abel, Springer, Berlin (2004), 749-770. · Zbl 1082.14005
[78] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613-670. · Zbl 0694.14001
[79] E. Witten, Phases of {N=2} theories in two dimensions, Nuclear Phys. B 403 (1993), no. 1-2, 159-222. · Zbl 0910.14020
[80] J. Włodarczyk, Birational cobordisms and factorization of birational maps, J. Algebraic Geom. 9 (2000), no. 3, 425-449. · Zbl 1010.14002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.