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The derived category of a GIT quotient. (English) Zbl 1354.14029

Given a variety \(X\) with an action of a reductive group \(G\), we may form a geometric invariant theory (GIT) quotient \(X /\!/ G\). The main theorem of this article gives, in a general setting, an equivalence between the derived category \(D^b (X /\!/ G)\) of the quotient, and an explicit subcategory \(W\) of the \(G\)-equivariant derived category of \(X\). This is an important and powerful result. The article includes applications to birational geometry, relating the derived categories of varieties linked by certain flips and flops, and also to derived categories of singularities, and hyper-Kähler reductions.
The main result may be viewed as a categorical analogue of Kirwan surjectivity for the canonical map \(H^*_G(X) \to H^*(X /\!/ G)\). It is also linked to the class of theorems in geometric quantization theory relating quantization and reduction, in particular the work of C. Teleman [Ann. Math. (2) 152, No. 1, 1–43 (2000; Zbl 0980.53102)]. The definition of the subcategory \(W\) is inspired by the physics of gauged linear \(\sigma\)-models, specifically the grade restriction rules of Herbst-Hori-Page, as interpreted mathematically by E. Segal [Commun. Math. Phys. 304, No. 2, 411–432 (2011; Zbl 1216.81122)].
The main theorem is easiest to state in the smooth setting: generalizations and refinements are indicated later. Take \(X\) a smooth projective-over-affine variety, with an action of a reductive group \(G\). By the usual GIT prescription, a linearization of this action determines an open semistable locus \(X^{\mathrm{ss}} \subseteq X\). Defining the GIT quotient \(X /\!/ G\) to be the quotient stack \([X^{\mathrm{ss}} / G]\), the author gives an equivalence \[ D^b(X /\!/ G) \cong W \subset D^b([X / G]) \] where the subcategory \(W\) is described as follows. There exists a GIT stratification of \(X-X^{\mathrm{ss}}\), determined by a finite set of distinguished one-parameter subgroups \(\lambda_i\) of \(G,\) and \(\lambda_i\)-fixed subvarieties \(Z_i\) of \(X\). Given \(F \in D^b([X / G])\), write \(\mathrm{wt}_i ( F )\) as shorthand for the set of \(\lambda_i\)-weights of the derived restriction of \(F\) to \(Z_i\). Then by definition \(W\) is the full subcategory of \(D^b([X / G])\) containing \(F\) such that \[ \mathrm{wt}_i ( F) \subseteq [w_i, w_i + \eta_i ), \] where the \(\eta_i\) are non-negative integers associated to the stratification, and integers \(w_i\) may be chosen freely.
A fundamental example is the quotient \(\mathbb{C}^n /\!/ \mathbb{G}_m\) of a vector space by dilation. For a suitable linearization, the semistable locus is \(\mathbb{C}^n -0\), so the quotient is \(\mathbb{P}^{n-1}\), and there is a restriction functor \(D^b([\mathbb{C}^n/\mathbb{G}_m]) \to D^b(\mathbb{P}^{n-1}) \). Letting \(\mathcal{O} \{ k \}\) denote the trivial line bundle on \(\mathbb{C}^n\) linearized with \(\mathbb{G}_m\)-weight \(k\), this restriction functor is an equivalence on the subcategory \(W\) generated by \(\mathcal{O} \{ k \}\) for \(k \in [w, w + n)\), with any choice of integer \(w\). These generators restrict to standard line bundles \(\mathcal{O} ( k )\) on \(\mathbb{P}^{n-1}\), recovering Beilinson’s celebrated exceptional collection. M. M. Kapranov’s collection for a Grassmannian [Invent. Math. 92, No. 3, 479–508 (1988; Zbl 0651.18008)] is also recovered (Example 2.14).
For a pair of different linearizations, the resulting quotients \(X /\!/_{\!+} G\) and \(X /\!/_{\!-} G\) may be related by a birational map. It is then natural to compare their derived categories, via the corresponding categories \(W_\pm\): in nice cases one may embed in the other, or they may be equivalent, and the same then follows for the derived categories \(D^b(X /\!/_{\!\pm} G)\). For \(\mathbb{G}_m\)-quotients in which the change in linearization only affects a single GIT stratum, a complete treatment is given (Proposition 4.2). For linearizations related by ‘balanced’ or ‘almost balanced’ wall-crossings, equivalences are shown (Proposition 4.5): in particular, these methods produce equivalences associated to Grassmannian flops (Example 4.12).
The main theorem in the general setting (Theorem 2.10) is more complicated to state. The smoothness assumption may be lifted under appropriate conditions on the stratification, in particular that the cotangent complex associated to (the inclusion of) each stratum has non-negative \(\lambda_i\)-weights when derived restricted to \(Z_i\). The result then is a semi-orthogonal decomposition \[ D^b([X/G]) = \left\langle U^<, W, U^\geq \right\rangle. \] As before \(D^b(X /\!/ G) \cong W \) for each choice of integers \(w_i\) as above, though the definition of \(W\) here is a little more technical. The components \(U\) consist of objects supported on \(X-X^{\mathrm{ss}}\) which we do not describe: the author explains how these \(U\) themselves have semi-orthogonal decomposition into explicit categories of equivariant objects on the \(Z_i\) (Amplification 2.11).
In the final section, the author develops further tools to deal with quotients of singular varieties. These are applied to derived categories of singularities, and the derived category of a hyper-Kähler quotient \(X /\!/\!/ G\) of an algebraic symplectic manifold \((X, \omega)\) by a Hamiltonian \(G\)-action. The author obtains equivalences associated to stratified Mukai flops (Example 5.4), and shows that for a pair of generic linearizations of an abelian hyper-Kähler quotient, the associated categories \(D^b(X /\!/\!/_{\!\pm} G)\) are equivalent (Corollary 5.9).
Note that closely-related questions to those studied here were addressed, at a similar time, by M. Ballard, D. Favero, and L. Katzarkov [“Variation of geometric invariant theory quotients and derived categories”, Preprint, arXiv:1203.6643].

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
19L47 Equivariant \(K\)-theory
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[1] Achar, Pramod N.; Treumann, David, Baric structures on triangulated categories and coherent sheaves, Int. Math. Res. Not. IMRN, 16, 3688-3743 (2011) · Zbl 1239.18009 · doi:10.1093/imrn/rnq226
[2] Arinkin, Dmitry; Bezrukavnikov, Roman, Perverse coherent sheaves, Mosc. Math. J., 10, 1, 3-29, 271 (2010) · Zbl 1205.18010
[3] Artin, M.; Zhang, J. J., Noncommutative projective schemes, Adv. Math., 109, 2, 228-287 (1994) · Zbl 0833.14002 · doi:10.1006/aima.1994.1087
[4] [BFK12] M. Ballard, D. Favero, and L. Katzarkov. \newblockVariation of geometric invariant theory quotients and derived categories. \newblockArXiv e-prints, March 2012. · Zbl 1432.14015
[5] Ben-Zvi, David; Francis, John; Nadler, David, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc., 23, 4, 909-966 (2010) · Zbl 1202.14015 · doi:10.1090/S0894-0347-10-00669-7
[6] [BO95] A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties. \newblockeprint arXiv:alg-geom/9506012.
[7] Bondal, A.; Orlov, D., Derived categories of coherent sheaves. Proceedings of the International Congress of Mathematicians, Vol. II , Beijing, 2002, 47-56 (2002), Higher Ed. Press, Beijing · Zbl 0996.18007
[8] Brosnan, Patrick, On motivic decompositions arising from the method of Bia\l ynicki-Birula, Invent. Math., 161, 1, 91-111 (2005) · Zbl 1085.14045 · doi:10.1007/s00222-004-0419-7
[9] Cautis, Sabin; Kamnitzer, Joel; Licata, Anthony, Coherent sheaves and categorical \(\mathfrak{sl}_2\) actions, Duke Math. J., 154, 1, 135-179 (2010) · Zbl 1228.14011 · doi:10.1215/00127094-2010-035
[10] Dolgachev, Igor V.; Hu, Yi, Variation of geometric invariant theory quotients, Inst. Hautes \'Etudes Sci. Publ. Math., 87, 5-56 (1998) · Zbl 1001.14018
[11] Donovan, Will; Segal, Ed, Window shifts, flop equivalences and Grassmannian twists, Compos. Math., 150, 6, 942-978 (2014) · Zbl 1354.14028 · doi:10.1112/S0010437X13007641
[12] Donovan, Will, Grassmannian twists on the derived category via spherical functors, Proc. Lond. Math. Soc. (3), 107, 5, 1053-1090 (2013) · Zbl 1283.14009 · doi:10.1112/plms/pdt008
[13] Harada, Megumi; Landweber, Gregory D., Surjectivity for Hamiltonian \(G\)-spaces in \(K\)-theory, Trans. Amer. Math. Soc., 359, 12, 6001-6025 (electronic) (2007) · Zbl 1128.53057 · doi:10.1090/S0002-9947-07-04164-5
[14] [HHP08] M. Herbst, K. Hori, and D. Page. Phases of \(n=2\) theories in \(1+1\) dimensions with boundary. \newblockeprint arXiv:hep-th/0803.2045, 2008.
[15] Herbst, M.; Hori, K.; Page, D., B-type D-branes in toric Calabi-Yau varieties. Homological mirror symmetry, Lecture Notes in Phys. 757, 27-44 (2009), Springer, Berlin · Zbl 1162.81033
[16] Kapranov, M. M., Derived category of coherent sheaves on Grassmann manifolds, Izv. Akad. Nauk SSSR Ser. Mat., 48, 1, 192-202 (1984) · Zbl 0564.14023
[17] Kawamata, Yujiro, Derived categories of toric varieties, Michigan Math. J., 54, 3, 517-535 (2006) · Zbl 1159.14026 · doi:10.1307/mmj/1163789913
[18] Kirwan, Frances Clare, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes 31, i+211 pp. (1984), Princeton University Press, Princeton, NJ · Zbl 0528.35039 · doi:10.1007/BF01145470
[19] Manetti, Marco, Differential graded Lie algebras and formal deformation theory. Algebraic geometry-Seattle 2005. Part 2, Proc. Sympos. Pure Math. 80, 785-810 (2009), Amer. Math. Soc., Providence, RI · Zbl 1190.14007 · doi:10.1090/pspum/080.2/2483955
[20] Orlov, D. O., Derived categories of coherent sheaves, and motives, Uspekhi Mat. Nauk. Russian Math. Surveys, 60 60, 6, 1242-1244 (2005) · Zbl 1146.18302 · doi:10.1070/RM2005v060n06ABEH004292
[21] Orlov, Dmitri, 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, 503-531 (2009), Birkh\"auser Boston, Inc., Boston, MA · Zbl 1200.18007 · doi:10.1007/978-0-8176-4747-6\_16
[22] Orlov, Dmitri, Matrix factorizations for nonaffine LG-models, Math. Ann., 353, 1, 95-108 (2012) · Zbl 1243.81178 · doi:10.1007/s00208-011-0676-x
[23] Rouquier, Rapha{\"e}l, Dimensions of triangulated categories, J. K-Theory, 1, 2, 193-256 (2008) · Zbl 1165.18008 · doi:10.1017/is007011012jkt010
[24] Segal, Ed, Equivalence between GIT quotients of Landau-Ginzburg B-models, Comm. Math. Phys., 304, 2, 411-432 (2011) · Zbl 1216.81122 · doi:10.1007/s00220-011-1232-y
[25] Shipman, Ian, A geometric approach to Orlov’s theorem, Compos. Math., 148, 5, 1365-1389 (2012) · Zbl 1253.14019 · doi:10.1112/S0010437X12000255
[26] Teleman, Constantin, The quantization conjecture revisited, Ann. of Math. (2), 152, 1, 1-43 (2000) · Zbl 0980.53102 · doi:10.2307/2661378
[27] Thaddeus, Michael, Geometric invariant theory and flips, J. Amer. Math. Soc., 9, 3, 691-723 (1996) · Zbl 0874.14042 · doi:10.1090/S0894-0347-96-00204-4
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