##
**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].

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].

Reviewer: Will Donovan (Chiba Kashiwa)

### 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 |

### Keywords:

geometry invariant theory (GIT); GIT quotient; variation of GIT; derived category of coherent sheaves; equivariant K-theory; semiorthogonal decomposition; exceptional collection; flop; flip; matrix factorization; derived category of singularities; hyper-Kähler reduction
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XMLCite

\textit{D. Halpern-Leistner}, J. Am. Math. Soc. 28, No. 3, 871--912 (2015; Zbl 1354.14029)

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