2.1 Elephants

The pullback along $f$ of a general hyperplane section through $p \in \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ is a partial crepant resolution of an ADE surface singularity, the so-called general elephant [Reference ReidRei83, (1.14)]

(2.A)

Let $\Delta$ be the Dynkin diagram associated to $\mathbb{C}^2/G$ , and let the composition

\begin{align*} Z \to Y\to \mathbb {C}^2/G \end{align*}

be the full minimal resolution. By the McKay correspondence, the exceptional curves ${\mathrm{C}}_i \subseteq Z$ are indexed by the nodes $i \in \Delta$ . We write

\begin{align*} \unicode{x2110} \subseteq \Delta \end{align*}

for the subset indexing those curves ${\mathrm{C}}_i \subseteq Z$ , which are contracted by the morphism $Z\to Y$ , so the complement $\unicode{x2110}^{\kern 0.5pt{\mathrm{c}}} = \Delta \setminus \unicode{x2110}$ indexes the curves that survive. In particular,

\begin{align*}\{ \mathrm {C}_i \mid i \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}} \}\end{align*}

forms the set of exceptional curves in both $Y$ and $\unicode{x1D4B3}$ , and the group ${\mathrm{A}}_1(Y)={\mathrm{A}}_1(\unicode{x1D4B3})$ is freely generated by their cycle classes.

The geometry of $\unicode{x1D4B3}$ will be studied by viewing it as the total space of a one-parameter deformation of $Y_{\unicode{x2110}}$ . This requires detailed control over the associated root theory, which we establish in the following subsections.

2.2 Root theory

For any Dynkin diagram $\Delta$ , let $\mathfrak{h}$ be the $\mathbb{R}$ -vector space based on the set of simple roots $\{\unicode{x03B1}_i \mid i \in \Delta \}$ so that

\begin{align*} \mathfrak {h}=\bigoplus _{i\in \Delta }\mathbb {R}\unicode {x03B1}_i, \end{align*}

and write $\Theta = \mathfrak{h}^\star$ for the dual. The Weyl group $W$ acts naturally on both $\mathfrak{h}$ and $\Theta$ . For every positive root $\unicode{x03B1} \in \mathfrak{h}$ , write $\unicode{x1D49F}_{\unicode{x03B1}}\subseteq \mathfrak{h}$ for the perpendicular hyperplane, and write $\mathsf{H}_{\unicode{x03B1}} \subseteq \Theta$ for the dual hyperplane.

For a given restricted positive root $\unicode{x03B2} \in \mathfrak{h}_{\unicode{x2110}}$ , there are in general many different positive roots $\unicode{x03B1} \in \mathfrak{h}$ such that $ \unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1})=\unicode{x03B2}$ . The following result controls the possible lifts and is a very mild generalisation of [Reference Bryan, Katz and LeungBKL01, Lemma 2.4]. It is used in the proof of Theorem2.6, which in turn is used to relate enumerative invariants to hyperplane arrangements in Section 3.

Proof. Since $\unicode{x2110}$ is fixed, to ease notation, set $\unicode{x03C0}=\unicode{x03C0}_{\unicode{x2110}}$ .

(1) $\Rightarrow$ (2) This is the only difficult part, and it proceeds by case analysis. Consider first the $A_n$ root system, where the positive roots are precisely the connected chains of $1$ s on the Dynkin graph

\begin{align*} \begin {array}{c@{\hspace {2pt}}c@{\hspace {2pt}}c@{\hspace {2pt}}c@{\hspace {2pt}}c@{\hspace {1pt}}c@{\hspace {2pt}}c@{\hspace {2pt}}c@{\hspace {2pt}}c@{\hspace {2pt}}c@{\hspace {2pt}}c@{\hspace {2pt}}} \unicode {x03B1}_{ij}:= \ 0 & \ldots & 0 & 1 & \ldots & 1 & 0 & \ldots & 0 \quad (1 \leq i \leq j \leq n).\\[-0.5mm] & & & i & & j \end {array} \end{align*}

By definition of the action of $W$ on $\mathfrak{h}$ , the reflection $s_i$ acts by

(2.B) \begin{align} s_i(\unicode{x03B1}_{ij})=\unicode{x03B1}_{i+1\, j} \qquad {\mathrm{for}}\ i\lt\ j, \end{align}

which has the effect of replacing the leftmost $1$ with a $0$ . Similarly $s_j(\unicode{x03B1}_{ij})=\unicode{x03B1}_{i\, j-1}$ for $i \lt j$ , which has the effect of replacing the rightmost $1$ with a $0$ .

Suppose now that we are given positive roots $\unicode{x03B1}_{ij}$ and $\unicode{x03B1}_{kl}$ , and assume without loss of generality that $i \leq k$ . Since $\unicode{x03C0}(\unicode{x03B1})=\unicode{x03C0}(\unicode{x03B1}^\prime )$ is non-zero, the two chains of $1$ s must overlap, and any position at which they do not overlap must be indexed by an element of $\unicode{x2110}$ . By iteratively applying (2.B), we can shorten both $\unicode{x03B1}_{ij}$ and $\unicode{x03B1}_{kl}$ using only elements of $W_{\unicode{x2110}}$ and force them to line up on the left, at the leftmost element belonging to the overlap. They can similarly be forced to line up on the right as well, proving the claim.

The case $D_n$ for $n \geq 4$ is similar in spirit, albeit more involved. As usual, write $\unicode{x03B1}_1,\ldots, \unicode{x03B1}_n$ for the simple roots; then the positive roots are given by the following linear combinations. Note that the support of each is a connected subset of the Dynkin graph.

\begin{align*} & \begin{array}{c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{1pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}l@{\hspace{2pt}}} & & & & & & & & q & & \qquad 1 \leq i \leq j \leq n-2,\\ \unicode{x03B1}_{ij}^{pq}:= \ 0 & \ldots & 0 & 1 & \ldots & 1 & 0 & \ldots & 0 & p & \qquad p,q\in \{0,1\},\\ & & & i & & j & & & & & \qquad (p,q) \neq (0,0) \Rightarrow j=n-2 \end{array}\\ \, & \begin{array}{c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{1pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}} & & & & & & & & 1\\ \unicode{x03B2}_{ij}:= \ 0 & \ldots & 0 & 1 & \ldots & 1 & 2 & \ldots & 2 & 1 & \qquad 1 \leq i \lt j \leq n-2 \\ & & & i & & & j \end{array} \end{align*}

In addition, the collection $\unicode{x03B1}_{ij}^{pq}$ includes two special cases $\unicode{x03B1}_{n-1}$ and $\unicode{x03B1}_n$ , which we interpret as $(p,q)=(1,0)$ and $(p,q)=(0,1)$ , with the string of $1$ s between $i$ and $j$ being empty.

For every $1 \leq i \leq n$ , the reflection $s_i$ acts on the coefficient in the $i$ th position by negating it and then by adding the sum of the coefficients in adjacent positions. The coefficients in all other positions are left unchanged.

Consider now positive roots $\unicode{x03B1}, \unicode{x03B1}^\prime$ , with $\unicode{x03C0}(\unicode{x03B1})=\unicode{x03C0}(\unicode{x03B1}^\prime )\neq 0$ . We work through the different cases. The first case is $\unicode{x03B1}=\unicode{x03B1}_{ij}^{pq}$ and $\unicode{x03B1}^\prime = \unicode{x03B1}_{kl}^{rs}$ . If $(p,q) \neq (r,s)$ , then we can use the reflections $s_{n-1},s_n$ to transform $\unicode{x03B1},\unicode{x03B1}^\prime$ until $(p,q)=(r,s)$ . For instance, if $(p,q)=(0,1)$ and $(r,s)=(0,0)$ , then, since $\unicode{x03C0}(\unicode{x03B1})=\unicode{x03C0}(\unicode{x03B1}^\prime )$ , we must have $n \in \unicode{x2110}$ . Applying $s_n \in W_{\unicode{x2110}}$ to $\unicode{x03B1}$ replaces $(p,q)=(0,1)$ , with $(p,q)=(0,0)$ . The other cases are similar, so we may assume that $(p,q)=(r,s)$ . Given this, we follow the strategy in the $A_n$ case. The only difference is when $j=n-2$ and $l \lt n-2$ , in which case $(p,q)=(r,s)=(0,0)$ since the chain of $1$ s is connected. Then $s_{n-2}\in W_{\unicode{x2110}}$ and $s_{n-2} (\unicode{x03B1}_{i\,n-2}^{00}) = \unicode{x03B1}_{i\, n-3}^{00}$ . This, and all other situations, then simply mirror the $A_n$ proof.

The next case is $\unicode{x03B1} = \unicode{x03B1}_{ij}^{pq}$ and $\unicode{x03B1}^\prime = \unicode{x03B2}_{kl}$ . If $i \leq k$ then necessarily $j \geq k$ since otherwise the supports do not overlap. In particular $s_i, s_{i+1}, \ldots, s_{k-1} \in W_{\unicode{x2110}}$ which gives

\begin{align*} (s_{k-1} \circ \cdots \circ s_i)(\unicode {x03B1}_{ij}^{pq}) = \unicode {x03B1}_{kj}^{pq}. \end{align*}

Similarly if $i \geq k$ we use $s_{k} \circ \cdots \circ s_{i-1}$ to achieve the same transformation. For the next step, notice that $s_{j+1},\ldots, s_{n-2} \in W_{\unicode{x2110}}$ , and applying these left-to-right gives

\begin{align*} (s_{n-2} \circ \cdots \circ s_{j+1})(\unicode {x03B1}_{kj}^{pq}) = \unicode {x03B1}_{k\,n-2}^{pq}. \end{align*}

As in the previous case, if $(p,q) \neq (1,1)$ then we may use the reflections $s_{n-1},s_n$ to transform $\unicode{x03B1}_{k\,n-2}^{pq}$ into $\unicode{x03B1}_{k\,n-2}^{11}$ . Now, going back from right-to-left gives the required

\begin{align*} (s_l \circ \cdots \circ s_{n-2})(\unicode {x03B1}_{k\,n-2}^{11}) = \unicode{x03B2}_{kl}. \end{align*}

The next case is $\unicode{x03B1}=\unicode{x03B2}_{ij}$ and $\unicode{x03B1}^\prime = \unicode{x03B2}_{kl}$ , where without loss of generality $i \geq k$ . But then $s_{i-1},\ldots, s_k \in W_{\unicode{x2110}}$ which is applied directly to give $(s_k \circ \cdots \circ s_{i-1})(\unicode{x03B2}_{ij}) = \unicode{x03B2}_{kj}$ . If $j \geq l$ , then $s_{j-1},\ldots, s_l \in W_{\unicode{x2110}}$ , and $(s_{l}\circ \cdots \circ s_{j-1})(\unicode{x03B2}_{kj}) = \unicode{x03B2}_{kl}$ . The case $j \leq l$ is similar.

This completes the proof for $D_n$ , except for some special cases involving $\unicode{x03B1}_{n-1}$ and $\unicode{x03B1}_n$ . These are more elementary, and so are left to the reader.

The remaining cases $E_6,E_7,E_8$ encompass finitely many possibilities. It is possible to verify these by hand, but it is also possible to employ computer algebra [Reference Bosma, Cannon and PlayoustBCP97]. Source code is available from the authors upon request. This completes the proof of (1) $\Rightarrow$ (2).

(2) $\Rightarrow$ (1) This holds since applying elements of $W_{\unicode{x2110}}$ to a given positive root cannot change the coefficients associated to elements of $\unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}$ .

(2) $\Rightarrow$ (3) If $s \in W$ and $v \in \mathfrak{h}$ then $s(\unicode{x1D49F}_v)=\unicode{x1D49F}_{s(v)}$ since $s$ preserves the Cartan pairing. In particular this applies when $v=\unicode{x03B1}$ is a positive root and $s(v)=\unicode{x03B1}^\prime$ is its image.

(3) $\Rightarrow$ (2) Suppose there is an element $s \in W_{\unicode{x2110}}$ such that $s(\unicode{x1D49F}_{\unicode{x03B1}})=\unicode{x1D49F}_{\unicode{x03B1}^\prime }$ . We then have $\unicode{x1D49F}_{\unicode{x03B1}^\prime } = \unicode{x1D49F}_{s(\unicode{x03B1})}$ and since both $\unicode{x03B1}^\prime$ and $s(\unicode{x03B1})$ are roots it follows that $s(\unicode{x03B1})$ equals either $\unicode{x03B1}^\prime$ or $-\unicode{x03B1}^\prime$ . We claim that $s(\unicode{x03B1})$ must be a positive root, thus ruling out the latter possibility. If $s=s_i$ for some $i \in \unicode{x2110}$ , then $s_i$ permutes the set of positive roots excluding $\unicode{x03B1}_i$ . But certainly $\unicode{x03B1} \neq \unicode{x03B1}_i$ since $\unicode{x03C0}(\unicode{x03B1}) \neq 0$ , so $s_i(\unicode{x03B1})$ must be a positive root. The argument for general $s$ follows by induction, since $\unicode{x03C0}(\unicode{x03B1})=\unicode{x03C0}(s_i(\unicode{x03B1}))$ for any $\unicode{x03B1}$ and any $s_i$ with $i \in \unicode{x2110}$ .

2.3 Hyperplane arrangements: finite and infinite

To the above data of a Dynkin diagram $\Delta$ and a subset $\unicode{x2110} \subseteq \Delta$ it is possible to associate two hyperplane arrangements encoding the set of restricted positive roots: one finite and one infinite [Reference Iyama and WemyssIW]. Recalling Notation 2.2, both arrangements live inside $\Theta _{\unicode{x2110}}\cong \mathbb{R}^{|\unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}|}$ .

The real vector space $\mathfrak{h}_{\unicode{x2110}}$ is dual to $\Theta _{\unicode{x2110}}$ , and so for each restricted root $0\neq \unicode{x03B2}=\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1})\in \mathfrak{h}_{\unicode{x2110}}$ we may consider the dual hyperplane

\begin{align*} \mathsf {H}_{\unicode{x03B2}}:= \{ (\vartheta _i) \mid \sum \unicode{x03B2}_i\vartheta _i=0 \} \subseteq \Theta _{\unicode{x2110}}. \end{align*}

Since there are only finitely many restricted roots, the collection of $\mathsf{H}_{\unicode{x03B2}}$ forms a finite hyperplane arrangement in $\Theta _{\unicode{x2110}}$ , which we refer to as the finite linear arrangement, namely

(2.C) \begin{align} \unicode{x210B}_{\unicode{x2110}} := \left \{ \mathsf{H}_{\unicode{x03B2}} \mid \unicode{x03B2} \text{ is a restricted positive root} \right \}. \end{align}

The above list includes repetitions, whenever two restricted roots are proportional. As in Example 1.1, we remember these repetitions by attaching a finite list of multiplicities to each hyperplane. We refer to this data as the enhanced finite arrangement.

To define the infinite arrangement, for each restricted positive root $\unicode{x03B2}$ , consider the infinite disjoint union of affine hyperplanes in $\Theta _{\unicode{x2110}}$ defined by

\begin{align*} \mathsf {H}_{\unicode{x03B2}}^{\mathsf {aff}} = \left\{ (\vartheta _i)\mid \sum _{i\in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}}\unicode{x03B2}_i\vartheta _i=z\mbox { for some } z\in \mathbb {Z} \right\}. \end{align*}

The infinite affine arrangement $\unicode{x210B}_{\unicode{x2110}}^{\mathsf{aff}}$ is then defined to be

(2.D) \begin{align} \unicode{x210B}_{\unicode{x2110}}^{\mathsf{aff}} := \bigcup _{\unicode{x03B2}} \mathsf{H}_{\unicode{x03B2}}^{\mathsf{aff}}. \end{align}

There is an inclusion $\unicode{x210B}_{\unicode{x2110}} \subseteq \unicode{x210B}_{\unicode{x2110}}^{\mathsf{aff}}$ , as $\unicode{x210B}_{\unicode{x2110}}$ can be recovered from $\unicode{x210B}_{\unicode{x2110}}^{\mathsf{aff}}$ as the subset of hyperplanes which pass through the origin. On the other hand to construct $\unicode{x210B}_{\unicode{x2110}}^{\mathsf{aff}}$ from the finite arrangement $\unicode{x210B}_{\unicode{x2110}}$ it is necessary to remember the multiplicities, since $\mathsf{H}_{2\unicode{x03B2}}^{\mathsf{aff}}\neq \mathsf{H}_{\unicode{x03B2}}^{\mathsf{aff}}$ .

2.4 Simultaneous partial resolution

The enumerative geometry of the $3$ -fold $\unicode{x1D4B3}$ will be studied by replacing $\unicode{x1D4B3}$ with a generic perturbation, a strategy employed by many authors [Reference MorrisonMor96, Reference WilsonWil99, Reference Bryan, Katz and LeungBKL01]. In the first instance this will allow us to qualitatively characterise the GV invariants, and extract the poles of the quantum product. However the real strength in this approach, and indeed our new contribution, is to use the wall crossing formula from [Reference Iyama and WemyssIW] to construct iterated flops via simultaneous (partial) resolution. This iteration step is harder, and so will be delayed until Section 4.

In this subsection we simply recall the necessary background and set notation, largely following [Reference Bryan, Katz and LeungBKL01, Section 2], together with [Reference BrieskornBri68, Reference PinkhamPin83, Reference FriedmanFri86, Reference Katz and MorrisonKM92].

2.4.1 Simultaneous resolution

With notation as in Section 2.2, given any Kleinian singularity $\mathbb{C}^2/G$ with associated Dynkin diagram $\Delta$ , consider the complex vector space $\mathfrak{h}_{\mathbb{C}}=\bigoplus _{i\in \Delta }\mathbb{C}\unicode{x03B1}_i$ , based by the simple roots. Write $\mathop{\mathrm{Spec}}\nolimits \unicode{x1D4B1}$ for a versal deformation of $\mathbb{C}^2/G$ , then as is very well known, base changing with respect to the Weyl group

gives $\mathop{\mathrm{Spec}}\nolimits \unicode{x1D4B2}\to \mathfrak{h}_{\mathbb{C}}$ , which admits a simultaneous resolution. Since $\dim \unicode{x1D4B2}\geq 3$ , there are in fact many such simultaneous resolutions, since minimal models are not unique.

In [Reference Katz and MorrisonKM92, Theorem 1] Katz–Morrison construct a particular simultaneous resolution, from a particular $\mathop{\mathrm{Spec}}\nolimits \unicode{x1D4B1}\to \mathfrak{h}_{\mathbb{C}}/W$ , for which positive roots and their hyperplanes control those curve classes that survive under deformation [Reference Katz and MorrisonKM92, Theorem 1(c)]. We will recap this result in greater generality in Theorem2.6 below, but for now write $\unicode{x1D4B5}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4B2}$ for this preferred resolution. Katz–Morrison refer to their particular choice of $\unicode{x1D4B5}$ as the standard simultaneous resolution [Reference Katz and MorrisonKM92, Section 6].

2.4.2 Simultaneous partial resolution

Given any subset $\unicode{x2110}\subseteq \Delta$ , consider the standard simultaneous resolution $\unicode{x1D4B5}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4B2}$ from Section 2.4.1. As explained in [Reference Katz and MorrisonKM92, above Theorem 3] following [Reference PinkhamPin83] it is possible to blow down $\unicode{x1D4B5}$ at the curves in $\unicode{x2110}$ and take the quotient by $W_{\unicode{x2110}}$ to obtain $\unicode{x1D4B4}_{\unicode{x2110}}$ , which sits in the following commutative diagram

(2.E)

with all squares cartesian. By construction, the fibre $(\mathsf{h}_{\unicode{x2110}}\circ \mathsf{g}_{\unicode{x2110}})^{-1}(0)$ is the partial resolution of $\mathbb{C}^2/G$ obtained from the full minimal resolution by blowing down the curves in $\unicode{x2110}$ . Namely, recalling Notation 2.1, $(\mathsf{h}_{\unicode{x2110}}\circ \mathsf{g}_{\unicode{x2110}})^{-1}(0)=Y_{\unicode{x2110}}$ .

In a similar way as in Section 2.4.1, the middle morphism $\mathsf{h}_{\unicode{x2110}}\colon \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4B1}_{\unicode{x2110}}\to \mathfrak{h}_{\mathbb{C}}/W_{\unicode{x2110}}$ admits simultaneous partial resolutions. Again these are not unique, however we will refer to the choice $\unicode{x1D4B4}_{\unicode{x2110}}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4B1}_{\unicode{x2110}}$ constructed above as the standard simultaneous partial resolution associated to $\unicode{x2110}$ . From our perspective, the point is that $\unicode{x1D4B4}_{\unicode{x2110}}$ is precisely the partial simultaneous resolution for which Theorem2.6 below holds.

2.5 Surface deformations via simultaneous partial resolution

Fix a subset $\unicode{x2110}\subseteq \Delta$ and consider the composition

\begin{align*} \mathsf {s}_{\unicode{x2110}} = \mathsf {h}_{\unicode{x2110}}\circ \mathsf {g}_{\unicode{x2110}}\colon \unicode{x1D4B4}_{\unicode{x2110}}\to \mathfrak {h}_{\mathbb {C}}/W_{\unicode{x2110}}\end{align*}

from (2.E). This is a versal deformation of the surface $Y_{\unicode{x2110}}$ .

Definition 2.5. The standard discriminant locus

\begin{align*} \unicode{x1D49F}_{\unicode{x2110}} \subseteq \mathfrak {h}_{\mathbb {C}}/W_{\unicode{x2110}} \end{align*}

is the set of points $p \in \mathfrak{h}_{\mathbb{C}}/W_{\unicode{x2110}}$ such that the fibre $\mathsf{s}_{\unicode{x2110}}^{-1}(p)$ contains a complete curve.

There is a similar definition of a discriminant locus associated to any simultaneous partial resolution: the word standard in Definition 2.5 emphasises the choice made in (2.E). The following discussion draws heavily on [Reference Katz and MorrisonKM92, Theorem 1] as used in [Reference Bryan, Katz and LeungBKL01, Proposition 2.2], while also incorporating Lemma 2.3 above to relate the resulting combinatorics to the enhanced movable cone.

To set notation, recall that $\unicode{x1D49F}_{\unicode{x03B1}}\subseteq \mathfrak{h}$ is the hyperplane perpendicular to $\unicode{x03B1}$ , and let $\unicode{x1D49F}_{\unicode{x03B1},\mathbb{C}}\subseteq \mathfrak{h}_{\mathbb{C}}$ denote its complexification. Recall from Notation 2.2 that for $\unicode{x2110} \subseteq \Delta$ there is a quotient map $\unicode{x03C0}_{\unicode{x2110}} \colon \mathfrak{h}\to \mathfrak{h}_{\unicode{x2110}}$ , where the vector space $\mathfrak{h}_{\unicode{x2110}}$ has basis $\{ \unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1}_i) \mid i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}\}$ , and that there is a natural identification

(2.F) \begin{align} \mathfrak{h}_{\unicode{x2110}} & \cong {\mathrm{A}}_1(Y_{\unicode{x2110}})_{\mathbb{R}}\\ \unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1}_i) & \mapsto {\mathrm{C}}_i.\nonumber \end{align}

Every restricted positive root $\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1}) \in \mathfrak{h}_{\unicode{x2110}}$ has non-negative integer coefficients, and so may be interpreted as a curve class $\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1})=\unicode{x03B2} \in {\mathrm{A}}_1(Y_{\unicode{x2110}})$ .

Proof. (1) If $\unicode{x2110}=\emptyset$ then by [Reference Katz and MorrisonKM92, Theorem 1(3)] (see also [Reference Bryan, Katz and LeungBKL01, Proposition 2.2]) there is a decomposition of the standard discriminant locus

\begin{align*} \unicode{x1D49F}_\emptyset = \bigcup _{\unicode {x03B1}} \unicode{x1D49F}_{\unicode {x03B1},\mathbb {C}} \subseteq \mathfrak {h}_{\mathbb {C}} \end{align*}

where the union is over all positive roots $\unicode{x03B1}$ . The analogous decomposition for general $\unicode{x2110}$ follows by considering the standard simultaneous resolution $\unicode{x1D4B5}$ of the standard simultaneous partial resolution $\unicode{x1D4B4}_{\unicode{x2110}}$ from 2.4

The map $\unicode{x1D4B5} \to \unicode{x1D4B4}_{\unicode{x2110}}$ is given by blowing down $\unicode{x1D4B5}$ at the curves in $\unicode{x2110}$ and then taking the quotient by $W_{\unicode{x2110}}$ .

Fixing a point $p \in \mathfrak{h}_{\mathbb{C}}$ , it follows that the fibre $\mathsf{s}_{\unicode{x2110}}^{-1}(\unicode{x03D5}_{\unicode{x2110}}(p))$ contains a complete curve if and only if the fibre $\mathsf{t}_{\unicode{x2110}}^{-1}(p)$ contains a complete curve which is not blown down. Again by [Reference Katz and MorrisonKM92, Theorem 1(3)], the fibre $\mathsf{t}_{\unicode{x2110}}^{-1}(p)$ contains a complete curve if and only if $p \in \unicode{x1D49F}_{\unicode{x03B1},\mathbb{C}}$ for some positive root $\unicode{x03B1}$ , and this curve is not blown down if and only if $\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1}) \neq 0$ . This produces the desired decomposition of $\unicode{x1D49F}_{\unicode{x2110}}$ . It then follows from Lemma 2.3 that the components $\unicode{x1D49F}_{\unicode{x03B1},\mathbb{C}}/\unicode{x1D4B2}_{\unicode{x2110}}$ are indexed by the restricted positive roots $\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1})$ , i.e. $\unicode{x1D49F}_{\unicode{x03B1},\mathbb{C}}/W_{\unicode{x2110}}=\unicode{x1D49F}_{\unicode{x03B1}^\prime, \mathbb{C}}/W_{\unicode{x2110}}$ if and only if $\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1})=\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1}^\prime )$ .

(2)(3) First recall what it means for a curve in $\mathsf{s}_{\unicode{x2110}}^{-1}(p)$ to have class $\unicode{x03B2} \in {\mathrm{A}}_1(Y_{\unicode{x2110}})$ . The inclusion of the central fibre $\mathsf{i}_0 \colon \mathsf{s}_{\unicode{x2110}}^{-1}(0) = Y_{\unicode{x2110}} \hookrightarrow \unicode{x1D4B4}_{\unicode{x2110}}$ induces an isomorphism

\begin{align*} \mathsf {i}_{0\star } \colon \mathrm {A}_1(Y_{\unicode{x2110}}) \xrightarrow {\sim } \mathrm {A}_1(\unicode{x1D4B4}_{\unicode{x2110}}). \end{align*}

Now consider an arbitrary fibre $\mathsf{s}_{\unicode{x2110}}^{-1}(p)$ with inclusion $\mathsf{i}_p \colon \mathsf{s}_{\unicode{x2110}}^{-1}(p) \hookrightarrow \unicode{x1D4B4}_{\unicode{x2110}}$ . If ${\mathrm{C}} \subseteq \mathsf{s}_{\unicode{x2110}}^{-1}(p)$ is a complete curve, then ${\mathrm{C}}$ has class $\unicode{x03B2} \in {\mathrm{A}}_1(Y_{\unicode{x2110}})$ if $\mathsf{i}_{p\star } {\mathrm{C}} = \mathsf{i}_{0\star } \unicode{x03B2}$ . The same definition applies to the full simultaneous resolution $\unicode{x1D4B5}$ . Both (2) and (3) are known for $\unicode{x1D4B5}$ by loc. cit., and the general case follows by tracking curve classes from $\unicode{x1D4B5}$ to $\unicode{x1D4B4}_{\unicode{x2110}}$ .

Remark 2.8. The description of the standard discriminant locus $\unicode{x1D49F}_{\unicode{x2110}}$ in Theorem 2.6 is a union over positive roots $\unicode{x03B1} \in \mathfrak{h}$ such that $\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1}) \neq 0$ . The complementary union of hyperplane quotients

\begin{align*} \unicode{x2130}_{\unicode{x2110}}=\bigcup _{\unicode {x03C0}_{\unicode{x2110}}(\unicode {x03B1})= 0}\,\,\unicode{x1D49F}_{\unicode {x03B1},\mathbb {C}}/W_{\unicode{x2110}} \end{align*}

parametrise points $p \in \mathfrak{h}_{\mathbb{C}}/W_{\unicode{x2110}}$ such that the fibre $\mathsf{s}_{\unicode{x2110}}^{-1}(p)$ is singular. Clearly

\begin{align*} \unicode{x1D49F}_\emptyset /W_{\unicode{x2110}} = \unicode{x1D49F}_{\unicode{x2110}} \cup \unicode{x2130}_{\unicode{x2110}}. \end{align*}

For a generic point $p \in \unicode{x2130}_{\unicode{x2110}}$ , the fibre $\mathsf{s}_{\unicode{x2110}}^{-1}(p)$ contains a single $A_1$ singularity. The locus $\unicode{x2130}_{\unicode{x2110}}$ will play a less central role than $\unicode{x1D49F}_{\unicode{x2110}}$ . The translation between our notation and that of [Reference Bryan, Katz and LeungBKL01, Section 2] is as follows: $\unicode{x1D49F}_{\unicode{x2110}} = D^{{\mathrm{curv}}}, \unicode{x2130}_{\unicode{x2110}} = D^{{\mathrm{sing}}}$ and $\unicode{x1D49F}_{\unicode{x2110}} \cup \unicode{x2130}_{\unicode{x2110}} = D$ .

2.6 $3$ -fold perturbations via surface deformations

Given the flopping contraction $\unicode{x1D4B3}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ , a choice of local equation for the hypersurface $\mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}/g \subseteq \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ produces a flat family $\mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD} \to \unicode{x1D49F} {\mathrm{isc}}$ over a formal disc, with central fibre an ADE surface singularity. By composition this produces a flat family $\unicode{x1D4B3} \to \unicode{x1D49F} {\mathrm{isc}}$ with central fibre the partial resolution $Y\cong Y_{\unicode{x2110}}$ of the ADE singularity [Reference ReidRei83]. This exhibits $\unicode{x1D4B3}$ , respectively $\mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ , as the total space of a one-parameter deformation of the surface $Y_{\unicode{x2110}}$ , respectively $\mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}/g$ . These deformations are induced by an associated classifying map

\begin{align*} \unicode {x03BC} \colon \unicode{x1D49F} \mathrm {isc} \to \mathfrak {h}_{\mathbb {C}}/W_{\unicode{x2110}} \end{align*}

where $\mathfrak{h}_{\mathbb{C}}/W_{\unicode{x2110}}$ is as described in the previous subsection, and the contraction $\unicode{x1D4B3} \to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ is obtained from the simultaneous partial resolution of Section 2.4 by base change

The central fibre of $\mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD} \to \unicode{x1D49F} {\mathrm{isc}}$ is the ADE singularity corresponding to $\Delta$ , so $\unicode{x03BC}(0) = 0$ . On the other hand, since $\unicode{x1D4B3}$ contains no complete curves outside of $Y$ the map $\unicode{x03BC}$ does not intersect the discriminant locus away from the origin, so $\unicode{x03BC}^{-1}(\unicode{x1D49F}_{\unicode{x2110}}) = \unicode{x03BC}^{-1}(0) = 0$ .

As in [Reference Bryan, Katz and LeungBKL01, below Lemma 2.7] there exists a one-parameter perturbation of $\unicode{x03BC}$

\begin{align*} (\unicode {x03BC}_t)_{t\in [0,\unicode {x025B}]} \colon \unicode{x1D49F} \mathrm {isc} \times [0,\unicode {x025B}] \to \mathfrak {h}_{\mathbb {C}}/W_{\unicode{x2110}} \end{align*}

such that $\unicode{x03BC}_0=\unicode{x03BC}$ and for $t \neq 0$ the following transversality condition is satisfied

(2.G) \begin{align} \unicode{x03BC}_{t}\ \mathrm{intersects}\ \unicode{x1D49F}_{\unicode{x2110}}\cup\ \unicode{x2130}_{\unicode{x2110}}\ \mathrm{transversely\ and\ away\ from\ codimension\ two\ strata} \end{align}

where $\unicode{x2130}_{\unicode{x2110}}$ is defined in Remark 2.8. Furthermore, making $\unicode{x025B}$ smaller if necessary, we can assume that $\unicode{x03BC}_t^{-1}(\unicode{x1D49F}_{\unicode{x2110}})$ is bounded away from the boundary of $ \unicode{x1D49F} {\mathrm{isc}}$ .

The spaces $\unicode{x1D4B3}_{t\neq 0}$ give generic perturbations of the target $\unicode{x1D4B3}_0=\unicode{x1D4B3}$ . The following is well-known, and will be used to reduce the enumerative geometry of $\unicode{x1D4B3}_t$ , locally, to that of the Atiyah flop.

Write $\mathfrak{X}$ for the $4$ -dimensional total space of the entire family $(\unicode{x03BC}_t)_{t}$ . Then as explained in [Reference WilsonWil92, Section 3], pulling back along inclusions of fibres induces isomorphisms

(2.H) \begin{align} {\mathrm{H}}^2(Y_{\unicode{x2110}};\mathbb{Z})\xleftarrow{\cong } {\mathrm{H}}^2(\unicode{x1D4B3};\mathbb{Z})\xleftarrow{\cong } {\mathrm{H}}^2(\mathfrak{X};\mathbb{Z})\xrightarrow{\cong } {\mathrm{H}}^2(\unicode{x1D4B3}_t;\mathbb{Z}) \end{align}

for any $t$ . Any class $L$ in ${\mathrm{H}}^2(\unicode{x1D4B3},\mathbb{Z})\cong \mathop{\mathrm{Pic}}\nolimits (\unicode{x1D4B3})$ thus induces an invertible sheaf $\unicode{x2112}$ on $\mathfrak{X}$ with $\unicode{x2112}|_{\unicode{x1D4B3}_0}=L$ . Similarly, pushing forward curve classes along the inclusion of fibres induces isomorphisms

(2.I) \begin{align} {\mathrm{A}}_1(Y_{\unicode{x2110}})\xrightarrow{\cong } {\mathrm{A}}_1(\unicode{x1D4B3}) \xrightarrow{\cong } {\mathrm{A}}_1(\mathfrak{X}) \xleftarrow{\cong } {\mathrm{A}}_1(\unicode{x1D4B3}_t) \end{align}

for any $t$ . Given $\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3})$ we abuse notation and let $\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3}_t)$ denote the image of $\unicode{x03B2}$ under the composition of the natural isomorphisms above. Further, combining (2.I) and (2.F) it makes sense to ask when curve classes are restricted roots.

3.1 Gopakumar–Vafa

Curve counting invariants of $\unicode{x1D4B3}$ will be defined using the perturbed target $\unicode{x1D4B3}_t$ (for some fixed $t \neq 0$ ) constructed in Section 2.6. Given a curve class $\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3})$ the associated genus-zero Gopakumar–Vafa (GV) invariant

\begin{align*} n_{\unicode{x03B2}} = n_{\unicode{x03B2},\unicode{x1D4B3}} \in \mathbb {Z}_{\geq 0} \end{align*}

is defined as the number of complete curves in $\unicode{x1D4B3}_t$ of class $\unicode{x03B2}$ . By Corollary 2.10 this is zero if $\unicode{x03B2}$ is not a restricted positive root, and otherwise is equal to the number of intersection points of $\unicode{x03BC}_t$ with the appropriate component of the discriminant locus, i.e.

(3.A) \begin{align} n_{\unicode{x03B2}} = \left|\unicode{x03BC}_t^{-1}(\unicode{x1D49F}_{\unicode{x03B1},\mathbb{C}}/W_{\unicode{x2110}})\right| \end{align}

where $\unicode{x03B1}$ is any positive root with $\unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1})=\unicode{x03B2}$ . This number is independent of the choice of small perturbation $\unicode{x03BC}_t$ .

In what follows, for a curve class $\unicode{x03B2}$ consider the dual hyperplane $\mathsf{H}_{\unicode{x03B2}}\subseteq \Theta _{\unicode{x2110}}$ .

Note that $\mathsf{H}_{\unicode{x03B2}}$ and $\mathsf{H}_{2\unicode{x03B2}}$ should be considered as different hyperplanes in the enhanced finite arrangement. See also the discussion in Section 2.3, and Remark 2.7.

3.2 Gromov–Witten

We refer to [Reference Cox and KatzCK99, Section 7] for an introduction to Gromov–Witten theory. For every non-zero curve class $\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3})$ there is an associated genus-zero Gromov–Witten (GW) invariant

\begin{align*} N_{\unicode{x03B2}}=N_{\unicode{x03B2},\unicode{x1D4B3}} \in \mathbb {Q} \end{align*}

defined as the virtual degree of the corresponding moduli space of stable maps to $\unicode{x1D4B3}$ . By deformation invariance this coincides with the virtual degree of the moduli space of stable maps to $\unicode{x1D4B3}_t$ for $t \neq 0$ , as constructed in Section 2.6.

The latter space decomposes as a disjoint union of spaces of stable maps to $\mathbb{P}^1$ , and applying the Aspinwall–Morrison multiple cover formula [Reference Aspinwall and MorrisonAM93, Reference VoisinVoi96] for the local invariants of $\unicode{x1D4AA}_{\mathbb{P}^1}(-1)\oplus \unicode{x1D4AA}_{\mathbb{P}^1}(-1)$ gives the following relationship between the GW invariants and the GV invariants from Section 3.1, namely

(3.B) \begin{align} N_{\unicode{x03B2}} = \sum _{d | \unicode{x03B2}} \dfrac{n_{\unicode{x03B2}/d}}{d^3}. \end{align}

More generally, given $k \geq 0$ and homogeneous classes $\unicode{x03B3}_1,\ldots, \unicode{x03B3}_k \in {\mathrm{H}}^\star (\unicode{x1D4B3};\mathbb{C})$ , the associated GW invariant with cohomological insertions at marked points is defined to be

\begin{align*} \langle \unicode {x03B3}_1,\ldots, \unicode {x03B3}_k \rangle ^{\unicode{x1D4B3}}_{0,k,\unicode{x03B2}} := \int _{[\overline {\unicode{x1D4A8}}_{0,k}(\unicode{x1D4B3},\unicode{x03B2})]^{\mathrm {virt}}} \prod _{i=1}^k \mathrm {ev}_i^\star \unicode {x03B3}_i \end{align*}

and provides a virtual count of rational curves in $\unicode{x1D4B3}$ of class $\unicode{x03B2}$ passing through the cycles $\unicode{x03B3}_1,\ldots, \unicode{x03B3}_k$ . Note that in particular $N_{\unicode{x03B2}} = \langle \rangle ^{\unicode{x1D4B3}}_{0,0,\unicode{x03B2}}$ . Since $\unicode{x1D4B3}$ is a Calabi–Yau $3$ -fold, the invariant vanishes unless the input data satisfies the dimension constraint

(3.C) \begin{align} \sum _{i=1}^k \deg \unicode{x03B3}_i = 2k. \end{align}

The cohomology of $\unicode{x1D4B3}$ is well-understood, see e.g. [Reference CaibărCai05, 5.2]. In particular

\begin{align*} \mathrm {H}^0(\unicode{x1D4B3}; \mathbb {C}) = \mathbb {C} \cdot \unicode {x1D7D9},\quad \mathrm {H}^1(\unicode{x1D4B3}; \mathbb {C})=0,\quad \mathrm {H}^2(\unicode{x1D4B3}; \mathbb {C}) = \mathop {\mathrm {Pic}}\nolimits \unicode{x1D4B3} \otimes \mathbb {C}. \end{align*}

Moreover, as we work in the complete local setting, by e.g. [Reference Van den BerghVdB04, 3.4.4] $\mathop{\mathrm{Pic}}\nolimits \unicode{x1D4B3}$ is dual to the group ${\mathrm{A}}_1(\unicode{x1D4B3})$ of curve classes, since there is a basis of divisor classes $\mathop{\mathrm{Pic}}\nolimits \unicode{x1D4B3} = \langle D_i \mid i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}} \rangle _{\mathbb{Z}}$ which satisfies $D_i \cdot {\mathrm{C}}_j = \unicode{x03B4}_{ij}$ .

Given a GW invariant $\langle \unicode{x03B3}_1,\ldots, \unicode{x03B3}_k \rangle ^{\unicode{x1D4B3}}_{0,k,\unicode{x03B2}}$ , if any $\unicode{x03B3}_i=\unicode{x1D7D9}$ then the invariant vanishes by the string equation. It follows from (3.C) that the invariant vanishes unless each $\unicode{x03B3}_i \in {\mathrm{H}}^2(\unicode{x1D4B3};\mathbb{C})$ . But then the $\unicode{x03B3}_i$ are divisors, and the $k$ -pointed invariants with divisorial insertions are related to the $0$ -pointed invariants by the divisor equation

\begin{align*} \langle D_{j_1}, \ldots, D_{j_k} \rangle ^{\unicode{x1D4B3}}_{0,k,\unicode{x03B2}} = \left ( \prod _{i=1}^k D_{j_i}\cdot \unicode{x03B2} \right ) N_{\unicode{x03B2}}. \end{align*}

In this way, the non-zero GW invariants are controlled entirely by the $N_{\unicode{x03B2}}$ , which by (3.B) are controlled entirely by the GV invariants $n_{\unicode{x03B2}}$ . The latter constitutes a finite list of numbers.

3.3 Quantum cohomology

As is well known, the GW invariants form the structure constants for quantum cohomology. The information defining quantum cohomology is equivalent to the quantum potential, defined in our setting as

(3.D) \begin{align} \Phi ^{\unicode{x1D4B3}}_{\mathsf{t}}( \unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) := \sum _{\substack{\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3})\\ \unicode{x03B2} \neq 0}} \,\sum _{k \geq 0} \,\,\dfrac{1}{k!} \langle \unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3, \mathsf{t}, \ldots, \mathsf{t} \rangle ^{\unicode{x1D4B3}}_{0,k+3,\unicode{x03B2}}. \end{align}

Here we exclude the case $\unicode{x03B2} = 0$ from the sum, since for non-compact $\unicode{x1D4B3}$ such invariants are not defined (see Remark 3.7). We view (3.D) as a family of multilinear maps

\begin{align*} \Phi ^{\unicode{x1D4B3}}_{\mathsf {t}} \colon \mathrm {H}^\star (\unicode{x1D4B3};\mathbb {C})^{\otimes 3} \to \mathbb {C} \end{align*}

parametrised by the formal variable $\mathsf{t} \in {\mathrm{H}}^\star (\unicode{x1D4B3};\mathbb{C})$ . By the earlier dimension arguments, we see that the quantum potential only depends on the component of $\mathsf{t}$ in the ${\mathrm{H}}^2(\unicode{x1D4B3};\mathbb{C})$ direction. Thus we may assume $\mathsf{t} \in {\mathrm{H}}^2(\unicode{x1D4B3};\mathbb{C})$ and write

\begin{align*} \mathsf {t} = (\mathsf {t}_i)_{i \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}} = \sum _{i \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}} \mathsf {t}_i D_i. \end{align*}

The parameter space for the quantum potential is thus co-ordinatised by the $\mathsf{t}_i$ . An alternative co-ordinate system is given by the Novikov parameters, defined by

\begin{align*} \mathsf {q}_i := \exp (\mathsf {t}_i). \end{align*}

The following result resembles expressions appearing in earlier work [Reference MorrisonMor96, Reference WilsonWil92], but is more explicit, being given in terms of canonical bases for ${\mathrm{H}}^2(\unicode{x1D4B3};\mathbb{C})$ and ${\mathrm{A}}_1(\unicode{x1D4B3})$ . This refined information will allow us in Corollary 3.4 to pinpoint the non-vanishing terms using Dynkin combinatorics and, in Corollary 5.13, to track the change in quantum potential under iterated flops.

Theorem 3.3. The quantum potential has a natural analytic continuation over the parameter space, given as a finite sum of terms indexed by the non-vanishing GV invariants

(3.E) \begin{align} \Phi ^{\unicode{x1D4B3}}_{\mathsf{t}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) = \sum _{\unicode{x03B2}=(m_i)} n_{\unicode{x03B2}} (\unicode{x03B3}_1 \cdot \unicode{x03B2}) (\unicode{x03B3}_2 \cdot \unicode{x03B2}) (\unicode{x03B3}_3 \cdot \unicode{x03B2}) \dfrac{\prod _{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}} \mathsf{q}_i^{m_i}}{1-\prod _{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}} \mathsf{q}_i^{m_i}}. \end{align}

The sum is over non-zero curve classes

\begin{align*}\unicode{x03B2}=(m_i)_{i \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}} = \sum m_i \mathrm {C}_i \in \mathrm {A}_1(\unicode{x1D4B3}).\end{align*}

Each term is a cubic polynomial in the input variables, multiplied by a specific rational function in the Novikov parameters and weighted by the GV invariant $n_{\unicode{x03B2}}$ . We will also use the term ‘quantum potential’ to refer to this analytic continuation.

Proof. Write the formal parameter $\mathsf{t}$ and the curve class $\unicode{x03B2}$ as sums

\begin{align*} \mathsf {t} = \sum _{i \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}} \mathsf {t}_i D_i, \qquad \unicode{x03B2} = \sum _{i \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}} m_i \mathrm {C}_i. \end{align*}

Applying the divisor equation together with the multiple cover formula (3.B) then gives

\begin{align*} \Phi ^{\unicode{x1D4B3}}_{\mathsf{t}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) & = \sum _{\unicode{x03B2}} \,\sum _{k \geq 0} \,\,\dfrac{1}{k!} \langle \unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3, \mathsf{t}, \ldots, \mathsf{t} \rangle ^{\unicode{x1D4B3}}_{0,k+3,\unicode{x03B2}}\\[3pt] & = \sum _{\unicode{x03B2}} \langle \unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3 \rangle ^{\unicode{x1D4B3}}_{0,3,\unicode{x03B2}} \,\bigg (\sum _{k\geq 0} \tfrac{(\mathsf{t} \cdot \unicode{x03B2})^k}{k!} \bigg )\\[3pt] & = \sum _{\unicode{x03B2}} N_{\unicode{x03B2}} (\unicode{x03B3}_1 \cdot \unicode{x03B2}) (\unicode{x03B3}_2 \cdot \unicode{x03B2}) (\unicode{x03B3}_3 \cdot \unicode{x03B2}) \exp \!\bigg ( \sum _{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}} m_i \mathsf{t}_i\bigg ) \\[3pt] & = \sum _{\unicode{x03B2}} n_{\unicode{x03B2}} \sum _{d \geq 1} \dfrac{1}{d^3} (\unicode{x03B3}_1 \cdot d\unicode{x03B2})(\unicode{x03B3}_2 \cdot d\unicode{x03B2})(\unicode{x03B3}_3 \cdot d\unicode{x03B2}) \exp \!\bigg ( \sum _{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}} dm_i \mathsf{t}_i\bigg ) \\[3pt] & = \sum _{\unicode{x03B2}} n_{\unicode{x03B2}} (\unicode{x03B3}_1 \cdot \unicode{x03B2}) (\unicode{x03B3}_2 \cdot \unicode{x03B2}) (\unicode{x03B3}_3 \cdot \unicode{x03B2}) \sum _{d \geq 1} \exp \!\bigg ( \sum _{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}} m_i \mathsf{t}_i\bigg )^d \\[3pt] & = \sum _{\unicode{x03B2}} n_{\unicode{x03B2}} (\unicode{x03B3}_1 \cdot \unicode{x03B2}) (\unicode{x03B3}_2 \cdot \unicode{x03B2}) (\unicode{x03B3}_3 \cdot \unicode{x03B2}) \sum _{d \geq 1} \big ( \Pi _{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}} \mathsf{q}_i^{m_i}\big )^d. \end{align*}

Note that this sum is finite, since $n_{\unicode{x03B2}}=0$ for all but finitely many $\unicode{x03B2}$ (Corollary 3.1). For fixed inputs $(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3)$ , the above is a formal power series in the Novikov parameters. It is the Taylor series for the following rational function, expanded about the point $(\mathsf{q}_i)_i = (0,\ldots, 0)$ , equivalently $(\mathsf{t}_i)_i = (-\infty, \ldots, -\infty )$ ,

\begin{align*} \Phi ^{\unicode{x1D4B3}}_{\mathsf{t}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) = \sum _{\unicode{x03B2}} n_{\unicode{x03B2}} (\unicode{x03B3}_1 \cdot \unicode{x03B2}) (\unicode{x03B3}_2 \cdot \unicode{x03B2}) (\unicode{x03B3}_3 \cdot \unicode{x03B2}) \dfrac{\prod _{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}} \mathsf{q}_i^{m_i}}{1-\prod _{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}} \mathsf{q}_i^{m_i}}. \end{align*}

This expression provides a natural analytic continuation of the quantum potential beyond the radius of convergence $\{ |\mathsf{q}_i| \lt 1 \mid i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}} \}$ .

Recall that after combining (2.I) with (2.F), we can ask which curve classes are restricted roots.

Corollary 3.4. Under the uniformly rescaled co-ordinates $\mathsf{p}_i := \mathsf{t}_i/2\unicode{x03C0}\sqrt{-1}$ on ${\mathrm{H}}^2(\unicode{x1D4B3};\mathbb{C})$ , the pole locus of the quantum potential is given by

\begin{align*} \bigcup_{\beta=(m_i)}\left\{\textstyle\sum_{i \in {{\unicode{x2110}}^{\kern 0.5pt\mathrm{c}}}} m_i \mathsf{p}_i \in {\mathbb{Z}} \right\}\end{align*}

where the union is over all restricted positive roots $\unicode{x03B2}$ . This is precisely the complexification of $\unicode{x210B}^{\mathsf{aff}}_{\unicode{x2110}}$ under the natural identification ${\mathrm{H}}^2(\unicode{x1D4B3};\mathbb{R})\cong \Theta _{\unicode{x2110}}$ dual to (2.F) .

Proof. The pole locus of (3.E) is the union of loci in the parameter space given by

where $\unicode{x03B2}=(m_i)_{i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}}=\sum m_i {\mathrm{C}}_i$ is such that $n_{\unicode{x03B2}}$ is non-zero. The first statement then follows from Corollary 3.1, since $n_{\unicode{x03B2}}$ is non-zero if and only if $\unicode{x03B2}$ is a restricted positive root. The second is an immediate consequence of the definition of $\unicode{x210B}^{\mathsf{aff}}_{\unicode{x2110}}$ in (2.D).

Example 3.5. For a single-curve flop with , the complexification of $\unicode{x210B}^{\mathsf{aff}}_{\unicode{x2110}}$ is

extended to infinity in both directions. The non-zero GV invariants are $n_{k{\mathrm{C}}}$ for $1 \leq k \leq 4$ .

5.1 Tracking GV invariants under flop

The benefit of Theorem4.4 is that both $\unicode{x1D4B3}$ and a flop $\unicode{x1D4B3}_i^+$ can be perturbed using essentially the same classifying map, and thus their curve invariants can be easily compared. This requires three combinatorial results.

Write $\mathsf{M}_i$ for the induced isomorphism in Lemma 5.1, so that the following diagrams commute.

(5.A)

In essence, the bases of $\mathfrak{h}_{\unicode{x03C9}_i(\unicode{x2110})}$ and $\mathfrak{h}_{\unicode{x2110}}$ really differ at only one element. Indeed, setting $e_j = \unicode{x03C0}_{\unicode{x2110}}(\unicode{x03B1}_j)$ , then $\{ e_j \mid j \not \in \unicode{x2110} \}$ is a basis for $\mathfrak{h}_{\unicode{x2110}}$ . On the other hand, for $\mathfrak{h}_{\omega _i(\unicode{x2110})}$ we abuse notation, setting $e_j=\unicode{x03C0}_{\unicode{x03C9}_i(\unicode{x2110})}(\unicode{x03B1}_j)$ whenever $j\notin \unicode{x2110}+i$ , and $e_i=\unicode{x03C0}_{\unicode{x03C9}_i(\unicode{x2110})}(\unicode{x03B1}_{\unicode{x03B9}_{\unicode{x2110}+i}(i)})$ . Then $\{e_j,e_i\mid j\notin \unicode{x2110}+i\}$ is a basis for $\mathfrak{h}_{\unicode{x03C9}_i(\unicode{x2110})}$ .

Lemma 5.2. For $i \in \unicode{x2110}^{\kern 0.5pt{\mathrm{c}}}$ the action of $\mathsf{M}_i$ is given in terms of the above bases as

\begin{align*} e_k \mapsto \begin {cases} e_k+\unicode {x03BB}_{k}e_i & \mbox {if }k\notin \unicode{x2110}+i\\ -e_i & \mbox {if }k=i \end {cases} \end{align*}

for some $\unicode{x03BB}_k \in \mathbb{Z}_{\geq 0}$ .

Proof. In (5.A), given any $\sum a_i\unicode{x03B1}_i$ , since $\ell _{\unicode{x2110}}\ell _{\unicode{x2110}+i}$ consists only of reflections $s_i$ with $i\in \unicode{x2110}+i$ , the map $\ell _{\unicode{x2110}}\ell _{\unicode{x2110}+i}$ cannot change the coefficient of any $a_j$ with $j\notin \unicode{x2110}+i$ . The claim that the induced map $\mathsf{M}_i$ sends $e_k\mapsto e_k+\unicode{x03BB}_{k}e_i$ if $k\notin \unicode{x2110}+i$ follows. We next claim that $\unicode{x03BB}_k$ is positive. In the decomposition of $\ell _{\unicode{x2110}}\ell _{\unicode{x2110}+i}\unicode{x03B1}_k$ into simple roots, there is at least some positive coefficient (namely, the coefficient of $\unicode{x03B1}_k$ , which is $1$ ). Hence, all coefficients must be positive, in particular, the coefficient of $\unicode{x03B1}_{\unicode{x03B9}_{\unicode{x2110}+i}(i)}$ . But under the induced map, this coefficient is what gives $\unicode{x03BB}_k$ in the claim.

Now, as in Lemma 5.1, for all Dynkin $\Gamma$ and all $i\in \Gamma$ , $\ell _\Gamma \unicode{x03B1}_i=-\unicode{x03B1}_{\unicode{x03B9}_\Gamma (i)}$ . Thus since $\unicode{x03B9}_{\unicode{x2110}+i}(i)\in \unicode{x2110}+i$ , it follows that

where we have used the facts that $\ell _{\unicode{x2110}}$ changes coefficients only in $\unicode{x2110}$ and that $\unicode{x03C0}_{\unicode{x2110}}$ forgets these.

The following is one of our main results. In order to obtain a unified statement, set $|\mathsf{M}_i\cdot \unicode{x03B2}|$ to be the curve class obtained from $\mathsf{M}_i\cdot \unicode{x03B2}$ by making every coefficient positive.

Theorem 5.4. With the notation as above, for any curve class $\unicode{x03B2}\in {\mathrm{A}}_1(\unicode{x1D4B3}_i^+)\cong \mathfrak{h}_{\unicode{x03C9}_i(\unicode{x2110})}$ ,

\begin{align*} n_{\unicode{x03B2},\unicode{x1D4B3}^+_i} & = \begin{cases} n_{\unicode{x03B2},\unicode{x1D4B3}} & \mbox{if }\unicode{x03B2}\in \mathbb{Z}e_i=\mathbb{Z}{\mathrm{C}}_i^+\\ n_{\kern 1pt\mathsf{M}_i\cdot \unicode{x03B2}, \unicode{x1D4B3}} & \mbox{else} \end{cases} \\ & = n_{\kern 1pt|\mathsf{M}_i\cdot \unicode{x03B2}|, \unicode{x1D4B3}}. \end{align*}

Proof. With respect to the notation in (4.A), perturbing $\unicode{x03BC}$ to $\unicode{x03BC}_t$ gives, by composition, a perturbation of $\nu$ to $\nu_t$ .

Set $w=\ell _{\unicode{x2110}}\ell _{\unicode{x2110}+i}$ . Then for any positive root $\unicode{x03B1}$ for which $\unicode{x03C0}_{\unicode{x03C9}_i(\unicode{x2110})}(\unicode{x03B1})=\unicode{x03B2}$ ,

\begin{align*} n_{\unicode{x03B2},\unicode{x1D4B3}^+_i} & =\left|\nu_t^{-1}(\unicode{x1D49F}_{\unicode{x03B1},\mathbb{C}}/W_{\omega _i(\unicode{x2110})})\right| \qquad \qquad \left( \mathrm{by}\ ({3.\mathrm{A}})\ \text{ applied to } \unicode{x1D4B3}_i^+\right)\\ & =\left|\unicode{x03BC}_t^{-1}(\unicode{x1D49F}_{w\cdot \unicode{x03B1},\mathbb{C}}/W_{\unicode{x2110}})\right|.\qquad \qquad ( \mathrm{since}\ \nu=(w \cdot )^{-1} \circ \unicode{x03BC}) \end{align*}

Now by (5.A) we have $\unicode{x03C0}_{\unicode{x2110}}(w\cdot \unicode{x03B1})=\mathsf{M}_i\circ \unicode{x03C0}_{\unicode{x03C9}_i(\unicode{x2110})}(\unicode{x03B1})=\mathsf{M}_i\cdot \unicode{x03B2}$ , so $w\cdot \unicode{x03B1}$ is a lift of $\mathsf{M}_i\cdot \unicode{x03B2}$ , albeit not necessarily a positive one.

Case 1. If $\unicode{x03B2}\notin \mathbb{Z}{\mathrm{C}}^+_i$ , then by Corollary 5.3 all entries of $\mathsf{M}_i\cdot \unicode{x03B2}$ are positive, and further as argued in the proof, $w\cdot \unicode{x03B1}$ is a positive root restricting to $\mathsf{M}_i\cdot \unicode{x03B2}$ . (3.A) then implies that $|\unicode{x03BC}_t^{-1}(\unicode{x1D49F}_{w\cdot \unicode{x03B1},\mathbb{C}}/W_{\unicode{x2110}})|=n_{\kern 1pt\mathsf{M}_i\cdot \unicode{x03B2}, \unicode{x1D4B3}}$ .

Case 2. If $\unicode{x03B2}\in \mathbb{Z}{\mathrm{C}}^+_i$ , then by Corollary 5.3, $\mathsf{M}_i\cdot \unicode{x03B2}=-\unicode{x03B2}$ . Arguing as above, it follows that $w\cdot \unicode{x03B1}$ is negative root restricting to $\mathsf{M}_i\cdot \unicode{x03B2}=-\unicode{x03B2}$ , and thus $-w\cdot \unicode{x03B1}$ is positive root restricting to $\unicode{x03B2}$ . But negating a root does not affect the hyperplane, and combining this fact with (3.A), it follows that

\begin{align*} |\unicode {x03BC}_t^{-1}(\unicode{x1D49F}_{w\cdot \unicode {x03B1},\mathbb {C}}/W_{\unicode{x2110}})|=|\unicode {x03BC}_t^{-1}(\unicode{x1D49F}_{-w\cdot \unicode {x03B1},\mathbb {C}}/W_{\unicode{x2110}})|=n_{\unicode{x03B2},\unicode{x1D4B3}}. \end{align*}

This covers both cases. For the final equality, note in case 1 that $|\mathsf{M}_i\cdot \unicode{x03B2}|=\mathsf{M}_i\cdot \unicode{x03B2}$ since all coefficients are already positive, and in case 2 note that $|\mathsf{M}_i\cdot \unicode{x03B2}|=|-\unicode{x03B2}|=\unicode{x03B2}$ .

Example 5.5. Consider the running Example 1.1. Then after flop of the right pink curve, by Theorem 4.4 and Example 4.2 we obtain . Hence, the restricted roots and thus the curve classes giving non-zero GV invariants, on the flopped space $\unicode{x1D4B3}_i^+$ , are as follows, where the hyperplanes are drawn in $\Theta _{\unicode{x03C9}_i(\unicode{x2110})}$ :

Write $1$ for the leftmost pink node, $2$ for the orange node, and $2'$ for the rightmost pink node (in $\unicode{x2110}$ ). Under this wall crossing, $\ell _{\unicode{x2110}}\ell _{\unicode{x2110}+i}$ is very large; however the morphism

\begin{align*} \mathsf {M}_i\colon \mathfrak {h}_{\unicode {x03C9}_i(\unicode{x2110})}\to \mathfrak {h}_{\unicode{x2110}} \end{align*}

is easily described: in the notation of Lemma 5.2, $\unicode{x03BB}_1=1$ and thus $\mathsf{M}_i$ sends $\unicode{x03BC}_1e_1+\unicode{x03BC}_2e_2\mapsto \unicode{x03BC}_1e_1+(\unicode{x03BC}_1-\unicode{x03BC}_2)e_{2'}$ . Under the dual transformation between the hyperplane arrangements in Example 1.1 and here, the pictures are drawn so that hyperplanes are sent to hyperplanes in such a way that the colours are preserved.

Indeed, $\mathsf{M}_i$ sends $01\mapsto 0-\!1$ , with all other restricted roots being permuted; e.g. $31\mapsto 32$ . In particular, by Theorem 5.4, the GV invariants on $\unicode{x1D4B3}_i^+$ can be obtained from the GV invariants on $\unicode{x1D4B3}$ as follows:

5.2 Tracking fundamental regions

The previous subsection tracked GV invariants from $\unicode{x1D4B3}$ to $\unicode{x1D4B3}^+_i$ . As with the movable cone, it is possible to fix $\unicode{x1D4B3}$ and track all other crepant resolutions back to $\unicode{x1D4B3}$ .

As notation, recall that the fixed $\unicode{x1D4B3}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ has an associated $\Theta _{\unicode{x2110}}$ in Notation 2.2, and recall from (5.A) that there is a map $\mathsf{M}_i^{-1}\colon \mathfrak{h}_{\unicode{x2110}}\to \mathfrak{h}_{\unicode{x03C9}_i(\unicode{x2110})}$ . Write

\begin{align*} \mathsf {N}_i\colon \Theta _{\unicode {x03C9}_i(\unicode{x2110})}\to \Theta _{\unicode{x2110}} \end{align*}

for the dual. Below, $\Theta _{\unicode{x2110}}$ will be temporarily be written $\Theta _{\unicode{x1D4B3}}$ , to allow for the flexibility of considering another crepant resolution $\unicode{x1D4B4}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ , which has associated $\Theta _{\unicode{x1D4B4}}$ .

Definition 5.7. Let $\unicode{x1D4B4}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ be a crepant resolution. Consider a chain of flops, each flopping a single irreducible curve, that links $\unicode{x1D4B4}$ to $\unicode{x1D4B3}$ , and the resulting maps

\begin{align*} \Theta _{\unicode{x1D4B4}}\xrightarrow {\mathsf {N}_{i_1}}\ldots \xrightarrow {\mathsf {N}_{i_t}}\Theta _{\unicode{x1D4B3}}. \end{align*}

The composition will be called the comparison map and will be written $\mathsf{N}\colon \Theta _{\unicode{x1D4B4}}\to \Theta _{\unicode{x1D4B3}}$ .

By [Reference Hirano and WemyssHW23, 4.8], the comparison map $\mathsf{N}$ is independent of the choice of chain of flops.

Example 5.10. Write $\unicode{x1D4B4}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ for the crepant resolution obtained after flop in Example 5.5. Then the region $\mathsf{N}(\mathsf{Fund}_{\unicode{x1D4B4}})$ is illustrated below, where for clarity we have illustrated the images of the $x$ and $y$ co-ordinates in Example 5.5 under the map $\mathsf{N}$ .

It is visually clear that both $\mathsf{Fund}_{\unicode{x1D4B3}}$ in (1.D) and $\mathsf{N}(\mathsf{Fund}_{\unicode{x1D4B4}})$ above individually generate $\unicode{x210B}_{\unicode{x2110}}^{\mathsf{aff}}$ , via translation, and that $\mathsf{Fund}_{\unicode{x1D4B3}}$ and $\mathsf{N}(\mathsf{Fund}_{\unicode{x1D4B4}})$ are different.

5.3 Crepant transformation conjecture

We make no attempt at a comprehensive summary of the Crepant Transformation Conjecture (CTC) and instead refer the reader to [Reference Coates and RuanCR13, Reference Coates, Iritani and JiangCIJ18, Reference Bryan and GraberBG09, Reference LeeLee11]. For a pair of smooth varieties related via a sequence of flops, the conjecture asserts that their quantum potentials should coincide, under a suitable identification of (co)homologies and analytic continuation in the Novikov parameters. There has been extensive work on this conjecture within both algebraic and symplectic geometry [Reference Li and RuanLR01, Reference McLeanMcL20, Reference Lee, Lin and WangLLW10, Reference Lee, Lin and WangLLW16a, Reference Lee, Lin and WangLLW16b, Reference Lee, Lin, Qu and WangLLQW16].

Here we prove the CTC for germs of isolated $3$ -fold flops, as a direct application of the expression for the quantum potential in Theorem3.3, together with the construction of flops via simultaneous partial resolutions in Theorem4.4. This gives the first algebraic-geometric proof of the CTC for flops of arbitrary type (for recent symplectic developments; see [Reference McLeanMcL20]).

As before, consider a curve ${\mathrm{C}}_i \subseteq \unicode{x1D4B3}$ , and let $\unicode{x1D4B3}_i^+$ be the flop of $\unicode{x1D4B3}$ at ${\mathrm{C}}_i$ . Recall that the following vector spaces are based on the sets of exceptional curves:

\begin{align*} \mathfrak {h}_{\unicode{x2110},\mathbb {C}}= \mathrm {A}_1(\unicode{x1D4B3})_{\mathbb {C}}= \langle {\mathrm{C}}_j \mid j \in {{\unicode{x2110}}^{\kern 0.5pt\mathrm{c}}} \rangle_{{\mathbb{C}}}, \qquad \mathfrak{h}_{\unicode{x03C9}_i({\unicode{x2110}}),{\mathbb{C}}} = {\mathrm{A}}_1({\unicode{x1D4B3}}_i^+)_{{\mathbb{C}}} = \left\langle {\mathrm{C}}_j,{\mathrm{C}}_i^+\mid j\notin{\unicode{x2110}}+i \right\rangle_{{\mathbb{C}}}.\end{align*}

where as in Lemma 5.2, we abuse notation by denoting the flopped curve ${\mathrm{C}}_i^+$ instead of ${\mathrm{C}}_{{\unicode{x03B9}_{\unicode{x2110}+i}(i)}}^+$ . As explained in Section 5.1, there is an explicit transformation matrix

\begin{align*} \mathsf {M}_i \colon \mathrm {A}_1\left(\unicode{x1D4B3}_i^+\right)_{\mathbb {C}} \to \mathrm {A}_1(\unicode{x1D4B3})_{\mathbb {C}}.\end{align*}

This is the complexification of the matrix $\mathsf{M}_i$ from earlier, but we use the same symbol. Let $\mathsf{N}_i$ be the matrix dual to $\mathsf{M}_i^{-1}$ , which can be viewed as the linear map

\begin{align*} \mathsf{N}_i \colon \mathrm{H}^2\left(\unicode{x1D4B3}_i^+;{\mathbb{C}}\right) \to \mathrm{H}^2(\unicode{x1D4B3};{\mathbb{C}})\end{align*}

with the property that $\mathsf{N}_i \unicode{x03B3} \cdot \unicode{x03B2} = \unicode{x03B3} \cdot \mathsf{M}_i^{-1} \unicode{x03B2}$ for $\unicode{x03B3} \in {\mathrm{H}}^2(\unicode{x1D4B3}_i^+;\mathbb{C})$ and $\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3})_{\mathbb{C}}$ . We notate the Novikov co-ordinates on the parameter spaces for the quantum potentials by

\begin{align*} \begin {array}{ll} \{ \mathsf {q}_j \mid j \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}\} & \text {on $\mathrm {A}_1(\unicode{x1D4B3})_{\mathbb {C}}$}, \\ \{ \mathsf {r}_j \mid j \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}} \} & \text {on $\mathrm {A}_1(\unicode{x1D4B3}_i^+)_{\mathbb {C}}$.} \end {array} \end{align*}

The matrix $\mathsf{M}_i^{-1}$ defines a monomial co-ordinate transformation relating the two sets of Novikov parameters. Writing monomials as

\begin{align*} \mathsf {q}^{\unicode{x03B2}} := \prod _{j \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}} \mathsf {q}_j^{m_j},\qquad \mathsf {r}^{\unicode{x03B2}} := \prod _{j \in \unicode{x2110}^{\kern 0.5pt\mathrm {c}}} \mathsf {r}_j^{m_j}\end{align*}

this is given by the equation

(5.B) \begin{align} \mathsf{q}^{\unicode{x03B2}} = \mathsf{r}^{\mathsf{M}_i^{-1} \unicode{x03B2}}. \end{align}

By Lemma 5.2, $\mathsf{q}_i = \mathsf{r}_i^{-1}$ whereas every other $\mathsf{q}_{j}$ is a monomial in the $\mathsf{r}_j$ with non-negative coefficients.

Lastly, recall the quantum potentials of $\unicode{x1D4B3}$ and $\unicode{x1D4B3}_i^+$ from Section 3.3, which to ease notation will be written $\Phi$ and $\Phi ^+$ , namely,

\begin{align*} \Phi _{\mathsf{q}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) & := \Phi _{\mathsf{q}}^{\unicode{x1D4B3}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3), \\ \Phi _{\mathsf{r}}^+(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) & := \Phi _{\mathsf{r}}^{\unicode{x1D4B3}_i^+}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3). \end{align*}

The equation (5.B) will be used to express the quantum potential of $\unicode{x1D4B3}$ in the variables $\mathsf{r}$ , and this will be denoted $\Phi _{\mathsf{r}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3)$ .

Corollary 5.13 (Crepant Transformation Conjecture). On the $\mathsf{r}$ parameter space, the quantum potentials of $\unicode{x1D4B3}$ and $\unicode{x1D4B3}_i^+$ coincide, up to the following explicit term, which does not depend on the Novikov parameters

(5.C) \begin{align} \Phi ^{+}_{\mathsf{r}}\big (\unicode{x03B3}_1, \unicode{x03B3}_2,\unicode{x03B3}_3\big ) - \Phi _{\mathsf{r}}(\mathsf{N}_i \unicode{x03B3}_1,\mathsf{N}_i \unicode{x03B3}_2,\mathsf{N}_i \unicode{x03B3}_3) = -(\unicode{x03B3}_1 \cdot {\mathrm{C}}_i^+)(\unicode{x03B3}_2 \cdot {\mathrm{C}}_i^+)(\unicode{x03B3}_3\cdot {\mathrm{C}}_i^+) \sum _{k \geq 1} k^3 n_{k{\mathrm{C}}_i,\unicode{x1D4B3}}. \end{align}

More precisely, the pole loci are transformed into each other via (5.B), and away from these, the analytic continuations constructed in Theorem3.3 coincide.

Remark 5.15. The expansion points for the quantum potentials differ, as

\begin{align*} ({\mathsf{r}}_j)_{j \in {{\unicode{x2110}}^{\kern 0.5pt\mathrm{c}}}} = (0,\ldots,0) \ \Leftrightarrow \ ({\mathsf{q}}_j)_{j \in {{\unicode{x2110}}^{\kern 0.5pt\mathrm{c}}}} = (0,\ldots,0,\infty,0,\ldots,0), \end{align*}

with $\infty$ in the $i$ th position. Thus, the term $\Phi _{\mathsf{r}}(\mathsf{N}_i \unicode{x03B3}_1,\mathsf{N}_i \unicode{x03B3}_2,\mathsf{N}_i \unicode{x03B3}_3)$ is analytically continued from $\mathsf{q}_i=0$ to $\mathsf{q}_i=\infty$ , the analytic continuation occurring precisely in the Novikov parameter corresponding to the flopped curv

Proof of 5.13. We explicitly match both sides, using our knowledge of the structure of the quantum potentials (Theorem3.3) and the behaviour of the GV invariants under the flop (Theorem5.4).

Separating curve classes according to whether or not they are a multiple of ${\mathrm{C}}_i$ , the quantum potential for $\unicode{x1D4B3}$ may be written as the sum of contributions

\begin{align*}\Phi_{{\mathsf{q}}}({\mathsf{N}}_i\gamma_1,{\mathsf{N}}_i\gamma_2,{\mathsf{N}}_i\gamma_3) & = \mathsf{G}_{{\mathsf{q}}}({\mathsf{N}}_i \gamma_1,{\mathsf{N}}_i \gamma_2, {\mathsf{N}}_i \gamma_3) + \mathsf{H}_{{\mathsf{q}}}({\mathsf{N}}_i \gamma_1,{\mathsf{N}}_i \gamma_2, {\mathsf{N}}_i \gamma_3),\end{align*}

where

\begin{align*} \mathsf{G}_{\mathsf{q}}(\mathsf{N}_i \unicode{x03B3}_1,\mathsf{N}_i \unicode{x03B3}_2, \mathsf{N}_i \unicode{x03B3}_3) & =\quad \, \sum _{k \geq 1} n_{k{\mathrm{C}}_i,\unicode{x1D4B3}} \, (\mathsf{N}_i\unicode{x03B3}_1 \cdot k{\mathrm{C}}_i)(\mathsf{N}_i\unicode{x03B3}_2 \cdot k{\mathrm{C}}_i) (\mathsf{N}_i\unicode{x03B3}_3 \cdot k{\mathrm{C}}_i) \dfrac{\mathsf{q}_i^k}{1-\mathsf{q}_i^k} \\ \mathsf{H}_{\mathsf{q}}(\mathsf{N}_i \unicode{x03B3}_1,\mathsf{N}_i \unicode{x03B3}_2, \mathsf{N}_i \unicode{x03B3}_3) & = \sum _{\substack{\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3}) \\ \unicode{x03B2} \neq k{\mathrm{C}}_i}} n_{\unicode{x03B2},\unicode{x1D4B3}} \, (\mathsf{N}_i\unicode{x03B3}_1\cdot \unicode{x03B2})(\mathsf{N}_i\unicode{x03B3}_2\cdot \unicode{x03B2})(\mathsf{N}_i\unicode{x03B3}_3\cdot \unicode{x03B2}) \dfrac{\mathsf{q}^{\unicode{x03B2}}}{1-\mathsf{q}^{\unicode{x03B2}}}. \end{align*}

Similarly, write the quantum potential of $\unicode{x1D4B3}_i^+$ as

\begin{align*} \Phi ^{+}_{\mathsf{r}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) & = \mathsf{G}^{+}_{\mathsf{r}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) +\mathsf{H}^{+}_{\mathsf{r}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3), \end{align*}

where

\begin{align*} \mathsf{G}^{+}_{\mathsf{r}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) & = \quad \, \sum _{k \geq 1} n_{k{\mathrm{C}}_i^+,\unicode{x1D4B3}_i^+} \, (\unicode{x03B3}_1 \cdot k{\mathrm{C}}_i^+)(\unicode{x03B3}_2 \cdot k{\mathrm{C}}_i^+) (\unicode{x03B3}_3 \cdot k{\mathrm{C}}_i^+) \dfrac{\mathsf{r}_i^{k}}{1-\mathsf{r}_i^{k}} \\ \mathsf{H}^{+}_{\mathsf{r}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) & = \sum _{\substack{\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3}) \\ \unicode{x03B2} \neq k{\mathrm{C}}_i}} n_{\unicode{x03B2},\unicode{x1D4B3}_i^+} \, (\unicode{x03B3}_1\cdot \unicode{x03B2})(\unicode{x03B3}_2\cdot \unicode{x03B2})(\unicode{x03B3}_3\cdot \unicode{x03B2}) \dfrac{\mathsf{r}^{\unicode{x03B2}}}{1-\mathsf{r}^{\unicode{x03B2}}}. \end{align*}

We begin with the $\mathsf{G}$ terms. Using (5.B) to change variables from $\mathsf{q}$ to $\mathsf{r}$ gives

\begin{align*}\mathsf{G}_{{\mathsf{r}}}({\mathsf{N}}_i\gamma_1,{\mathsf{N}}_i\gamma_2,{\mathsf{N}}_i\gamma_3) & = \sum_{k \geq 1} n_{k{\mathrm{C}}_i,{\unicode{x1D4B3}}} \, ({\mathsf{N}}_i\gamma_1 \cdot k{\mathrm{C}}_i)({\mathsf{N}}_i\gamma_2 \cdot k{\mathrm{C}}_i) ({\mathsf{N}}_i\gamma_3 \cdot k{\mathrm{C}}_i) \dfrac{{\mathsf{r}}_i^{-k}}{1-{\mathsf{r}}_i^{-k}} \\& = (\gamma_1 \cdot {\mathrm{C}}_i^+) (\gamma_2 \cdot {\mathrm{C}}_i^+) (\gamma_3 \cdot {\mathrm{C}}_i^+) \sum_{k \geq 1} k^3 n_{k{\mathrm{C}}_i,{\unicode{x1D4B3}}} \dfrac{1}{1-r_i^k}.\end{align*}

where the second equality follows from $\mathsf{N}_i \unicode{x03B3} \cdot {\mathrm{C}}_i = \unicode{x03B3} \cdot \mathsf{M}_i^{-1} {\mathrm{C}}_i = - \unicode{x03B3} \cdot {\mathrm{C}}_i^+$ and the equality

\begin{align*} \dfrac {r_i^{-k}}{1-r_i^{-k}} = \dfrac {1}{r_i^k - 1}. \end{align*}

Using $n_{k{\mathrm{C}}_i,\unicode{x1D4B3}} = n_{k{\mathrm{C}}_i^+,\unicode{x1D4B3}_i^+}$ by Theorem5.4, the difference $\mathsf{G}^{+}_{\mathsf{r}}(\unicode{x03B3}_1,\unicode{x03B3}_2,\unicode{x03B3}_3) - \mathsf{G}_{\mathsf{r}}(\mathsf{N}_i\unicode{x03B3}_1,\mathsf{N}_i\unicode{x03B3}_2,\mathsf{N}_i\unicode{x03B3}_3)$ is equal to

\begin{align*} & (\unicode{x03B3}_1 \cdot {\mathrm{C}}_i^+)(\unicode{x03B3}_2 \cdot {\mathrm{C}}_i^+)(\unicode{x03B3}_3\cdot {\mathrm{C}}_i^+) \sum _{k \geq 1} k^3 n_{k{\mathrm{C}}_i,\unicode{x1D4B3}} \left ( \dfrac{r_i^k}{1-r_i^k} - \dfrac{1}{1-r_i^k} \right ) \\ = & -(\unicode{x03B3}_1 \cdot {\mathrm{C}}_i^+)(\unicode{x03B3}_2 \cdot {\mathrm{C}}_i^+)(\unicode{x03B3}_3\cdot {\mathrm{C}}_i^+) \sum _{k \geq 1} k^3 n_{k{\mathrm{C}}_i,\unicode{x1D4B3}}. \end{align*}

We next examine the $\mathsf{H}$ terms. Note that for $\unicode{x03B2} \in {\mathrm{A}}_1(\unicode{x1D4B3})$ , we have $\unicode{x03B2} \in \mathbb{Z} {\mathrm{C}}_i$ if and only if $\mathsf{M}_i^{-1}\unicode{x03B2} \in \mathbb{Z} {\mathrm{C}}_i^+$ . Consequently,

\begin{align*}\mathsf{H}_{{\mathsf{r}}}({\mathsf{N}}_i \gamma_1,{\mathsf{N}}_i \gamma_2,{\mathsf{N}}_i\gamma_3) & = \sum_{\substack{\beta \in {\mathrm{A}}_1({\unicode{x1D4B3}}) \\ \beta \neq k{\mathrm{C}}_i}} n_{\beta,{\unicode{x1D4B3}}} \, ({\mathsf{N}}_i\gamma_1\cdot\beta)({\mathsf{N}}_i\gamma_2\cdot\beta)({\mathsf{N}}_i\gamma_3\cdot\beta) \dfrac{{\mathsf{r}}^{{\mathsf{M}}_i^{-1} \beta}}{1-{\mathsf{r}}^{{\mathsf{M}}_i^{-1} \beta}} \\& = \sum_{\substack{\beta \in {\mathrm{A}}_1({\unicode{x1D4B3}}_i^+) \\ \beta \neq k{\mathrm{C}}_i^+}} n_{{\mathsf{M}}_i \beta,{\unicode{x1D4B3}}} \, ({\mathsf{N}}_i \gamma_1 \cdot {\mathsf{M}}_i \beta)({\mathsf{N}}_i \gamma_2 \cdot {\mathsf{M}}_i \beta)({\mathsf{N}}_i \gamma_3 \cdot {\mathsf{M}}_i \beta) \dfrac{r^{\beta}}{1-r^{\beta}} \\& = \sum_{\substack{\beta \in {\mathrm{A}}_1({\unicode{x1D4B3}}_i^+) \\ \beta \neq k{\mathrm{C}}_i^+}} n_{\beta,{\unicode{x1D4B3}}_i^+} \, (\gamma_1 \cdot \beta)(\gamma_2 \cdot \beta)(\gamma_3 \cdot \beta) \dfrac{r^{\beta}}{1-r^{\beta}} \\& = \mathsf{H}^{+}_{{\mathsf{r}}}(\gamma_1,\gamma_2,\gamma_3).\end{align*}

where the penultimate equality holds since $\mathsf{N}_i \unicode{x03B3} \cdot \mathsf{M}_i \unicode{x03B2} = \unicode{x03B3} \cdot \mathsf{M}_i^{-1} \mathsf{M}_i \unicode{x03B2} = \unicode{x03B3} \cdot \unicode{x03B2}$ and $n_{\mathsf{M}_i \unicode{x03B2},\unicode{x1D4B3}} = n_{\unicode{x03B2},\unicode{x1D4B3}_i^+}$ again by Theorem5.4. Combining the comparison of the $\mathsf{G}$ terms with the comparison of the $\mathsf{H}$ terms gives (5.C), as required.

5.4 The contraction algebra under flop

The flopping contraction $\unicode{x1D4B3}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ has an associated contraction algebra ${\mathrm{A}}_{{\mathrm{con}}}$ , defined using noncommutative deformation theory [Reference Donovan and WemyssDW16, Reference Donovan and WemyssDW19]. After flopping a single curve ${\mathrm{C}}_i$ to obtain $\unicode{x1D4B3}_i^+\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ , noncommutative deformation theory associates to this another contraction algebra, written $\nu_i{\mathrm{A}}_{{\mathrm{con}}}$ . The algebra $\nu_i{\mathrm{A}}_{{\mathrm{con}}}$ can be intrinsically obtained from ${\mathrm{A}}_{{\mathrm{con}}}$ via a certain mutation procedure, and in fact ${\mathrm{A}}_{{\mathrm{con}}}$ and $\nu_i{\mathrm{A}}_{{\mathrm{con}}}$ are derived equivalent [Reference AugustAug20b]. Both ${\mathrm{A}}_{{\mathrm{con}}}$ and $\nu_i{\mathrm{A}}_{{\mathrm{con}}}$ are finite dimensional algebras [Reference Donovan and WemyssDW16, 2.13].

Fix the GV invariants $n_{\unicode{x03B2}}$ associated to $\unicode{x1D4B3}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}$ , and then Toda’s dimension formula (see A.4) asserts that

\begin{align*} \dim _{\mathbb {C}}\mathrm {A}_{\mathrm {con}}=\sum _{\unicode{x03B2}\in \mathrm {A}_1(\unicode{x1D4B3})}n_{\unicode{x03B2}} \big (\,\unicode{x03B2}\cdot \unicode {x1D7D9}\big )^2 \end{align*}

In many, but not all, cases, the dimension of ${\mathrm{A}}_{{\mathrm{con}}}$ is in fact enough to recover the $n_{\unicode{x03B2}}$ . The next result asserts that the numbers $n_{\unicode{x03B2}}$ associated to ${\mathrm{A}}_{{\mathrm{con}}}$ , together with the matrix $\mathsf{M}^{-1}_i$ , completely determine the dimension of $\nu_i{\mathrm{A}}_{{\mathrm{con}}}$ .

Corollary 5.16. Under mutation at vertex $i$ , equivalently flop at ${\mathrm{C}}_i$ ,

\begin{align*} \dim_{\mathbb{C}}\nu_i{\mathrm{A}_{{\mathrm{con}}}}=\sum_{\beta\in {\mathrm{A}}_1({\unicode{x1D4B3}})}n_\beta \big(\,(\mathsf{M}^{-1}_i\beta)\cdot \mathbb{1}\big)^2 \end{align*}

where $\mathsf{M}_i$ is the explicit matrix in (5.A) .

Proof. Combining previous results, it follows that

where the point is that by Corollary 5.3, the sign issue doesn’t matter once we square.

In particular, it is possible to compute the dimension of $\nu_i{\mathrm{A}}_{{\mathrm{con}}}$ without first having to present it.

Example 5.17. As in [Reference Smith and WemyssSW23], consider the $cA_2$ example $\unicode{x1D4AD}_k={\mathbb{C}[\![u,v,x,y]\!] \over uv-xy(x^k+y)}$ for $k\geq 1$ and the specific crepant resolution $\unicode{x1D4B3}\to \mathop{\mathrm{Spec}}\nolimits \unicode{x1D4AD}_k$ described in [Reference Smith and WemyssSW23, 3.1], obtained first by blowing up $(u,y)$ and then $(u,x)$ . In this case, as explained in [Reference Smith and WemyssSW23, Subsection 6.2] ${\mathrm{A}}_{{\mathrm{con}}}$ can be presented as

We can immediately read off the GV invariants, namely, $n_{1,0}=1$ , $n_{0,1}=1$ , and $n_{1,1}=k$ . Thus $\dim _{\mathbb{C}}{\mathrm{A}}_{{\mathrm{con}}}=n_{1,0}\cdot (1+0)^2+n_{0,1}\cdot (0+1)^2+n_{1,1}\cdot (1+1)^2$ , which equals $1+1+4k=4k+2$ .

We now flop the right-hand curve. In this Type $A$ example, $\mathsf{M}_i^{-1}$ sends $(1,0)\mapsto (1,1)$ , $(1,1)\mapsto (1,0)$ and $(0,1)\mapsto (0,-1)$ . Thus, by Corollary 5.16,

\begin{align*} \dim _{\mathbb {C}}\nu_i\mathrm {A}_{\mathrm {con}}=n_{1,0}\cdot (1+1)^2+n_{0,1}\cdot (0+1)^2+n_{1,1}\cdot (0-1)^2, \end{align*}

which equals $4+1+k=k+5$ . In particular, $\nu_i{\mathrm{A}}_{{\mathrm{con}}}\ncong {\mathrm{A}}_{{\mathrm{con}}}$ provided that $k\geq 2$ .