Flops and clusters in the homological minimal model programme.

*(English)*Zbl 1390.14012A central problem in birational geometry of 3-folds is to construct all minimal models, (nonsingular spaces with birational morphisms) \(X_i\rightarrow\text{Spec}R\) of a singular space \(\text{Spec}R\), and to pass between them via flops in an effective manner.

The classical way to construct a minimal model, is to take \(\text{Proj}(R)\) of an appropriate finitely generated graded ring. This variety comes with an ample line bundle, not giving sufficient information for flopping between minimal models.

In this article, the author looks for a bundle on a variety giving necessary information. This means a bundle containing many summands, whose determinant bundle recovers the classically obtained ample bundle. Ideally, these are close to tilting bundles, for which the conditions are known to be satisfied.

Passing between minimal models is a hard problem in general. One solution is to achieve a GIT chamber decomposition which together contains all projective minimal models. Another is to find a curve, flop, compute all the geometry explicitly, and repeat. Both of these methods are known, and demands a lot of complicated computations.

This article illustrates in the cases where a larger tilting bundle exists, that the extra information encoded in the tilting algebra (the endomorphism ring) can be used to produce an effective homological method to pass between the minimal models, both in finding the floppable curves, and to produce the flop. This setup gives a map for navigating the through the GIT chambers, giving a finer control than the derived category approaches to GIT.

Given a crepant projective birational morphism \(X\rightarrow\text{Spec}(R)\) with one-dimensional fibres, \(R\) a 3-dimensional normal Gorenstein complete local ring (over \(\mathbb C\)), and \(X\) with only Gorenstein terminal singularities. Then associate a noncommutative ring \(\Lambda=\text{End}_R(N)\) for some reflexive \(R\)-module \(N\). This induces a derived equivalence \(\Psi_X:\text{D}^b(\text{coh}X)\rightarrow \text{D}^b(\text{mod}\Lambda)\). Notice that the author include the usual representation of \(\Lambda\) as a quiver with relations in the exposition. In the case where \(X\) is smooth, the loops encode the normal bundle of the curves. It is claimed that the article contains two key new ideas: The first says that certain factors of the algebra \(\Lambda\) encode noncommutative deformations of the curves and detects the floppable curves. The second says that when curves flop, we should not view the flop as a variation of GIT, but as a change in the algebra via a universal property whilst keeping the GIT stability constant.

Suppose that \(f:X\rightarrow\text{Spec}R\) is a crepant projective birational morphism where \(R\) is complete local with at most 1-dimensional fibres. Choose \(\bigcup_{i\in I}C_i\) in \(X\) and contract them to give \(g:X\rightarrow X_{\text{con}}\) supposing that \(g\) is an isomorphism away from \(\bigcup_{i\in I}C_i\). The author defines the mutation functor (very roughly) as \(\mu_I M:=M_{I^c}\oplus K_I\) where \(M_{I^c}\) is the complement of \(M_I\) and \(K_I=\ker({a_i\cdot}).\)

It is proved that the mutation functor is isomorphic to the inverse of the Bridgeland-Chen flop functor when the curves are floppable, Viewing the flop via this universal property gives new control to the process. The mutated algebra contains the information needed to iterate without the need of explicitly calculate the geometry at each step. This implies the GIT chamber structures and wall crossing together with many results in noncommutative minimal models: In particular, it induces an Auslander-McKay correspondence in dimension 3. The author describes explicitly the algorithm for jumping between the minimal models of \(\text{Spec}R\). The process is called the Homological MMP.

The first result, proved by iteration, is that \(\bigcup_{i\in I}C_i\) contracts to a point without contracting a divisor if and only if \(\dim_{\mathbb C}\Lambda_I<\infty.\)

The next step is to choose a subset of curves \(\{C_i\mid i\in I\}\) for an index set \(I\). Each curve \(C_i\) correspond to an indecomposable summand \(N_i\) of an \(R\)-module \(N\), and the tilting algebra is replaced with its mutated algebra \(\nu_I\Lambda:=\text{End}_R(\nu_i N).\) The mutation handles the cases with loops, 2-cycles, and no superpotential (saying that it can be generalized to singular minimal models). When \(C_i\) flops, denote the flop by \(X^+\).

The second main result states that: (1) The irreducible curve \(C_i\) flops if and only if \(\nu_i N\neq N\). (2) If \(\Gamma\) denotes the natural algebra associated to the flop \(X^+\), then \(\Gamma\cong\nu_i\Gamma.\) (3) If \(X\rightarrow\text{Spec}R\) is a minimal model, and \(\dim_{\mathbb C}\Lambda_i<\infty\), then \(\Phi_i\cong\Psi_{X^+}\circ\mathsf{Flop}\circ\Psi^{-1}_X\) where \(\mathsf{Flop}\) is the inverse of the flop functor of Bridgeland-Chen. One sees that the first part of the result gives a lower bound on the number of minimal models, while the second part allows to iterate.

The above describes the theory in the article of independent interest. In addition the article includes applications. For the applications to GIT, let \(\Gamma=\text{End}_R(N)\) be any algebra with \(\Gamma\in\text{CM}R.\) The dimension vector given by the ranks of the summands of \(N\) are denoted \(\mathsf{rk}\). When \(N\) is a generator, the GIT chamber decomposition \(\Theta(\Gamma)\) associated to \(\Gamma\) has coordinates \(\vartheta_i\) for \(i\neq 0\), \(\vartheta_0\) corresponding to the summand \(R\) of \(N\). \(C_+(\Gamma)=\{\vartheta\in\Theta(\Gamma)\mid\vartheta_i>0\text{ for all }i>0\},\) and \(\mathcal M_{\mathsf{rk},\phi}(\Lambda)\) denotes the moduli space of \(\phi\)-semistable \(\Lambda\)-modules of dimension vector \(\mathsf{rk}\). This review cannot include all applications, but an example is the first corollary: Choose \(C_i\) and suppose \(\dim_{\mathbb C}\Lambda_i<\infty\) (so that \(C_i\) flops). Then (1) \(\mathcal M_{\mathsf{rk},\phi}(\Lambda)\cong X\) for all \(\phi\in C_+(\Lambda)\). (2) \(\mathcal M_{\mathsf{rk},\phi}(\nu_i\Lambda)\cong X^+\) for all \(\phi\in C_+(\nu_i\Lambda)\). Thus one can view the flop as changing the algebra, but keeping the GIT chamber structure fixed. Mutation induces a derived equivalence, and then it is possible to track the moduli space back across the equivalence to obtain the flop as a moduli space on the original algebra. Thus a second corollary to the theory is a moduli-tracking theorem.

I would like to give one more of the corollaries given in the introduction, to illustrate the strength of the theory. It states verbatim that there exists a connected path in the GIT chamber decomposition of \(\Lambda\) where every minimal model of \(\text{Spec}R\) can be found, and each wall crossing in this path corresponds to the flop of a singular curve.

The other main application of the theory is a generalized Auslander-McKay correspondence. This lifts the Auslander-McKay correspondence from dimension 2 to 3-fold compound du Val singularities, and states that when \(R\) is a complete local cDV singularity there is a one-to-one correspondence from the set of basic maximal modification \(R\)-module generators to the set of minimal models \(f_i:X_i\rightarrow\text{Spec}R\). The article includes a study on the properties of this extended correspondence.

It is fair to remark that the search for the more ample bundles includes several areas of noncommutative geometry, including noncommutative deformations. Most of these theories are recalled and explored in an explicit way in the article, and the results is as always from this author very nice and important, leading to new questions and conjectures listed at the end of the article.

The classical way to construct a minimal model, is to take \(\text{Proj}(R)\) of an appropriate finitely generated graded ring. This variety comes with an ample line bundle, not giving sufficient information for flopping between minimal models.

In this article, the author looks for a bundle on a variety giving necessary information. This means a bundle containing many summands, whose determinant bundle recovers the classically obtained ample bundle. Ideally, these are close to tilting bundles, for which the conditions are known to be satisfied.

Passing between minimal models is a hard problem in general. One solution is to achieve a GIT chamber decomposition which together contains all projective minimal models. Another is to find a curve, flop, compute all the geometry explicitly, and repeat. Both of these methods are known, and demands a lot of complicated computations.

This article illustrates in the cases where a larger tilting bundle exists, that the extra information encoded in the tilting algebra (the endomorphism ring) can be used to produce an effective homological method to pass between the minimal models, both in finding the floppable curves, and to produce the flop. This setup gives a map for navigating the through the GIT chambers, giving a finer control than the derived category approaches to GIT.

Given a crepant projective birational morphism \(X\rightarrow\text{Spec}(R)\) with one-dimensional fibres, \(R\) a 3-dimensional normal Gorenstein complete local ring (over \(\mathbb C\)), and \(X\) with only Gorenstein terminal singularities. Then associate a noncommutative ring \(\Lambda=\text{End}_R(N)\) for some reflexive \(R\)-module \(N\). This induces a derived equivalence \(\Psi_X:\text{D}^b(\text{coh}X)\rightarrow \text{D}^b(\text{mod}\Lambda)\). Notice that the author include the usual representation of \(\Lambda\) as a quiver with relations in the exposition. In the case where \(X\) is smooth, the loops encode the normal bundle of the curves. It is claimed that the article contains two key new ideas: The first says that certain factors of the algebra \(\Lambda\) encode noncommutative deformations of the curves and detects the floppable curves. The second says that when curves flop, we should not view the flop as a variation of GIT, but as a change in the algebra via a universal property whilst keeping the GIT stability constant.

Suppose that \(f:X\rightarrow\text{Spec}R\) is a crepant projective birational morphism where \(R\) is complete local with at most 1-dimensional fibres. Choose \(\bigcup_{i\in I}C_i\) in \(X\) and contract them to give \(g:X\rightarrow X_{\text{con}}\) supposing that \(g\) is an isomorphism away from \(\bigcup_{i\in I}C_i\). The author defines the mutation functor (very roughly) as \(\mu_I M:=M_{I^c}\oplus K_I\) where \(M_{I^c}\) is the complement of \(M_I\) and \(K_I=\ker({a_i\cdot}).\)

It is proved that the mutation functor is isomorphic to the inverse of the Bridgeland-Chen flop functor when the curves are floppable, Viewing the flop via this universal property gives new control to the process. The mutated algebra contains the information needed to iterate without the need of explicitly calculate the geometry at each step. This implies the GIT chamber structures and wall crossing together with many results in noncommutative minimal models: In particular, it induces an Auslander-McKay correspondence in dimension 3. The author describes explicitly the algorithm for jumping between the minimal models of \(\text{Spec}R\). The process is called the Homological MMP.

The first result, proved by iteration, is that \(\bigcup_{i\in I}C_i\) contracts to a point without contracting a divisor if and only if \(\dim_{\mathbb C}\Lambda_I<\infty.\)

The next step is to choose a subset of curves \(\{C_i\mid i\in I\}\) for an index set \(I\). Each curve \(C_i\) correspond to an indecomposable summand \(N_i\) of an \(R\)-module \(N\), and the tilting algebra is replaced with its mutated algebra \(\nu_I\Lambda:=\text{End}_R(\nu_i N).\) The mutation handles the cases with loops, 2-cycles, and no superpotential (saying that it can be generalized to singular minimal models). When \(C_i\) flops, denote the flop by \(X^+\).

The second main result states that: (1) The irreducible curve \(C_i\) flops if and only if \(\nu_i N\neq N\). (2) If \(\Gamma\) denotes the natural algebra associated to the flop \(X^+\), then \(\Gamma\cong\nu_i\Gamma.\) (3) If \(X\rightarrow\text{Spec}R\) is a minimal model, and \(\dim_{\mathbb C}\Lambda_i<\infty\), then \(\Phi_i\cong\Psi_{X^+}\circ\mathsf{Flop}\circ\Psi^{-1}_X\) where \(\mathsf{Flop}\) is the inverse of the flop functor of Bridgeland-Chen. One sees that the first part of the result gives a lower bound on the number of minimal models, while the second part allows to iterate.

The above describes the theory in the article of independent interest. In addition the article includes applications. For the applications to GIT, let \(\Gamma=\text{End}_R(N)\) be any algebra with \(\Gamma\in\text{CM}R.\) The dimension vector given by the ranks of the summands of \(N\) are denoted \(\mathsf{rk}\). When \(N\) is a generator, the GIT chamber decomposition \(\Theta(\Gamma)\) associated to \(\Gamma\) has coordinates \(\vartheta_i\) for \(i\neq 0\), \(\vartheta_0\) corresponding to the summand \(R\) of \(N\). \(C_+(\Gamma)=\{\vartheta\in\Theta(\Gamma)\mid\vartheta_i>0\text{ for all }i>0\},\) and \(\mathcal M_{\mathsf{rk},\phi}(\Lambda)\) denotes the moduli space of \(\phi\)-semistable \(\Lambda\)-modules of dimension vector \(\mathsf{rk}\). This review cannot include all applications, but an example is the first corollary: Choose \(C_i\) and suppose \(\dim_{\mathbb C}\Lambda_i<\infty\) (so that \(C_i\) flops). Then (1) \(\mathcal M_{\mathsf{rk},\phi}(\Lambda)\cong X\) for all \(\phi\in C_+(\Lambda)\). (2) \(\mathcal M_{\mathsf{rk},\phi}(\nu_i\Lambda)\cong X^+\) for all \(\phi\in C_+(\nu_i\Lambda)\). Thus one can view the flop as changing the algebra, but keeping the GIT chamber structure fixed. Mutation induces a derived equivalence, and then it is possible to track the moduli space back across the equivalence to obtain the flop as a moduli space on the original algebra. Thus a second corollary to the theory is a moduli-tracking theorem.

I would like to give one more of the corollaries given in the introduction, to illustrate the strength of the theory. It states verbatim that there exists a connected path in the GIT chamber decomposition of \(\Lambda\) where every minimal model of \(\text{Spec}R\) can be found, and each wall crossing in this path corresponds to the flop of a singular curve.

The other main application of the theory is a generalized Auslander-McKay correspondence. This lifts the Auslander-McKay correspondence from dimension 2 to 3-fold compound du Val singularities, and states that when \(R\) is a complete local cDV singularity there is a one-to-one correspondence from the set of basic maximal modification \(R\)-module generators to the set of minimal models \(f_i:X_i\rightarrow\text{Spec}R\). The article includes a study on the properties of this extended correspondence.

It is fair to remark that the search for the more ample bundles includes several areas of noncommutative geometry, including noncommutative deformations. Most of these theories are recalled and explored in an explicit way in the article, and the results is as always from this author very nice and important, leading to new questions and conjectures listed at the end of the article.

Reviewer: Arvid Siqveland (Kongsberg)

##### MSC:

14A22 | Noncommutative algebraic geometry |

14E30 | Minimal model program (Mori theory, extremal rays) |

##### Keywords:

birational geometry; minimal model; flops; flopping; tilting bundle; GIT chamber decomposition; crepant projective birational morphism; mutation; mutation functor; wall crossing; Auslander-McKay correspondence; homological MMP; derived equivalence; compound du Val singularities; maximal modification module; MM module
Full Text:
DOI

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