3.1 Spherical twists

The functor $\Phi $ of Eq. (1.2) is the restriction to of the projection to the admissible subcategory in $D^b(\mathcal {A})$ . It is an equivalence:

Figure 1

The spherical functor F.

Proof. The right adjoint of F is $R=\operatorname {\mathrm {Hom}}(E_x,-)\oplus \operatorname {\mathrm {Hom}}(E_y,-)$ , and as a consequence of Remark 3.2 there is also a left adjoint $L=RS[1]$ . The cotwist $C=\operatorname {\mathrm {cone}}(1\to RF)[-1]$ is the following bimodule over $\mathbb {C}\oplus \mathbb {C}$ :

$$ \begin{align*} \begin{bmatrix} 0 & \operatorname{\mathrm{Hom}}(E_x,E_y) \\ \operatorname{\mathrm{Hom}}(E_y,E_x) & 0 \end{bmatrix} = \begin{bmatrix} 0 & \mathbb{C}[-2] \\ \mathbb{C}[-2] & 0 \end{bmatrix}. \end{align*} $$

This is a shift of the symmetry swapping the two points – an autoequivalence – and satisfies $R=CL[1]$ because

$$ \begin{align*} \operatorname{\mathrm{Hom}}(E_x,-) &= \operatorname{\mathrm{Hom}}(-,S(E_x))^{\vee_X} \\ &= \operatorname{\mathrm{Hom}}(-,E_y[1])^{\vee_X} = \operatorname{\mathrm{Hom}}(E_y,S(-))[-1]. \end{align*} $$

Hence F is spherical by the two-out-of-four theorem [Reference Anno and LogvinenkoAL, Theorem 1.1].

It follows that the twist $T=\operatorname {\mathrm {cone}}(FR\to 1)$ is an autoequivalence, with inverse the dual twist $T^{-1}=\operatorname {\mathrm {cone}}(1\to FL)[-1]$ . As $L=RS[1]$ , this is given by

$$ \begin{align*} T^{-1}(G) = \operatorname{\mathrm{cone}}\biggl(G \to \bigl(\operatorname{\mathrm{Hom}}(G,E_x)^{\vee_{\mathbb{C}}}\otimes E_x\bigr) \oplus\bigl(\operatorname{\mathrm{Hom}}(G,E_y)^{\vee_{\mathbb{C}}}\otimes E_y\bigr)\biggr)[-1], \end{align*} $$

which agrees with the projection to when .

Hence $\Phi $ is an equivalence, as $T^{-1}$ is an autoequivalence. By symmetry the same functor $T^{-1}$ also gives the other direction of the flop. We write $\Phi =T^{-1}$ for this extension of both functors as a slight abuse of notation.

By [Reference BarbacoviBar, Theorem 4.1.3] there is a simple way of expressing the flop-flop autoequivalence $\Phi ^2|_{D^b(\mathcal {A}_x)}$ as a spherical twist. Since $D^b(\mathcal {A}_x)\subset D^b(\mathcal {A})$ is the complement of an exceptional object, it is the dual twist around a spherical object: the image of $E_x$ in . Similar holds for $D^b(\mathcal {A}_y)$ . Explicitly, these spherical objects are and .

3.2 Crepancy

In view of Remarks 3.2 and 3.4, we make the following definition.

Recall that strong crepancy was the condition $S=\mathrm {id}_{\mathscr {C}}$ , while weak crepancy is the condition that $S(\ker (\pi _*))\subset \ker (\pi _*)$ by [Reference Kuznetsov and ShinderKS2, Lemma 5.8], provided we revise our definition of ‘strong categorical resolution’ to also require $\pi _*$ to be a categorical contraction [Reference Kuznetsov and ShinderKS2, Definition 5.1]. It is then clear that strong crepancy implies fair crepancy with index 0, and fair crepancy of any index implies weak crepancy. The reverse implications do not hold in general.

If we assume that $\mathscr {C}$ is suitably birational, so that $\operatorname {\mathrm {Hom}}_{\mathscr {C}}(x,y)$ for $x,y\in \ker (\pi _*)$ is a module supported at the singularity in X, then for an isolated Gorenstein singularity the relative Serre functor on $\ker (\pi _*)$ is a shift of the absolute Serre functor (as in Remark 3.2). In this situation fair crepancy of index n is equivalent to requiring that $\ker (\pi _*)$ is $(n+\dim X)$ -Calabi–Yau, implying that $\ker (\pi _*)$ has no nontrivial semiorthogonal decompositions. The resolution then cannot be made smaller via semiorthogonal decomposition, assuming $\mathscr {C}$ is connected, so we find it reasonable to expect that fairly crepant categorical resolutions do not factor nontrivially through further categorical resolutions. Note that ‘minimal’ categorical resolutions are conjectured to have a universal factorization property [Reference Bondal and OrlovBO], and in particular should be unique, although the correct definitions to ensure this are unclear.

Assuming uniqueness, nonvanishing of the index of crepancy would be a sufficient nonexistence criterion for (geometric) crepant resolutions of rational singularities, and also other strongly crepant categorical resolutions, such as NCCRs. We find results consistent with this prediction in Remarks 3.12 and 3.13.

3.3 General nodal curves

It was shown in [Reference Burban and DrozdBD] that $\mathcal {A}$ gives a categorical resolution for any nodal or cuspidal curve. We briefly note that exceptional objects with the same properties as $E_x$ and $E_y$ exist in $D^b(\mathcal {A})$ for all nodal curves.

Indeed, suppose X is a curve with a node at p, and take the normalization $\pi :\widetilde X\to X$ . Then $\pi ^{-1}(p)$ consists of two reduced points $q_1$ and $q_2$ , and these give simple modules $E_1=\pi _*\mathcal {O}_{q_1}$ and $E_2=\pi _*\mathcal {O}_{q_2}$ over $\mathcal {A}=\operatorname {\mathrm {End}}_X(\mathcal {O}_X\oplus \pi _*\mathcal {O}_{\widetilde X})$ .

Proof. We assume without loss of generality that X is affine. The algebra $\mathcal {A}$ has projective modules $P_X$ and $P_{\widetilde X}$ , with

(3.3) $$ \begin{align} \begin{aligned} \operatorname{\mathrm{End}}(P_X) &= \mathcal{O}_X, \qquad \operatorname{\mathrm{Hom}}(P_X,P_{\widetilde X}) = \mathcal{O}_{\widetilde X}, \\ \operatorname{\mathrm{End}}(P_{\widetilde X}) &= \mathcal{O}_{\widetilde X}, \qquad \operatorname{\mathrm{Hom}}(P_{\widetilde X},P_X) = \mathcal{I}. \end{aligned} \end{align} $$

Here $\mathcal {I}$ is the ideal cutting out $\operatorname {\mathrm {Sing}}(X)$ , equal to the conductor $\operatorname {\mathrm {Ann}}_X(\mathcal {O}_{\widetilde X}/\mathcal {O}_X)$ . This is all from [Reference Burban and DrozdBD, Proposition 2.2]. The points $q_i\in {\widetilde X}$ are divisors $\{s_i=0\}$ , and $g=s_1s_2$ satisfies $\mathcal {I}=\mathcal {O}_{\widetilde X}\cdot g$ . We then have the following resolutions:

$$ \begin{align*} &P_{\widetilde X} \xrightarrow{(-g,\,s_2)^T} P_X\oplus P_{\widetilde X} \xrightarrow{(1,\,s_1)} P_{\widetilde X} \to E_1, \\ &P_{\widetilde X} \xrightarrow{(-g,\,s_1)^T} P_X\oplus P_{\widetilde X} \xrightarrow{(1,\,s_2)} P_{\widetilde X} \to E_2, \end{align*} $$

with maps understood via Eq. (3.3). Using these (1) is a direct computation, and we also find that $(E_i)^{\vee _{\mathcal {A}}}=E_j[-2]$ , where $\{i,j\}=\{1,2\}$ and $E_j$ is viewed as a left-module. Then (2) follows from $\operatorname {\mathrm {Hom}}_X(\mathcal {O}_p,\mathcal {O}_X)=\mathcal {O}_p[-1]$ .

Such a resolution is fairly crepant of index 2, but not strongly crepant since $2\ne 0$ . This extends the description of [Reference CattaniCGL⁺ ] for $\dim (X)\ge 2$ , continuing the pattern of 2- or 3-spherical objects by parity of dimension.Footnote ² The zero-dimensional node also follows this pattern; see Example 3.8.

Remark 3.12. In particular, fairly crepant resolutions exist for isolated nodal singularities of all dimensions. The indices of these resolutions are displayed in Table 1. Note that the index is 0 in exactly the two dimensions where geometric crepant resolutions exist; the constructed categorical resolutions are equivalent to the geometric resolutions in these cases, and are therefore strongly crepant.

Table 1

Indices of crepancy for categorical resolutions of nodes.

4.1 The exoflop

We apply the idea of [Reference AspinwallAsp] to our situation as follows. The derived category of $X=\{xy=0\}$ is equivalent to a category of matrix factorizations $D^b(\mathbb {A}^3,xyz)$ , where the R-charge weights are $|x|=|y|=0$ and $|z|=2$ . This is an instance of Knörrer periodicity ([Reference KnörrerKnö], cf. [Reference ShipmanShi]), which relates the derived category of a complete intersection – or more generally factorizations on a complete intersection – to factorizations on the associated vector bundle. Now $\widetilde X$ lies inside the blowup $\operatorname {\mathrm {Tot}}_{\mathbb {P}^1_{x:y}}\mathcal {O}(-1)$ of the plane, cut out by the section $xy$ of $\mathcal {O}(2)$ , and by Knörrer periodicity it is also derived equivalent to an LG-model. We get a square

where z is the coordinate on $\mathcal {O}(-2)$ . This LG-model has a weighted flip:

with $\mathbb {P}^{1:2}$ viewed as an orbifold. By [Reference Ballard, Favero and KatzarkovBFK] there is a decomposition

(4.1) $$ \begin{align} D^b(\overline U,xyz) = \langle D^b(\mathrm{pt}),D^b(\widetilde X)\rangle, \end{align} $$

where the exceptional object is the sheafy matrix factorization $\mathcal {O}_{\mathbb {P}^{1:2}}(-1)$ on the zero section.Footnote ⁴ Here $D^b(\widetilde X)$ is embedded via Knörrer periodicity followed by the flip.

The key observation is that $\overline U$ is a partial compactification of the original LG-model $(\mathbb {A}^3,xyz)$ , which is the open set $\{w\ne 0\}$ . Restriction to this open set gives a functor $D^b(\overline U,xyz)\to D^b(X)$ , and the adjoint pushforward is defined for factorizations supported at $\{z=0\}$ , which corresponds to ${\mathfrak {Perf}}(X)$ under Knörrer periodicity. Restriction and pushforward compose to the identity, and $D^b(\overline U,xyz)$ is smooth (from compactification of the critical locus, or just Eq. (4.1)), so this is a categorical resolution. It can be checked to be equivalent to $D^b(\mathcal {A})$ .Footnote ⁵

4.2 Removing orbifold points

To better understand these LG-models, we should consider the critical locus of the superpotential (Fig. 2). For $(\mathbb {A}^3,xyz)$ this is the node itself in the plane $\{z=0\}$ , together with a vertical $\mathbb {A}^1_z$ branch through the singularity, with positive R-charge. This indicates the unbounded Ext groups supported at the singularity of X; indeed, the open set $\{z\ne 0\}$ of the LG-model is equivalent by Knörrer periodicity to $D^b(\mathbb {A}^1_z-\{0\})$ , where $|z|=2$ is positively graded. The nontrivial part of the compactification lies in this open set, and if we apply Knörrer periodicity to $\{z\ne 0\}\subset \overline U$ we instead get $D^b([\mathbb {A}^1_w/\mathbb {Z}_2])$ , because the branch $\mathbb {A}^1_z$ has been compactified to an orbifold $\mathbb {P}^{1:2}$ . This compactification of the positively graded part of the critical locus is why we get a relatively proper categorical resolution.

A more obvious compactification just uses $\mathbb {P}^1$ . We can achieve it algebraically by removing an exceptional object, motivated by the decomposition $D^b([\mathbb {A}^1_w/\mathbb {Z}_2])=\langle \mathcal {O}_0,\mathcal {O}\rangle =\langle D^b(\mathrm {pt}),D^b(\mathbb {A}^1_{w^2})\rangle $ for the chart $\{z\ne 0\}$ . There are two choices, since there are two twists of the skyscraper sheaf in $[\mathbb {A}^1/\mathbb {Z}_2]$ , giving two exceptional objects $\mathcal {O}_{\{x=w=0\}}$ and $\mathcal {O}_{\{y=w=0\}}$ in $D^b(\overline U,xyz)$ . Their orthogonal complements give two smaller categorical resolutions, matching § 3.

Figure 2

The critical locus of $(\mathbb {A}^3,xyz)$ , and a chart near $z=\infty $ for the partial compactification with an orbifold point.

4.3 Canonical bundles

The Serre functor on the LG-model is just given by the canonical bundle $\mathcal {O}(-1)$ , by a local version of [Reference Favero and KellyFK, Theorem 1.2]. Weak crepancy holds since the line bundle is trivial near $\{z=0\}$ , using Remark 2.6 together with the fact that ${\mathfrak {Perf}}(X)$ corresponds to factorizations supported at $\{z=0\}$ . Of course, this local condition does not imply global triviality (i.e., strong crepancy). The orbifold point at $\{z=\infty \}$ locally obstructs strong crepancy, contributing a local nontriviality to the canonical bundle, but there is also the global obstruction that the toric compactifications of $(\mathbb {A}^3,xyz)$ are not Calabi–Yau, so the smaller resolution is still not strongly crepant (cf. Remark 4.2).

The kernel subcategory consists of factorizations which restrict to zero on the open set $\{z\ne 0\}$ , that is, those with support at $\{z=\infty \}$ , and so fair crepancy is equivalent to local triviality of the canonical bundle near $\{z=\infty \}$ . This fails only for the orbifold compactification, where the isotropy group gives an obstruction.

4.4 The A-side

Let us briefly sketch an equivalent picture under homological mirror symmetry. For more details, the reader should consult [Reference Lekili and PolishchukLP1] and [Reference Lekili and PolishchukLP2].

Our starting point is a standard result, that the LG-model $(\mathbb {A}^3,xyz)$ is equivalent to a wrapped Fukaya category $\mathcal {W}(\Sigma )$ , where $\Sigma $ is the pair-of-pants surface.Footnote ⁷ It was further shown in [Reference Lekili and PolishchukLP1] that the categorical resolution $D^b(\mathcal {A})$ is equivalent to a partially wrapped Fukaya category $\mathcal {W}(\Sigma ;\Lambda )$ , where two stops $\Lambda =\{s_x,s_y\}$ have been added to one of the boundary components (the one with positive grading). Generators realizing the equivalence with $D^b(\mathcal {A})$ are horizontal lines in Fig. 3. The forgetful functor $\mathcal {W}(\Sigma ;\Lambda )\to \mathcal {W}(\Sigma )$ is a localization, which contracts two exceptional objects ( $E_x$ and $E_y$ ) given by small arcs bounding the two stops (also depicted).

The two smaller resolutions then correspond to removing just one of these stops, giving the equivalent categories $\mathcal {W}(\Sigma ;s_x)$ and $\mathcal {W}(\Sigma ;s_y)$ , with $\mathcal {W}(\Sigma ;\Lambda )$ providing a roof via the forgetful functors. Adjoint to these forgetful functors are the inclusions as admissible subcategories, which can be described on objects (purely algebraically) by choosing a lift to the roof, and then projecting to the orthogonal complement of the relevant exceptional object via a mapping cone. This reproduces the spherical twists of § 3; the arcs near the stop in the single-stopped case are spherical objects, and in the double-stopped case they are the exceptional objects of Lemma 3.3.

Figure 3

Objects in $\mathcal {W}(\Sigma ;s_x)$ , $\mathcal {W}(\Sigma ;\Lambda )$ and $\mathcal {W}(\Sigma ;s_y)$ .

These spherical twists are examples of the ‘wrap-once’ autoequivalence for a swappable stop from [Reference SylvanSyl], which is always the twist for a natural associated spherical functor – the Orlov functor – and is defined in terms of a ‘swapping’ self-isotopy of the stopping divisor. This is a full circuit around the boundary for our single-stopped surface, but can be a half twist in the double-stopped case (swapping the two stops), which explains why one autoequivalence is the square of the other.