Variation of geometric invariant theory quotients and derived categories.

*(English)*Zbl 1432.14015The 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:

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.”

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 \(\mathcal L_t=\mathcal L_-^{\frac{1-t}{2}}\otimes\mathcal L_+^{\frac{1+t}{2}}.\)

- 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.”

Reviewer: Arvid Siqveland (Kongsberg)

##### 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 |

##### Keywords:

projective toric variety; one-parameter subgroup; wall-crossing; semi-orthogonal decomposition; singularity categories; full exceptional collection; GIT; windows; geometric invariant theory; fan structure; bounded derived category of coherent sheaves; equivariant ample line bundle; VGIT; reductive linear algebraic group; variations of GIT structures; birational methods; blow-ups at smooth centres; linearisation of a group action
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\textit{M. Ballard} et al., J. Reine Angew. Math. 746, 235--303 (2019; Zbl 1432.14015)

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