Stability conditions for contraction algebras

Abstract

This paper gives a description of the full space of Bridgeland stability conditions on the bounded derived category of a contraction algebra associated to a $3$ -fold flop. The main result is that the stability manifold is the universal cover of a naturally associated hyperplane arrangement, which is known to be simplicial and in special cases is an ADE root system. There are four main corollaries: (1) a short proof of the faithfulness of pure braid group actions in both algebraic and geometric settings, the first that avoid normal forms; (2) a classification of tilting complexes in the derived category of a contraction algebra; (3) contractibility of the stability space associated to the flop; and (4) a new proof of the $K(\unicode{x3c0} \,,1)$ -theorem in various finite settings, which includes ADE braid groups.

1 Introduction

This paper provides a description of the entire stability manifold for a large class of finite-dimensional algebras, namely contraction algebras associated to $3$ -fold flops, in terms of their g-vector fans. To establish this purely algebraic result, which takes place in the derived category of a finite-dimensional algebra, requires both algebraic techniques through silting-discreteness and also, more remarkably, geometric techniques through Fourier–Mukai transforms. The main corollaries are algebraic, topological and geometric.

1.1 Change of categories

In birational geometry, $3$ -fold flops play a vital role in the minimal model programme, providing a method to pass between the different minimal models of a given $3$ -fold singularity. Contraction algebras [DW1, DW3] are an uncountable class of symmetric, finite-dimensional algebras introduced as a tool to study this geometry. To each $3$ -fold flopping contraction $f\colon X\to \operatorname {Spec} R$ , where X has at worst Gorenstein terminal singularities, there is an associated contraction algebra $\Lambda _{\mathrm {con}}$ , which recovers all known numerical invariants of the flop [DW1, HT, W]. A common theme to which this paper contributes is that results for contraction algebras imply results for $3$ -fold flops.

To explain this connection, note that associated to f is the subcategory

of $\mathop {\mathrm {D}^b}(\operatorname {coh} X)$ . This subcategory is small enough in the sense that it has desirable properties such as finite-dimensional Hom-spaces and the 3-Calabi–Yau property whenever X is smooth, but it is large enough that many problems involving X, some of which are introduced in Section 1.3 below, can be solved by replacing $\mathop {\mathrm {D}^b}(\operatorname {coh} X)$ with

.

The third (and easiest) category associated to this situation is the bounded derived category $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ . This is very different from , but the transfer between them is provided by noncommutative deformation theory, which gives a universal sheaf [DW3] and a functor

(1.1)

whose image lies in . The main idea is that establishing results on $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is in fact easier, and this in turn establishes results on . To do this, we use the functor given by equation (1.1) twice: first to bring Fourier–Mukai transforms into algebra to prove the main result on $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ and then to transfer the consequences back to , and thus also X, in a series of corollaries.

1.2 Main result

The change in category from to $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ brings one major advantage, namely that $\Lambda _{\mathrm {con}}$ is silting-discrete [Au1, 4.12]. This fact has two happy consequences. First, we are able to describe the full space of stability conditions on $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , not just a component. Second, it is known that all silting-discrete algebras have contractible stability manifolds [PSZ], and thus $\operatorname {Stab}{}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is contractible before we even begin.

Alas, the change in category also brings one major disadvantage. Whilst contractibility comes for free, moving to finite-dimensional algebras means that we lose the technology of Fourier–Mukai transforms, so controlling standard equivalences becomes significantly more difficult. An age-old algebraic problem rears its head: knowing whether a given autoequivalence that is the identity on simples is globally the identity functor. Happily, in our setting, we tame this problem by appealing to a commutative diagram that uses equation (1.1) to intertwine our algebraic equivalences with the geometric flop functors, where we can use a standard Fourier–Mukai argument (see Theorem 4.3).

The tilting theory of $\Lambda _{\mathrm {con}}$ is controlled by a real simplicial hyperplane arrangement [Au2], which can be described in many ways. On one hand, arises as the g-vector fan of $\Lambda _{\mathrm {con}}$ , and on the other, it arises naturally as the ample cone of $f\colon X\to \operatorname {Spec} R$ . Our main result is the following.

Theorem 1.1 (4.10).

If $\Lambda _{\mathrm {con}}$ is the contraction algebra of f, then the natural map

is a regular cover. Furthermore, $\operatorname {Stab}{}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is contractible, so this cover is universal.

The chambers of can be labelled by the contraction algebras of flopping contractions reached from f by iterated flops, and each path $\unicode{x3b1} $ in the skeleton graph of can be assigned a standard derived equivalence $F_{\unicode{x3b1} }$ . With this notation, the Galois group for the cover in Theorem 1.1 is the image of a pure braid group under the group homomorphism sending $\unicode{x3b1} \mapsto F_{\unicode{x3b1} }$ .

1.3 Corollaries

The universality in Theorem 1.1 allows us to both extract new results and simplify others in the literature. The first consequence is the following, which gives a short proof of [Au2, 1.4].

Corollary 1.2 (5.2).

The homomorphism sending $\unicode{x3b1} \mapsto F_{\unicode{x3b1} }$ is injective.

Combining the topological Corollary 1.2 with the algebraic-geometric Theorem 4.3 allows us to side-step algebra automorphism issues and fully classify one-sided tilting complexes in $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ . The following is a strengthening of [Au2, 7.2].

Corollary 1.3 (5.3).

There is a bijection between morphisms in the Deligne groupoid of ending at the vertex associated to $\Lambda _{\mathrm {con}}$ and the set of isomorphism classes of basic one-sided tilting complexes of $\Lambda _{\mathrm {con}}$ .

A more surprising consequence of the universality in Theorem 1.1 is the faithfulness of the geometric action from [DW3]. The logic is straightforward: given any element in the kernel of the geometric action, there is an element in the kernel of the algebraic action, and we can appeal directly to Corollary 1.2. As such, the following recovers the main result of the paper [HW1].

Corollary 1.4 (5.5).

The homomorphism , which sends $\unicode{x3b1} $ to the corresponding composition of Bridgeland-Chen flop functors, is injective.

We can further extend this to establish results on the stability manifold of the associated subcategory

. It was recently established in [HW2] that there is a component of the space of stability conditions,

and a regular covering map

with Galois group equal to the image of

. Given the faithfulness of this action from Corollary 1.4, which is implied from the silting-discrete contractibility theorem, we are also able to deduce that

is contractible. The logic again is straightforward: universal covers are unique. The proof of the following corollary is the only part of the paper where we use any prior results about

.

Corollary 1.5 (5.6).

is contractible.

The original proof of Corollary 1.5 in [HW2] relies on knowing the universal cover of is contractible: in other words, knowing the $K(\unicode{x3c0} \,,1)$ -conjecture holds for (or, more precisely, its associated braid group). Since the arrangement associated to f can be obtained from the root system of an ADE Dynkin diagram by intersecting the reflecting hyperplanes with a certain subspace, it is always simplicial, and hence it is well-known that the $K(\unicode{x3c0} \,,1)$ -conjecture holds in this case [D1]. However, our approach via silting-discreteness allows us to bypass this result and actually reprove it without using [B3, BT] or normal forms. For example, if $\Lambda _{\mathrm {con}}$ is a contraction algebra with associated being an ADE root system, then , and it follows from Theorem 1.1 that the composition

$$\begin{align*}\operatorname{Stab}\mathop{\mathrm{D}^b}(\Lambda_{\mathrm{con}})\to \mathfrak{h}_{\mathrm{reg}}\to \mathfrak{h}_{\mathrm{reg}}/W \end{align*}$$

is also a covering map.

Corollary 1.6 (5.1).

The $K(\unicode{x3c0} \,,1)$ -conjecture holds for all ADE braid groups.

We recall the $K(\unicode{x3c0} \,,1)$ -conjecture in Section 5 and remark that we actually reprove it for all intersection arrangements obtained from an ADE root system by intersecting the reflecting hyperplanes with a subspace generated by a choice of the coordinate vectors. This class of intersection arrangements is a bit eclectic: as a consequence, we prove $K(\unicode{x3c0} \,,1)$ for the Coxeter groups $I_n$ for $n=3,4,5,6,8$ but none of the other n. The paper [IW2] describes in more detail the types of arrangements that can arise; see Remark 2.1.

1.4 Conventions

Throughout, we work over the field of complex numbers. Given a noetherian ring A, modules will be right modules unless specified, and $\operatorname {mod} A$ denotes the category of finitely generated A-modules. We use the functional convention for composing arrows, so $f\circ g$ means g then f. With this convention, given a ring R, an R-module M is a left $\operatorname {End}_R(M)$ -module. Furthermore, $\operatorname {Hom}_R(M,N)$ is a right $\operatorname {End}_R(M)$ -module and a left $\operatorname {End}_R(N)$ -module, in fact a bimodule.

2 Wall crossing and functorial composition

Throughout this paper, $f\colon X\to \operatorname {Spec} R$ is a fixed $3$ -fold flopping contraction, where X has at worst Gorenstein terminal singularities and R is complete local. Necessarily, R is an isolated cDV singularity [R]. Associated to f is a rigid Cohen–Macaulay (=CM) module N, an algebra and a contraction algebra $\Lambda _{\mathrm {con}}$ . We first briefly review these notions, mainly to set notation.

2.1 Rigid modules

Since R is Gorenstein, recall that

For $M\in \operatorname {CM} R$ , we say that M is basic if there are no repetitions in its Krull–Schmidt decomposition into indecomposables. We call M a generator if one of these indecomposable summands is R, and we call M rigid if $\operatorname {Ext}^1_R(M,M)=0$ . We say $M\in \operatorname {CM} R$ is maximal rigid if it is rigid, and furthermore it is maximal with respect to this property (see [IW1, 4.1]).

By the Auslander–McKay correspondence for cDV singularities, there is a one-to-one correspondence between flopping contractions $Y\to \operatorname {Spec} R$ , up to R-isomorphism, and basic rigid generators in $\operatorname {CM} R$ [W, 4.13]. In particular, both sets are finite. For our fixed flopping contraction $f\colon X\to \operatorname {Spec} R$ , the corresponding basic rigid CM generator across the bijection will be denoted N throughout this paper. The contraction algebra of f may then be defined as the stable endomorphism algebra [DW1, DW3].

The set of basic rigid CM generators carries an operation called mutation. Indeed, given such an L, with indecomposable summand $L_i\ncong R$ , there is the so-called exchange sequence

(2.1) $$ \begin{align} 0\to L_i\to\bigoplus_{j\neq i}{L_j}^{\oplus b_{ij}} \to K_i\to 0 \end{align} $$

and

[W, (A.A)]. Given the summand $K_i$ of $\unicode{x3bd} _iL$ , we can mutate again and obtain $\unicode{x3bd} _i\unicode{x3bd} _iL$ . It is a general fact for isolated cDV singularities that $\unicode{x3bd} _i\unicode{x3bd} _iL\cong L$ [W, 4.20(1)], and moreover in the second exchange sequence

$$\begin{align*}0\to K_i\to\bigoplus_{j\neq i}{L_j}^{\oplus c_{ij}} \to L_i\to 0 \end{align*}$$

we have $b_{ij}=c_{ij}$ for all $i,j$ [W, 5.22]. Furthermore, there are equivalences of derived categories

We will abuse notation and notate both as

and refer to them as the mutation functors.

Given our fixed basic rigid CM generator N corresponding to f, write $\operatorname {Mut}_0(N)$ for the set of basic rigid CM generators that can be obtained from N by iteratively mutating at indecomposable summands, noting that, in this paper, we will never mutate at the summand R. Geometrically, across the Auslander–McKay correspondence, this corresponds to all R-schemes that can be obtained from X by iteratively flopping curves.

2.2 Hyperplanes and labels

We fix a decomposition $N=R\oplus N_1\oplus \ldots \oplus N_n$ and will often implicitly declare $N_0=R$ . For every $L\in \operatorname {Mut}_0(N)$ and each indecomposable summand $L_i$ of L, there is an exact sequence

(2.2) $$ \begin{align} 0\to \bigoplus_{j=0}^n N_j^{\oplus a_{ij}}\to \bigoplus_{j=0}^n N_j^{\oplus b_{ij}}\to L_i\to 0 , \end{align} $$

where the first and second terms do not share any indecomposable summands. In the setting where $N\in \operatorname {CM} R$ is maximal rigid, this fact is very well known; in the setting here for rigid generators, we rely instead on [IW2, 9.29, 5.6]. In any case, consider the cone

It is clear that

, which we denote throughout by $C_+$ . Furthermore, the chambers $C_L$ , as L varies over $\operatorname {Mut}_0(N)$ , sweep out the chambers of a simplicial hyperplane arrangement

[HW1, IW2].

Remark 2.1. The above arrangement has several equivalent descriptions; it can be obtained as the g-vector fan of $\Lambda _{\mathrm {con}}$ [Au2, 7.1] or as the moveable cone of the flopping contraction f [HW1, 4.6(3)] (where, as described in the following remark, both of these results are extended to the rigid setting using [IW2, 9.29]). Using this last description, HomMMP [W, 5.24, 5.25] gives the description of via intersecting a subspace of an ADE root system with the reflecting hyperplanes. In other words, all the above are intersection arrangements of ADE root systems and, as such, are both finite and simplicial.

Remark 2.2. The three key properties appearing here and later are the following:

1. 1. $\unicode{x3bd} _i\unicode{x3bd} _iL \cong L$ for any basic rigid CM generator.

2. 2. Moreover, $b_{ij}=c_{ij}$ in the corresponding exchange sequences.

3. 3. Every $L\in \operatorname {Mut}_0(N)$ has a two-term approximation by N, as in equation (2.2), ensuring that $\operatorname {Hom}_R(L,N)$ is a tilting bimodule for $\Lambda $ .

The first two follow as a consequence of HomMMP [W, 4.20(1), 5.22], since flopping is an involution. In the maximal rigid setting, (3) is known, but to prove (3) in the general rigid setting requires a small part of [IW2]. Once (1)–(3) hold, [Au2, 4.9(2)] extends all the results of [HW1, Au2] to the setting of rigid modules, and this is what we use below. A reader wishing only to work in the maximal rigid setting or in the smooth setting may safely disregard references to [IW2].

As explained by the general Coxeter-style labelling of walls and chambers in [IW2, §9.7], the fixed decomposition $N=R\oplus N_1\oplus \ldots \oplus N_n$ induces an ordering on the summands of all other elements L of $\operatorname {Mut}_0(N)$ such that each local wall crossing has a label $s_i$ and corresponds to mutation at the ith summand. In this way, there is a compatible labelling on the edges of the 1-skeleton of . We illustrate this in an example.

Example 2.3. There exists a two-curve flop over a $cD_4$ singularity with the simplicial hyperplane arrangement shown in Figure 1. Each local wall crossing is labelled by $s_i$ for $i=1,2$ , and under crossing wall $s_i$ the ith summand is mutated. For clarity, in each chamber, we have not written the zeroth summand R. Note that $C_N=C_+$ is the top-right chamber.

Figure 1 Exchange graph and $1$ -skeleton for a certain two-curve $cD_4$ singularity.

Fixing an ordering of the summands of N not only fixes an ordering of the projective modules of $\Lambda =\operatorname {End}_R(N)$ and, via the pairing between simples and projectives, an ordering of its simples , but also fixes an ordering on the projectives and simples of $\operatorname {End}_R(L)$ for all $L\in \operatorname {Mut}_0(N)$ .

To fix notation, for $L\in \operatorname {Mut}_0(N)$ , suppose that the induced ordering on the summands of L is $L=L_0\oplus L_1\oplus \ldots \oplus L_n$ , where $L_0= R$ . There is an induced ordering on the projectives of $\operatorname {End}_R(L)$ and again via the pairing between simples and projectives, an ordering of its simples . The simples for the contraction algebra $\operatorname {\underline {End}}_R(L)$ are .

2.3 Standard equivalences for simple wall crossings

Let $L\in \operatorname {Mut}_0(N)$ , with associated and contraction algebra . To ease notation, suppose that , with contraction algebra $\mathrm {B}_{\mathrm {con}}$ , and consider the good truncation

(2.3)

which is a complex of $\mathrm {B}_{\mathrm {con}}$ - $\mathrm {A}_{\mathrm {con}}$ bimodules. In this basic rigid setting, it is already known that the complex is a two-sided tilting complex [Au2, 3.3].

Theorem 2.4. Suppose that $L,M \in \operatorname {Mut}_0(N)$ , which correspond to the flopping contractions $X_L\to \operatorname {Spec} R$ and $X_M\to \operatorname {Spec} R$ , say. Then the following are equivalent:

1. 1. $C_L$ and $C_M$ share a codimension one wall in .

2. 2. $M\cong \unicode{x3bd} _iL$ for some $i\neq 0$ .

3. 3. $X_L$ and $X_M$ are related by a flop at a single irreducible curve.

In this case, set

,

and

,

. Then the mutation operations intertwine via the following commutative diagram

where

is the mutation functor, $\operatorname {\mathsf {Flop}}_i$ is the inverse of the Bridgeland–Chen flop functor [B1, C],

and

are the standard projective generators of zero perverse sheaves [VdB] and $\operatorname {res}$ is restriction of scalars induced from the ring homomorphisms $\mathrm {A}\to \mathrm {A}_{\mathrm {con}}$ and $\mathrm {B}\to \mathrm {B}_{\mathrm {con}}$ .

Proof. The statement (1) $\Leftrightarrow $ (2) is clear; see also [HW2, 3.6]. The statement (2) $\Leftrightarrow $ (3) is [W, 4.13], where since we are mutating only at indecomposable summands and R is isolated, [W, 4.20(1)] overrides the caveat in the latter part of [W, 4.13]. The left-hand diagram commutes by [Au2, 1.1], and the right-hand diagram commutes by [W, 4.2].

2.4 K-theory

Since $\Lambda _{\mathrm {con}}$ is a finite-dimensional algebra, consider the Grothendieck group

which is well-known to be a free abelian group with basis given by the ordered simples

. The notation $\mathsf {G}_0$ is chosen since we are including in the Grothendieck group all modules, whereas $K_0$ often only deals with vector bundles and thus projectives.

Continuing the notation from Theorem 2.4, for any $L\in \operatorname {Mut}_0(N)$ , consider its associated contraction algebra $\mathrm {A}_{\mathrm {con}}=\operatorname {\underline {End}}_R(L)$ and its mutations

. Abusing notation slightly, consider the standard equivalences

where the functor from right to left is induced by the mutation $\unicode{x3bd} _i L \to \unicode{x3bd} _i \unicode{x3bd} _i L \cong L$ . These and their inverses induce the following four isomorphisms on K-theory:

(2.4)

As in Subsection 2.2, write for the ordered simples of $\mathrm {A}_{\mathrm {con}}$ . For lack of suitable alternatives, also write for the correspondingly ordered simples in $\unicode{x3bd} _i\mathrm {A}_{\mathrm {con}}$ .

Lemma 2.5. With the notation in equation (2.1), $\mathsf {F}_i\colon \mathsf {G}_0(\mathrm {A}_{\mathrm {con}})\to \mathsf {G}_0(\unicode{x3bd} _i\mathrm {A}_{\mathrm {con}})$ sends

(2.5)

Proof. The fact that is [W, 4.15]. Theorem 2.4 then implies that maps, under restriction of scalars, to . It follows that in $\mathop {\mathrm {D}^b}(\mathrm {A}_{\mathrm {con}})$ ; see, for example, [Au2, 6.6]. This establishes the top row.

For the second row, applying $\operatorname {Hom}_R(L,-)$ to equation (2.1) and using the rigidity of L gives an exact sequence

Applying

to this, with $t\neq i$ , yields

Further, it is clear that

is zero if $j\neq t$ and equals $\mathbb {C}$ if $j=t$ . Combining, we see that

is a module, filtered by $b_{it}$ copies of

and one copy of

. Again, the left-hand side of the commutative diagram in Theorem 2.4 then shows that

must also be a module, filtered by $b_{it}$ copies of

and one copy of

. The second row follows.

Remark 2.6. Applying the above to $L=N$ , basing $\mathsf {G}_0(\Lambda _{\mathrm {con}})$ by the ordered simples

, the above transformation given by equation (2.5) assembles into an $n\times n$ matrix, with coefficients in $\mathbb {Z}$ , representing the map $\mathsf {F}_i\colon \mathbb {Z}^n\to \mathbb {Z}^n$ . As is standard in linear algebra, the dual map

where $\mathsf {G}_0(\Lambda _{\mathrm {con}})^*$ has dual basis $\textbf {e}_1,\ldots ,\textbf {e}_n$ , say, is given by the transpose matrix. But the transpose is precisely the transformation

$$\begin{align*}\textbf{e}_t\mapsto \left\{ \begin{array}{cl} \textbf{e}_t& \mbox{if } t\neq i,\\ -\textbf{e}_i+\sum_{j\neq i}b_{ij}\textbf{e}_j & \mbox{if } t=i, \end{array} \right. \end{align*}$$

which is precisely the transformation

seen in moduli tracking [W, §5] or in K-theory of projectives in [HW2, 3.2]. In particular, it will be convenient to think of the dual basis $\textbf {e}_1,\ldots ,\textbf {e}_n$ as being the basis

of

, where $\operatorname {per}\Lambda $ is the full subcategory of $\mathop {\mathrm {D}^b}(\Lambda )$ consisting of perfect complexes. Then the transformation $\mathsf {F}_i^*$ can be identified with these transformations elsewhere in the literature, and we can use those results freely. Note that the hyperplane arrangement

from Subsection 2.2 is defined in terms of $\textbf {e}_i$ and so naturally lives in $\mathsf {G}_0(\Lambda _{\mathrm {con}})^*$ .

By the above remark, the proof of the following does follow as the dual of [HW2, 3.2]. It is, however, instructive to give a direct proof.

Lemma 2.7. All four isomorphisms in equation (2.4) are given by the same matrix, namely the one from equation (2.5), and this matrix squares to the identity.

Proof. By equation (2.5), the matrices are controlled by the numbers $b_{ij}$ appearing in the relevant exchange sequences. Say the top $\mathsf {F}_i$ is controlled by numbers $b_{ij}$ , and the bottom $\mathsf {F}_i$ is controlled by numbers $c_{ij}$ . That the two matrices labelled $\mathsf {F}_i$ are the same is simply the statement that $b_{ij}=c_{ij}$ , which has already been explained in Subsection 2.1. Given this fact that $b_{ij}=c_{ij}$ , we see that $\mathsf {F}_i\mathsf {F}_i=\mathrm {Id}$ by simply observing

which is clearly the identity. Applying $\mathsf {F}_i^{-1}$ to each side of the equation $\mathsf {F}_i\mathsf {F}_i=\mathrm {Id}$ gives $\mathsf {F}_i^{-1}=\mathsf {F}_i$ , and all statements follow.

2.5 Groupoids

As in Subsection 2.2, associated to every contraction algebra is a hyperplane arrangement . As is standard, there is an associated graph defined as follows.

Definition 2.8. The vertices of are the chambers – that is, the connected components – of . There is a unique arrow $a \colon v_1\to v_2$ from chamber $v_1$ to chamber $v_2$ if the chambers are adjacent; otherwise there is no arrow.

By definition, if there is an arrow $a \colon v_1\to v_2$ , then there is a unique arrow $b\colon v_2\to v_1$ with the opposite direction of a. For an arrow $a\colon v_1\to v_2$ , set and .

A positive path of length n in

is a formal symbol

$$\begin{align*}p=a_n\circ \ldots\circ a_2\circ a_1 \end{align*}$$

whenever there exists a sequence of vertices $v_0,\ldots ,v_n$ of

and arrows $a_i\colon v_{i-1}\to v_i$ in

. Set

,

,

, and write $p\colon s(p)\to t(p)$ . If $q=b_m\circ \ldots \circ b_2 \circ b_1$ is another positive path with $t(p)=s(q)$ , we consider the formal symbol

and call it the composition of p and q.

Definition 2.9. A positive path $\unicode{x3b1} $ is called minimal if there is no shorter positive path in with the same start and end points as $\unicode{x3b1} $ .

Following [D2, p7], let $\sim $ denote the smallest equivalence relation, compatible with morphism composition, that identifies all morphisms that arise as positive minimal paths with the same source and target. Then consider the free category

on the graph

, whose morphisms correspond to directed paths, and its quotient category

called the category of positive paths.

Definition 2.10. The Deligne groupoid is the groupoid defined as the groupoid completion of : that is, a formal inverse is added for every morphism in .

It is a very well known fact [D1, P1, P3, S] (see also [P2, 2.1]) that for any vertex , there is an isomorphism , where . This fact is general and is significantly weaker and easier to establish than statements involving $K(\unicode{x3c0} \,,1)$ . We will use this fact implicitly throughout.

2.6 Composition and K-theory

For any

, say $\unicode{x3b1} ={s_{i_t}}\circ \ldots \circ {s_{i_1}}$ , consider

The following is known and easy to establish just using the tilting order, in the case when the modules are maximal rigid (e.g., if X is smooth). In our more general situation of rigid generators, the same proof works, but it relies on some recent advances in [IW2].

Proposition 2.11. Let $\unicode{x3b1} \colon C_L\to C_M$ be a positive minimal path. Set , , and . Then the following hold:

1. 1. is functorially isomorphic to $\operatorname {{\textbf {R}}Hom}_{\mathrm {A}}(\operatorname {Hom}_R(L,M),-)$ .

2. 2. $F_{\unicode{x3b1} }$ is functorially isomorphic to , where is the two-sided tilting complex $ \unicode{x3c4} _{\scriptscriptstyle { \geq -1}} ( \mathrm {B}_{\mathrm {con}} \otimes ^{ \mathrm{\mathbf{L}}}_{\mathrm {B}} \operatorname {Hom}_R(L,M) \otimes ^{ \mathrm{\mathbf{L}}}_{\mathrm {A}} \mathrm {A}_{\mathrm {con}})$ .

In particular, all positive minimal paths with the same start and end points are functorially isomorphic.

Proof. (1) When N is maximal rigid, this is precisely [Au2, 4.9(1)], [HW1, 4.6]. In the more general setting here with $L,M\in \operatorname {Mut}_0(N)$ , then certainly $L\in \operatorname {Mut}_0(M)$ . Since R is isolated cDV, it follows from the combinatorial and geometric description of mutation of rigid modules in [IW2, 9.25, 9.29] that $\operatorname {Hom}_R(L,M)$ is a tilting $\operatorname {End}_R(M)$ - $\operatorname {End}_R(L)$ -bimodule of projective dimension one when viewed as a right $\operatorname {End}_R(L)$ -module. This is the key technical condition, explained in [Au2, 4.9(2)] and not available when [HW1] was written, that now allows us to use the main result [HW1, 4.6] freely in the more general setting here.

(2) When N is maximal rigid, this is [Au2, 4.12]. The more general statement required here follows by (1) and, again, [Au2, 4.9(2)], which asserts, given (1), that we are able to apply the main result [Au2, 4.12] to the setting of rigid objects.

For any two positive minimal paths $\unicode{x3b1} $ and $\unicode{x3b2} $ with the same start and end points, the above proposition shows that there is a functorial isomorphism $F_{\unicode{x3b1} }\cong F_{\unicode{x3b2} }$ . Hence the association $\unicode{x3b1} \mapsto F_{\unicode{x3b1} }$ descends to a functor from . Since $F_{\unicode{x3b1} }$ is already an equivalence, this in turn formally descends to a functor from . Using the same logic, the assignment also descends to a functor from and then a functor from .

Furthermore, for every

, using the notation from the second sentence in Proposition 2.11, the following diagram commutes

(2.6)

just by composition, since by [Au2, 1.1], respectively [Au2, 3.2], both of the following commute:

Later, the following is one of the crucial ingredients in establishing that $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is a covering space. Recall that for the standard derived equivalence $F_{\unicode{x3b1} }$ associated to a path $\unicode{x3b1} $ , the induced map on the K-theory is denoted $\mathsf {F}_{\unicode{x3b1} }$ .

Proposition 2.12. Suppose that $\unicode{x3b2} \colon C\to D$ is a positive minimal path in .

1. 1. If $\unicode{x3b1} \colon C\to D$ is any positive path, then $\mathsf {F}_{\unicode{x3b1} }=\mathsf {F}_{\unicode{x3b2} }$ .

2. 2. If , then $\mathsf {F}_{\unicode{x3b1} }=\mathrm {Id}$ .

3. 3. If , then $\mathsf {F}_{\unicode{x3b1} }=\mathsf {F}_{\unicode{x3b3} }$ .

Proof. Having established Lemma 2.7, this is now word-for-word identical to [HW2, 4.8]. Note that this proof is elementary and does not require Deligne normal form.

2.7 The dual composition

For each $L \in \operatorname {Mut}_0(N)$ , consider

and recall that

. Choose a positive minimal path $\unicode{x3b2} \colon C_L \to C_+$ , which in turn gives rise to a derived equivalence

This derived equivalence is independent of the choice of positive minimal path, by Proposition 2.11. It induces an isomorphism $\mathsf {K}_0(\operatorname {per}\Lambda _L)\to \mathsf {K}_0(\operatorname {per}\Lambda )$ on the K-theory of perfect complexes, so write

in $\mathsf {K}_0(\operatorname {per}\Lambda )\cong \mathbb {Z}^{n+1}$ , where

and

. Since $\unicode{x3b1} $ is a sequence of mutations that do not involve mutating the zeroth summand, the zeroth summand is fixed at each stage. Hence this isomorphism descends to an isomorphism

Basing the first by

and the second by

, the matrix representing the isomorphism is

for $1\leq i,j\leq n$ . By Remark 2.6, later we will think of these bases as $\textbf {e}^{\prime }_1,\ldots ,\textbf {e}^{\prime }_n$ of $\mathsf {G}_0(\operatorname {\underline {End}}_R(L))^*$ and $\textbf {e}_1,\ldots ,\textbf {e}_n$ of $\mathsf {G}_0(\Lambda _{\mathrm {con}})^*$ , respectively.

Remark 2.13. The above description of is in terms of projectives of the ambient $\operatorname {End}_R(L)$ since this is convenient later. There is, however, a much more intrinsic description of that does not rely on this larger algebra, via the two-term tilting complexes of the contraction algebra $\Lambda _{\mathrm {con}}$ . In particular, in the language of g-vectors, , where $g^{L_i}$ is the g-vector of the two-term complex of $\Lambda _{\mathrm {con}}$ associated to the rigid object $L_i$ via the bijection [Au1, 2.18]. We do not use this description below.

3 Stability and t-structures

3.1 Stability generalities

Throughout this subsection, denotes a triangulated category whose Grothendieck group is a finitely generated free $\mathbb {Z}$ -module.

Proposition 3.1 [B2, 5.3].

To give a stability condition on

is equivalent to giving a bounded t-structure

with heart

and a group homomorphism

, called the central charge, such that for all

, the complex number $Z(E)$ lies in the semi-closed upper half-plane

and where furthermore Z must satisfy the Harder–Narasimhan property.

Write for the set of locally-finite stability conditions on . We do not define these here, as below this condition is automatic for all stability conditions on $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , since all hearts of bounded t-structures will be equivalent to finite-dimensional modules on some finite-dimensional algebra.

Theorem 3.2 [B2, 1.2].

The space

has the structure of a complex manifold, and the forgetful map

is a local isomorphism onto an open subspace of

.

Any triangle equivalence

induces a natural map

defined by

, where $\unicode{x3c6} ^{-1}$ is the corresponding isomorphism on K-theory

induced by the functor

. In this way, the group

of isomorphism classes of autoequivalences of

acts on

.

3.2 t-structures for $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$

The contraction algebra $\Lambda _{\mathrm {con}}$ is a silting-discrete symmetric algebra [Au1, 3.3, 4.12]. Being symmetric, the technical condition of being silting-discrete is equivalent [AM, 2.11] to there being only finitely many basic tilting complexes between P and $P[1]$ (with respect to the silting order $\leq $ ), for every tilting complex P obtained by iterated irreducible left mutation from the free module $\Lambda _{\mathrm {con}}$ . Geometrically, for each such P, this set is finite since it is in bijection with R-schemes obtained by iterated flops of irreducible curves starting from X, which is well-known to be finite.

This fact has the following remarkable consequence.

Proposition 3.3. Suppose that

is the heart of a bounded t-structure on $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ . Then

for some $L\in \operatorname {Mut}_0(M)$ and for some

, where

and $F_{\unicode{x3b1} }$ is the derived equivalence from Subsection 2.6 associated to $\unicode{x3b1} $ .

Proof. Since $\Lambda _{\mathrm {con}}$ is silting-discrete, necessarily has finite length [PSZ]. Furthermore, by the bijections in [KY, §5], there exists a silting complex T in $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ such that, in the notation of [KY, §5.4],

(3.1)

Since $\Lambda _{\mathrm {con}}$ is symmetric, silting equals tilting, so T is a tilting complex. It is already known (see [Au1, 2.12]) that every tilting complex T in $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ can be obtained as a composition of mutations from $\Lambda _{\mathrm {con}}$ , so say $T\cong \unicode{x3bc} _{\unicode{x3b2} }\Lambda _{\mathrm {con}}$ for some . Set ; then [Au1, 3.10(1)] gives $F_{\unicode{x3b2} }(\unicode{x3bc} _{\unicode{x3b2} }\Lambda _{\mathrm {con}})\cong \mathrm {B}_{\mathrm {con}}$ , and hence $F_{\unicode{x3b2} }(T)\cong \mathrm {B}_{\mathrm {con}}$ .

Thus, applying $F_{\unicode{x3b2} }$ to equation (3.1),

so applying $F_{\unicode{x3b2} }^{-1}=F_{\unicode{x3b2} ^{-1}}$ shows that

. Since

, the result follows.

Recall that inside $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ are the simples , which base the K-theory $\mathsf {G}_0(\Lambda _{\mathrm {con}})$ . In a similar way, the simple modules of $\operatorname {\underline {End}}_R(L)$ base its Grothendieck group.

Corollary 3.4. If $\unicode{x3c3} \in \operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , then for some and some Z satisfying for all $i=1,\ldots ,n$ .

Proof. By Proposition 3.3, every abelian heart is of the form for some , and hence every point of is of the form . To be a stability condition is equivalent to the map sending all simples of to $\mathbb {H}$ . Since the simples of are of the form , it follows that is a stability condition precisely when Z satisfies for all i.

The general action of $\operatorname {Auteq}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ on $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ simplifies somewhat if we restrict to those standard equivalences given by

. The functorial assignment $\unicode{x3b1} \to F_{\unicode{x3b1} }$ defines a group homomorphism

and we set $\mathop {\mathsf {PBr}}\nolimits $ to be the image of this homomorphism. Then using Corollary 3.4 to describe the points of $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , the action of $F_{\unicode{x3b2} }\in \mathop {\mathsf {PBr}}\nolimits $ on $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is

(3.2)

since $\mathsf {F}_{\unicode{x3b2} }=\mathop {\mathrm {Id}}\nolimits $ by Proposition 2.12(2).

4 Stability on contraction algebras as a universal cover

In order to realise $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ as a universal cover, fix the isomorphism

$$\begin{align*}\operatorname{Hom}_{\mathbb{C}}(\mathsf{G}_0(\Lambda_{\mathrm{con}}),\mathbb{C}) \xrightarrow{\sim}\mathbb{C}^n \end{align*}$$

given by , where are the simples of $\Lambda _{\mathrm {con}}$ . Composing this with the forgetful map from Theorem 3.2, we thus obtain

(4.1) $$ \begin{align} p\colon \operatorname{Stab}\mathop{\mathrm{D}^b}(\Lambda_{\mathrm{con}})\to\operatorname{Hom}_{\mathbb{C}}(\mathsf{G}_0(\Lambda_{\mathrm{con}}),\mathbb{C}) \xrightarrow{\sim}\mathbb{C}^n. \end{align} $$

Combining with Corollary 3.4, p sends an arbitrary point to . In this section, we will show that p is a regular covering map onto its image. To do this, it will be convenient to also consider the stability manifolds of the other contraction algebras of R and to track information between them.

Lemma 4.1. For any

, the following diagram commutes

where the topmost vertical arrows are the forgetful maps, $\mathsf {F}_{\unicode{x3b1} }^{-1}$ is the image in K-theory of the inverse of the functor $F_{\unicode{x3b1} }$ defined in Subsection 2.6 and

is defined in Subsection 2.7. The right-hand vertical composition is p.

Proof. The top square commutes by definition of $(\mathsf {F}_{\unicode{x3b1} })_*$ . For the bottom square, by Proposition 2.12(3) applied to $\unicode{x3b1} ^{-1}$ , we have $\mathsf {F}_{\unicode{x3b1} }^{-1}=\mathsf {F}_{\unicode{x3b1} ^{-1}}=\mathsf {F}_{\unicode{x3b2} }$ , where $\unicode{x3b2} $ is a positive minimal path $C_+ \to C_L$ . Writing $\unicode{x3b2} =s_{i_t}\circ \ldots \circ s_{i_1}$ , then the middle map is the composition

(4.2)

By Remark 2.6, each step is just the tracking of the projectives basing

under the mutation functors. Hence equation (4.2) is precisely

(4.3)

Consider the path $\overline {\unicode{x3b2} }=s_{i_1}\circ \ldots \circ s_{i_t}\colon C_L\to C_+$ . Being the opposite path to $\unicode{x3b2} $ , it follows that $\overline {\unicode{x3b2} }$ is also positive minimal. But then, by Proposition 2.11, there is a functorial isomorphism

Hence

, the image of this functor in

, realises equation (4.3).

As is standard, consider the subset of $\mathbb {C}^n$

Corollary 4.2. For any point of $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , which is necessarily of the form

for some

,

Proof. The first statement is Corollary 3.4. By definition,

, hence

The final statement that

again follows from Corollary 3.4.

4.1 A covering map

As in Section 3.2, let $\mathop {\mathsf {PBr}}\nolimits $ be the image of the homomorphism

sending $\unicode{x3b2} \mapsto F_{\unicode{x3b2} }$ . Then $F_{\unicode{x3b2} }\in \mathop {\mathsf {PBr}}\nolimits $ acts on the space of stability conditions via the action in equation (3.2). In this subsection, we will establish that this action is free and properly discontinuous, so that $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}}) \to \operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})/\mathop {\mathsf {PBr}}\nolimits $ is a covering map.

The following is one of our main technical results. It establishes a condition under which elements of $\mathop {\mathsf {PBr}}\nolimits $ , inside the autoequivalence group of $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , are the identity. The proof is via Fourier–Mukai techniques. Forgetting the ambient geometry is thus a bad idea: it seems extremely difficult to establish the following result in a purely algebraic manner.

Theorem 4.3. Suppose that satisfies $F_{\unicode{x3b1} }(\Lambda _{\mathrm {con}}) \cong \Lambda _{\mathrm {con}}$ . Then there is a functorial isomorphism $F_{\unicode{x3b1} } \cong \mathop {\mathrm {Id}}\nolimits $ .

Proof. By the assumption, the standard equivalence $F_{\unicode{x3b1} }$ is induced by the one-sided tilting complex $\Lambda _{\mathrm {con}}$ . By the usual lifting argument (see, e.g., [RZ, 2.3]), the bimodule complex defining $F_{\unicode{x3b1} }$ must be isomorphic to ${}_{1}({\Lambda _{\mathrm {con}}})_{\unicode{x3b6} }$ as bimodules, for some algebra automorphism $\unicode{x3b6} \colon \Lambda _{\mathrm {con}}\to \Lambda _{\mathrm {con}}$ . Hence $F_{\unicode{x3b1} }$ is induced by this algebra automorphism.

Since $F_{\unicode{x3b1} }$ induces a Morita equivalence, it must take simples to simples. Furthermore, as $F_{\unicode{x3b1} }$ is the identity on K-theory $\mathsf {G}_0(\Lambda _{\mathrm {con}})$ by Proposition 2.12, $F_{\unicode{x3b1} }$ must fix all simples.

Now consider the commutative diagram

where the left-hand side is equation (2.6) and the right-hand side can be obtained by iterating the right-hand side of Theorem 2.4. Since $F_{\unicode{x3b1} }$ fixes simples, the left-hand commutative diagram implies that

fixes the simples

. By [VdB, 3.5.8], across the right-hand commutative diagram, this in turn implies that $\operatorname {\mathsf {Flop}}_{\unicode{x3b1} }$ fixes the sheaves

, where each $\mathrm {C}_i\cong \mathbb {P}^1$ . Since the flop functor and its inverse both map

to

[B1, (4.4)] and commute with the pushdown to $\operatorname {Spec} R$ (see, e.g., [DW1, 7.16]), by the standard Fourier–Mukai argument (see [W, 4.3], which itself is based on [T]), $\operatorname {\mathsf {Flop}}_{\unicode{x3b1} }\cong h_*$ for some isomorphism $h \colon X\to X$ that commutes with the pushdown. But this isomorphism is the identity on the dense open set obtained by removing the exceptional locus, and hence it must be the identity.

It follows that $\operatorname {\mathsf {Flop}}_{\unicode{x3b1} }\cong \mathop {\mathrm {Id}}\nolimits $ and hence

. Restricting the left-hand commutative diagram to $\operatorname {mod} \Lambda _{\mathrm {con}}$ , we obtain a commutative diagram

where $\operatorname {mod} _{\{1,\ldots ,n\}}\Lambda $ denotes those $\Lambda $ -modules with a finite filtration by the simples

. In particular, $F_{\unicode{x3b1} }$ restricted to $\operatorname {mod} \Lambda _{\mathrm {con}}$ is functorially isomorphic to the identity. Hence $\unicode{x3b6} $ is an inner automorphism (see, e.g., [L, 2.8.16]), which in turn implies that $F_{\unicode{x3b1} }\cong \mathop {\mathrm {Id}}\nolimits $ .

Corollary 4.4. For each $x\in \operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , there is an open neighbourhood of x such that for all $1\neq F_{\unicode{x3b2} }\in \mathop {\mathsf {PBr}}\nolimits $ .

Proof. Consider the open neighbourhood

of x defined by

where $d(-,-)$ is the metric on stability conditions introduced in [B2, Section 6]. Suppose that

for some $F_{\unicode{x3b2} }\in \mathop {\mathsf {PBr}}\nolimits $ . We will show that $F_{\unicode{x3b2} }\cong \mathop {\mathrm {Id}}\nolimits $ .

As the two open balls of radius $\frac {1}{4}$ intersect, every must satisfy $d\bigl (y,(F_{\unicode{x3b2} })_*y\bigr )<1$ . Furthermore, the central charges of y and $(F_{\unicode{x3b2} })_*y$ are equal by Proposition 2.12(2) and the top commutative diagram in Lemma 4.1. Using [B2, Lemma 6.4], it follows immediately that $y=(F_{\unicode{x3b2} })_*y$ , for every .

In particular, by Corollary 3.4, say for some . Since , the property $(F_{\unicode{x3b2} })_*(x)=x$ implies that , so $F_{\unicode{x3b2} }$ restricts to an equivalence . In turn, this implies that the composition

$$\begin{align*}F_{\unicode{x3b1}^{-1}\unicode{x3b2}\unicode{x3b1}}=F_{\unicode{x3b1}}^{-1}\circ F_{\unicode{x3b2}}\circ F_{\unicode{x3b1}}\colon\mathop{\mathrm{D}^b}(\Gamma_{\mathrm{con}})\to\mathop{\mathrm{D}^b}(\Gamma_{\mathrm{con}}), \end{align*}$$

where , restricts to an equivalence $\operatorname {mod} \Gamma _{\mathrm {con}}\to \operatorname {mod} \Gamma _{\mathrm {con}}$ . It follows that $F_{\unicode{x3b1} ^{-1}\unicode{x3b2} \unicode{x3b1} }(\Gamma _{\mathrm {con}})$ must then be a basic tilting module, given that $\Gamma _{\mathrm {con}}$ is. Since $\Gamma _{\mathrm {con}}$ is symmetric, the only such module is $\Gamma _{\mathrm {con}}$ , so $F_{\unicode{x3b1} ^{-1}\unicode{x3b2} \unicode{x3b1} }(\Gamma _{\mathrm {con}})\cong \Gamma _{\mathrm {con}}$ . By Proposition 4.3 applied to the contraction algebra $\Gamma _{\mathrm {con}}$ , we conclude that $F_{\unicode{x3b1} ^{-1}\unicode{x3b2} \unicode{x3b1} }\cong \mathop {\mathrm {Id}}\nolimits $ , and hence $F_{\unicode{x3b2} }\cong \mathop {\mathrm {Id}}\nolimits $ .

Corollary 4.5. The map $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}}) \to \operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})/\mathsf {PBr}$ is the universal covering map, with Galois group $\mathsf {PBr}$ .

Proof. $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is contractible since contraction algebras are silting-discrete [Au1, 4.12], and silting-discrete algebras have contractible stability manifolds [PSZ]. In particular, $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is path connected, so the given map is a regular covering map by Corollary 4.4 together with the standard [H, 1.40(a)(b)]. The covering is clearly universal, since $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is contractible and hence simply connected.

4.2 The regular cover to the complexified complement

In this subsection, we will establish that p induces an isomorphism

(4.4)

Combining with Corollary 4.5 will then prove that equation (4.1) is the universal covering map onto its image, with Galois group $\mathop {\mathsf {PBr}}\nolimits $ . We will establish equation (4.4) in two steps: first by showing that p has image , and then by establishing that equation (4.4) is well-defined and injective.

Our proof makes use of the following key combinatorial result, which is folklore when is an ADE root system. In our mildly more general setting here, the proof, which is basically the same, can be found in [HW2, Appendix A].

Proposition 4.6. With notation as above, the following hold:

1. 1. If $\unicode{x3b1} $ and $\unicode{x3b2} $ terminate at $C_+$ , then

   

2. 2. There is a disjoint union

   

The above combinatorics lead directly to the following. In the special case when is an ADE root system, the following result is already implicit in [B3] and [T].

Corollary 4.7. The image of p is .

Proof. By Proposition 4.6(2), all the sets avoid the complexified hyperplanes, so by Corollary 4.2, the image of p lies in . Further, given any , we may write for some $L \in \operatorname {Mut}_0(N)$ and some $h\in \mathbb {H}_+$ , again by Proposition 4.6(2). Since the standard heart in $\mathop {\mathrm {D}^b}(\operatorname {\underline {End}}_R(L))$ has finite length, we can find a stability condition $\unicode{x3c3} \in \operatorname {Stab} \mathop {\mathrm {D}^b}(\operatorname {\underline {End}}_R(L) )$ , which maps to h via the left-hand vertical composition in Lemma 4.1. Then for any , the commutative diagram in Lemma 4.1 shows that $(F_{\unicode{x3b1} })_*(\unicode{x3c3} )\in \operatorname {Stab} \mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}} )$ maps, under p, to z.

The following shows that equation (4.4) is both well-defined and injective.

Lemma 4.8. For $\unicode{x3c3} _1,\unicode{x3c3} _2\in \operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , then

Proof. Note that ( $\Leftarrow $ ) is clear since, by Proposition 2.12(2), the action in equation (3.2) of a pure braid does not affect the central charge of a stability condition.

For ( $\Rightarrow $ ), recall that by Corollary 3.4, we can assume that and , where and . If $p(\unicode{x3c3} _1)=p(\unicode{x3c3} _2)$ , then certainly $Z_1 =Z_2$ since p is simply the forgetful map followed by an isomorphism.

Furthermore, by Corollary 4.2, we see that

, since the intersection contains $p(\unicode{x3c3} _1)=p(\unicode{x3c3} _2)$ . Hence by Proposition 4.6(1), it follows that $s(\unicode{x3b1} )=s(\unicode{x3b2} )$ , and thus we can consider the composition

. Then $F_{\unicode{x3b3} }\in \mathop {\mathsf {PBr}}\nolimits $ and

proving the statement.

Corollary 4.9. The map

induces a homeomorphism

Proof. The map p is surjective by Corollary 4.7, and by Lemma 4.8 it induces the bijection in the statement. The induced map is itself a homeomorphism by the definition of the quotient topology and the fact that p is a local homeomorphism.

The following is our main result.

Corollary 4.10. The map is the universal cover, with Galois group $\mathop {\mathsf {PBr}}\nolimits $ . Furthermore, $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is contractible.

Proof. The first statement is obtained by composing the universal cover from Corollary 4.5 with the homeomorphism from Corollary 4.9. As already stated, the second part follows from [PSZ] and [Au1, 4.12].

5 Corollaries

In this section, we prove the five main corollaries stated in the introduction. For an ADE root system, it is well-known [B4] that $\unicode{x3c0} _1(\mathfrak {h}_{\mathrm {reg}}/W)$ is isomorphic to the associated ADE braid group. Recall that the $K(\unicode{x3c0} \,,1)$ -conjecture for ADE braid groups, which is already a theorem in this setting, asserts that the universal cover of $\mathfrak {h}_{\mathrm {reg}}/W$ is contractible.

Corollary 5.1. The $K(\unicode{x3c0} \,,1)$ -conjecture holds for all ADE braid groups.

Proof. As in [T, §3] or [KM, §4.3], we may choose a flopping contraction for which the hyperplane arrangement

is an ADE root system $\mathfrak {h}$ . It is well-known that the complexified complement

. Write W for the Weyl group, which is finite, and thus clearly the covering map $ \mathfrak {h}_{\mathrm {reg}}\to \mathfrak {h}_{\mathrm {reg}}/W$ has finite fibres. It follows that the composition

is then also a covering map. As above, it is well-known that $\unicode{x3c0} _1(\mathfrak {h}_{\mathrm {reg}}/W)$ is the braid group. By Corollary 4.10, the fact that $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is contractible implies that it is simply connected. Hence the composition is the universal cover, and furthermore the universal cover is contractible.

Corollary 5.2. The homomorphism sending $\unicode{x3b1} \mapsto F_{\unicode{x3b1} }$ is injective.

Proof. As in Sections 3 and 4, by definition $\mathop {\mathsf {PBr}}\nolimits $ is the image of this homomorphism. Since p is a regular covering map, as is standard [H, 1.40(c)], there is a short exact sequence of groups

(5.1)

where as before takes $\unicode{x3b1} \mapsto F_{\unicode{x3b1} }$ . However, since by Corollary 4.10 $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is contractible, its fundamental group is trivial.

For , set , which is necessarily a tilting complex for $\Lambda _{\mathrm {con}}$ since equivalences map tilting complexes to tilting complexes.

Corollary 5.3. The map $\unicode{x3b1} \mapsto T_{\unicode{x3b1} }$ is a bijection from morphisms in the Deligne groupoid ending at $C_+$ to the set of basic one-sided tilting complexes of $\Lambda _{\mathrm {con}}$ , up to isomorphism.

Proof. Surjectivity is already known. Indeed, since $\Lambda _{\mathrm {con}}$ is silting-discrete, by [Au1, 3.16(2)], every standard derived equivalence from $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is, up to algebra automorphism, isomorphic to $F_{\unicode{x3b2} }$ for some . In particular, every one-sided tilting complex for $\Lambda _{\mathrm {con}}$ is isomorphic to $F_{\unicode{x3b2} }^{-1}(\operatorname {\underline {End}}_R(L))$ for some . Since and $F_{\unicode{x3b2} }^{-1}=F_{\unicode{x3b2} ^{-1}}$ , surjectivity of the map follows.

The content is that the map is also injective. Suppose that and are such that $T_{\unicode{x3b1} } \cong T_{\unicode{x3b2} }$ , where $L,M \in \operatorname {Mut}_0(N)$ . Then by definition,

(5.2) $$ \begin{align} F_{\unicode{x3b1}}(\operatorname{\underline{End}}_R(L)) \cong F_{\unicode{x3b2}}(\operatorname{\underline{End}}_R(M)) , \end{align} $$

and hence . In particular, choosing any central charge Z, we have and hence . But Corollary 4.2 then implies that and intersect, which by Proposition 4.6 implies that $L \cong M$ .

Set . Applying $F_{\unicode{x3b2} }^{-1}=F_{\unicode{x3b2} ^{-1}}$ to equation (5.2) gives $F_{\unicode{x3b2} ^{-1} \circ \unicode{x3b1} }(\mathrm {B}_{\mathrm {con}}) \cong \mathrm {B}_{\mathrm {con}}$ . Thus by applying Theorem 4.3 to the composition , we deduce that there is an isomorphism $F_{\unicode{x3b2} ^{-1} \circ \unicode{x3b1} } \cong \mathop {\mathrm {Id}}\nolimits $ . Corollary 5.2 applied to the contraction algebra $\mathrm {B}_{\mathrm {con}}$ then shows that $\unicode{x3b2} ^{-1} \circ \unicode{x3b1} $ must be the identity, and hence $\unicode{x3b1} =\unicode{x3b2} $ in the Deligne groupoid.

Remark 5.4. If we instead assign to a path $\unicode{x3b1} \colon C_+ \to C_L$ the tilting complex $F_{\unicode{x3b1} }^{-1}(\operatorname {\underline {End}}_R(L))$ , we equivalently obtain a bijection between the paths in the Deligne groupoid that start at $C_+$ and basic one-sided tilting complexes of $\Lambda _{\mathrm {con}}$ .

Corollary 5.5. The homomorphism sending $\unicode{x3b1} \mapsto \operatorname {\mathsf {Flop}}_{\unicode{x3b1} }$ is injective.

Proof. Suppose that $\unicode{x3b1} $ belongs to the kernel, so $\operatorname {\mathsf {Flop}}_{\unicode{x3b1} }=\mathop {\mathrm {Id}}\nolimits $ . Since is functorially isomorphic to $\operatorname {\mathsf {Flop}}_{\unicode{x3b1} }$ after tilting by Theorem 2.4, necessarily . The left-hand part of the commutative diagram in Theorem 2.4 then implies that $F_{\unicode{x3b1} }(\Lambda _{\mathrm {con}})$ maps, under restriction of scalars, to $\Lambda _{\mathrm {con}}$ . It follows that $F_{\unicode{x3b1} }(\Lambda _{\mathrm {con}})\cong \Lambda _{\mathrm {con}}$ in $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ ; see, for example, [Au2, 6.6]. By Lemma 4.3, there is a functorial isomorphism $F_{\unicode{x3b1} }\cong \mathop {\mathrm {Id}}\nolimits $ . By Corollary 5.2, we see that $\unicode{x3b1} =1$ , proving the statement.

Corollary 5.6. With the notation as in the introduction, is contractible.

Proof. By [HW2], there is a regular covering map with Galois group G equal to the image of . But by Corollary 5.5, the map is an isomorphism. By the corresponding version of equation (5.1), is trivial, so the cover is universal. Universal covers are unique, so by Corollary 4.10, it follows that is contractible.