2.1 Gauged Landau-Ginzburg models and their factorizations

Let $\operatorname {\mathbb {C}}$ be an algebraically closed field of characteristic zero. Let X be a smooth variety over $\operatorname {\mathbb {C}}$ and G an affine algebraic group that acts on X. Take W to be a G-invariant section of an invertible G-equivariant sheaf $\mathcal {L}$ , that is, $W \in \Gamma (X, \mathcal {L})^G$ . We call the data $(X, G, W)$ a gauged Landau-Ginzburg (LG) model. To a gauged LG model, there is the absolute derived category $\operatorname {D}^{\operatorname {abs}}[X,G,W]$ associated to it, which is the analogue of a variety’s bounded derived category of coherent sheaves.

We roughly outline its definition as follows. Good resources include [Reference Ballard, Favero and KatzarkovBFK14, Reference HiranoHir17].

Definition 2.1. A factorization is the data $\operatorname {\mathcal {E}} = (\operatorname {\mathcal {E}}_{0}, \operatorname {\mathcal {E}}_1, \phi ^{\operatorname {\mathcal {E}}}_0, \phi _1^{\operatorname {\mathcal {E}}})$ where $\operatorname {\mathcal {E}}_0$ and $\operatorname {\mathcal {E}}_1$ are G-equivariant quasi-coherent sheaves and

$$ \begin{align*}\operatorname{\mathcal{E}}_0 \stackrel{\phi_0^{\operatorname{\mathcal{E}}}}{\longrightarrow} \operatorname{\mathcal{E}}_1 \stackrel{\phi_1^{\operatorname{\mathcal{E}}}}{\longrightarrow} \operatorname{\mathcal{E}}_0 \otimes_{\operatorname{\mathcal{O}}_X} \mathcal{L} \end{align*} $$

are morphisms such that $\phi ^{\operatorname {\mathcal {E}}}_1 \circ \phi ^{\operatorname {\mathcal {E}}}_0 = W$ and $(\phi _0^{\operatorname {\mathcal {E}}} \otimes \mathcal {L}) \circ \phi _{1}^{\operatorname {\mathcal {E}}} = W$ .

For two factorizations $\operatorname {\mathcal {E}}$ and $\mathcal {F}$ , there is a complex $\operatorname {\mathrm {Hom}}(\operatorname {\mathcal {E}}, \mathcal {F})$ of morphisms from $\operatorname {\mathcal {E}}$ to $\mathcal {F}$ defined as follows. We have the graded vector space

$$ \begin{align*}\operatorname{\mathrm{Hom}}(\operatorname{\mathcal{E}}, \mathcal{F})^\bullet := \bigoplus_{n\in \operatorname{\mathbb{Z}}} \operatorname{\mathrm{Hom}}(\operatorname{\mathcal{E}}, \mathcal{F})^n \end{align*} $$

with differential $d^i: \operatorname {\mathrm {Hom}}(\operatorname {\mathcal {E}}, \mathcal {F})^i \to \operatorname {\mathrm {Hom}}(\operatorname {\mathcal {E}}, \mathcal {F})^{i+1}$ given by $d^i(f) = \phi ^{\mathcal {F}}_{\star + i} \circ f - (-1)^i f \circ \phi ^{\operatorname {\mathcal {E}}}_{\star }$ where

$$ \begin{align*}\begin{aligned} \operatorname{\mathrm{Hom}}(\operatorname{\mathcal{E}}, \mathcal{F})^{2m} &:= \operatorname{\mathrm{Hom}}(\operatorname{\mathcal{E}}_1, \mathcal{F}_1\otimes \mathcal{L}^{\otimes m}) \oplus \operatorname{\mathrm{Hom}}(\operatorname{\mathcal{E}}_0, \mathcal{F}_0\otimes \mathcal{L}^{\otimes m}) \\ \operatorname{\mathrm{Hom}}(\operatorname{\mathcal{E}}, \mathcal{F})^{2m+1} &:= \operatorname{\mathrm{Hom}}(\operatorname{\mathcal{E}}_1, \mathcal{F}_0\otimes \mathcal{L}^{\otimes m}) \oplus \operatorname{\mathrm{Hom}}(\operatorname{\mathcal{E}}_0, \mathcal{F}_1\otimes \mathcal{L}^{\otimes m+1}) \\ \end{aligned}\end{align*} $$

This yields a dg category $\operatorname {Fact }(X,G,W)$ . Denote by $\operatorname {fact }(X,G,W)$ the full dg-subcategory of this dg category whose components are coherent.

This category has a subcategory of acyclic complexes. Given $\operatorname {Fact }(X,G,W)$ , consider the subcategory $Z^0\operatorname {Fact }(X,G,W)$ with the same objects but only degree zero morphisms. Given a complex of objects in $Z^0\operatorname {Fact }(X,G,W)$ , one can construct a new object ${\mathcal {T} \in \operatorname {Fact }(X,G,W)}$ in a natural way by taking direct sums and arranging the morphisms in a natural way (see (2.1) of [Reference Favero and KellyFK18] for details). Take $\operatorname {Acyc}(X,G,W)$ to be the full subcategory of $\operatorname {Fact }(X,G,W)$ consisting of all totalizations of bounded exact complexes in $Z^0\operatorname {Fact }(X, G, W)$ and let $\operatorname {acyc}(X,G,W) = \operatorname {Acyc}(X,G,W) \cap \operatorname {fact }(X,G,W)$ .

The absolute derived category $\operatorname {D}^{\operatorname {abs}}[X,G,W]$ can be thought of as the derived category of the gauged LG model $(X,G,W)$ . To justify this claim, we must introduce some context and notation.

We have the following theorem, which has appeared in various forms due to Orlov [Reference OrlovOrl06], Isik [Reference Umut IsikIsi13], Shipman [Reference ShipmanShi12], and, in its most general form, Hirano [Reference HiranoHir17].

Theorem 2.4 (Proposition 4.8 of [Reference HiranoHir17]).

There exists an equivalence of categories

$$ \begin{align*}\Omega: \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[Z/G]) \stackrel{\sim}{\longrightarrow} \operatorname{D}^{\operatorname{abs}}[\operatorname{tot} \operatorname{\mathcal{E}}^\vee, G \times \operatorname{\mathbb{G}_m}, W]. \end{align*} $$

2.2 Partial compactifications yielding categorical resolutions

In effect, this subsection gives a mathematical introduction to the ‘exo’ part of the exoflop, following [Reference Favero and KellyFK18]. We first recall a categorical resolution of singularities, and provide sufficient criteria for their existence using gauged LG models. We use the definition stated in loc. cit., but the one given by Kuznetsov [Reference KuznetsovKuz08, Definition 3.2] could be used throughout if the reader prefers. Let Z be a variety with a G-action and $\mathcal {D}$ an admissible subcategory of $\operatorname {D}^{\operatorname {b}}(\operatorname {coh }[Z/G])$ . We denote by $\mathcal {D}^{\operatorname {perf}}$ the full subcategory of $\mathcal {D}$ consisting of G-equivariant perfect complexes on Z.

Definition 2.6 (Definition 3.1 of [Reference Favero and KellyFK18]).

Let $\tilde {\mathcal {D}}$ be the homotopy category of a homologically smooth and proper pretriangulated dg category. A pair of exact functors

$$ \begin{align*} F&:\tilde{\mathcal{D}} \to \mathcal{D}\\ G&:\mathcal{D}^{\operatorname{perf}} \to \tilde{\mathcal{D}} \end{align*} $$

is a categorical resolution of singularities if G is left adjoint to F and the natural morphism of functors $\operatorname {Id}_{\mathcal {D}^{\operatorname {perf}}} \to FG$ is an isomorphism. We say the categorical resolution is crepant if G is right adjoint to F.

We first discuss step (exo) and its interactions with the absolute derived category. A good resource for this can be found in [Reference Favero and KellyFK18, Section 3]. Consider a variety U equipped with an action by a linearly reductive group G, a character $\chi $ of G and a section W of $\operatorname {\mathcal {O}}_U(\chi )$ . Let

$$ \begin{align*}i: V \longrightarrow U \end{align*} $$

be a G-equivariant open immersion.

We then have the following functors

(2.1) $$ \begin{align}\begin{aligned} i_*: \operatorname{D}^{\operatorname{abs}}[V,G,W]_{\operatorname{rel} U} &\longrightarrow \operatorname{D}^{\operatorname{abs}}[U,G,W]; \\ i^*: \operatorname{D}^{\operatorname{abs}}[U,G,W] &\longrightarrow\operatorname{D}^{\operatorname{abs}}[V,G,W]; \end{aligned}\end{align} $$

where $i_*$ is both left and right adjoint to $i^*$ . We can use this to build crepant categorical resolutions, after giving a geometric context to the categories $\operatorname {D}^{\operatorname {abs}}[V,G,W]_{\operatorname {rel} U}$ and $\operatorname {D}^{\operatorname {abs}}[V,G,W]$ . We do this via the following variant of Theorem 2.4. Recall Notation 2.3.

Lemma 2.8 (Lemma 3.6 of [Reference Favero and KellyFK18]).

Assume Y admits a G-ample line bundle. The equivalence of categories

$$ \begin{align*}\Omega: \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[Z/G]) \to \operatorname{D}^{\operatorname{abs}}[\operatorname{tot} \operatorname{\mathcal{E}}^\vee, G \times \operatorname{\mathbb{G}_m}, W] \end{align*} $$

restricts to an equivalence between the full subcategory of perfect objects $\operatorname {Perf} [Z/G]$ and the full subcategory of $\operatorname {D}^{\operatorname {abs}}[\operatorname {tot} \operatorname {\mathcal {E}}^\vee , G \times \operatorname {\mathbb {G}_m}, W]$ with objects supported on the zero section of $\operatorname {\mathcal {E}}^\vee $ .

Note that if the zero section is already proper, then the image of the subcategory of perfect objects will not intersect with the partial compactification. Thus, to find a crepant categorical resolution, we find an appropriate partial compactification of $\operatorname {tot} \operatorname {\mathcal {E}}^\vee $ .

3.1 Cox stacks

Let $\Sigma $ be a fan in $N_{\operatorname {\mathbb {R}}}$ . We define a quotient stack $\mathcal {X}_{\Sigma }$ associated to the fan $\Sigma $ following the Cox construction as follows (see, e.g., [Reference Cox, Little and SchenckCLS11, Section 5.1] for the standard construction). Let $\nu = \{u_{\rho } \ | \ \rho \in \Sigma (1)\} \subseteq N$ , where $u_\rho $ is the primitive lattice generator of the ray $\rho \in \Sigma (1)$ . Consider the vector space $\operatorname {\mathbb {R}}^{\Sigma (1)}$ with elementary $\operatorname {\mathbb {Z}}$ -basis vectors $e_\rho $ for each $\rho \in \Sigma (1)$ . We construct a new fan

(3.1) $$ \begin{align} \operatorname{Cox}(\Sigma) := \{ \operatorname{Cone}(e_\rho \ | \ \rho \in \sigma) \ | \ \sigma \in \Sigma\} \end{align} $$

This fan is a subfan of the standard fan for $\mathbb {A}^{| \Sigma (1)|}$ , hence its associated toric variety is an open subset of affine space.

There is a canonical group acting on $U_\Sigma $ . Consider the right exact sequence

(3.2) $$ \begin{align} M \stackrel{f_{\nu}}{\longrightarrow} \operatorname{\mathbb{Z}}^{\nu} \stackrel{\pi}{\longrightarrow} \operatorname{coker}(f_\nu) \longrightarrow 0 \end{align} $$

where $f_{\nu }(m) := \sum _{\rho \in \Sigma (1)} \langle u_\rho , m\rangle e_\rho $ . Applying the functor $\operatorname {\mathrm {Hom}}(-,\mathbb {G}_m)$ to this sequence yields the left exact sequence

$$\begin{align*}1\rightarrow \operatorname{\mathrm{Hom}}(\operatorname{coker}(f_\nu),\mathbb{G}_m)\xrightarrow{\widehat{\pi}}\mathbb{G}_m^{\nu} \xrightarrow{\widehat{f_\nu}}\mathbb{G}_m^d. \end{align*}$$

Define

$$\begin{align*}S_{\Sigma}:=\operatorname{\mathrm{Hom}}(\operatorname{coker}(f_\nu),\mathbb{G}_m). \end{align*}$$

We note that $S_{\Sigma }$ acts on the open set $U_\Sigma $ constructed above.

Definition 3.2. Define the Cox stack associated to $\Sigma $ to be

$$\begin{align*}\mathcal{X}_\Sigma:=[U_\Sigma/S_{\Sigma}]. \end{align*}$$

3.2 Toric vector bundles

Let $\Sigma $ be a fan in $N_{\operatorname {\mathbb {R}}}$ . For each ray $\rho \in \Sigma (1)$ with primitive generator $u_\rho $ , denote by $D_\rho $ the torus-invariant divisor associated to it. Recall that any torus-invariant Weil divisor D on $X_\Sigma $ can be written as a linear combination

$$\begin{align*}D=\sum_{\rho\in \Sigma(1)}a_\rho D_\rho, \end{align*}$$

for some $a_\rho \in \operatorname {\mathbb {Z}}$ . Consider now a collection of r such divisors, $D_i=\sum _{\rho \in \Sigma (1)}a_{i\rho }D_\rho $ , with $a_{i\rho }\in \operatorname {\mathbb {Z}}$ .

Write $e_1, \dots , e_r$ for the elementary $\operatorname {\mathbb {Z}}$ -basis for $\operatorname {\mathbb {R}}^r$ , and we denote by $e_1^*, \dots , e_r^*$ its dual basis.

For any cone $\sigma \in \Sigma $ , define the cone

$$\begin{align*}\sigma_{D_1,\dots,D_r}:=\operatorname{Cone}(\{u_\rho-a_{1\rho}e_1-\dots-a_{r\rho}e_r\vert \rho\in\sigma(1)\}\cup\{e_i\mid i\in\{1,\dots,r\})\subseteq N_{\operatorname{\mathbb{R}}}\times \operatorname{\mathbb{R}}^r. \end{align*}$$

By considering the collection $\{\sigma _{D_1,\dots ,D_r}\mid \sigma \in \Sigma \}$ of these cones, together with their proper faces, we form the fan $\Sigma _{D_1,\dots ,D_r}\subseteq N_{\operatorname {\mathbb {R}}}\times \operatorname {\mathbb {R}}^r$ . Note that the rays of $\Sigma _{D_1,\dots ,D_r}$ are given by

$$ \begin{align*} \bar\rho := \operatorname{Cone}( u_\rho-a_{1\rho}e_1-\dots-a_{r\rho}e_r) \text{ for all } \rho \in \Sigma(1) \end{align*} $$

and $\epsilon _i := \operatorname {Cone}(e_i)$ for $i \in \{1, \dots , r\}$ . The toric variety associated to the fan $\Sigma _{D_1,\dots ,D_r}$ is the total space of the vector bundle $\bigoplus _{i=1}^r \operatorname {\mathcal {O}}(D_i)$ over $X_\Sigma $ (see, e.g., Proposition 7.3.1 of [Reference Cox, Little and SchenckCLS11]). The analogous proposition for toric stacks is the following.

Proposition 3.3 (Proposition 5.16 in [Reference Favero and KellyFK17]).

Let $D_1,\dots ,D_r$ be torus-invariant Weil divisors on $X_\Sigma $ defined as above. There is an isomorphism of stacks

$$\begin{align*}\mathcal{X}_{\Sigma_{D_1,\dots,D_r}}\simeq \operatorname{tot}\left(\bigoplus_{i=1}^r\operatorname{\mathcal{O}}_{\mathcal{X}_\Sigma}(D_i)\right). \end{align*}$$

Recall that the set of global functions on a toric variety $\Sigma $ in $N_{\operatorname {\mathbb {R}}}$ can be computed as follows. Take the support $\sigma = |\Sigma |$ and compute the dual cone $\sigma ^\vee \subseteq M_{\operatorname {\mathbb {R}}}$ . Then for any $m \in \sigma ^\vee \cap M$ , we obtain a monomial

(3.3) $$ \begin{align} x^m := \prod_{\rho \in \Sigma(1)} x_\rho^{\langle m, u_\rho\rangle} \end{align} $$

that is a global function on $X_\Sigma $ . Any algebraic map $X_\Sigma \to \operatorname {\mathbb {C}}$ can be written as a linear combination of $x^m$ for some $m \in \sigma ^\vee \cap M$ . While we will often consider $\Sigma $ to be complete, hence only have constant global functions, the supports of the fans $\Sigma _{D_1,\dots ,D_r}$ will be strictly convex cones, which will have have maximal dimension dual cones.

We now restrict to $\Sigma $ being a complete fan in $N_{\operatorname {\mathbb {R}}}$ . The polytope associated to the torus-invariant Weil divisor $D_i= \sum _{\rho \in \Sigma (1)} a_{\rho i} D_\rho $ is given by

$$ \begin{align*} P_{D_i} = \{ m \in M_{\operatorname{\mathbb{R}}} \ | \ \langle m, u_\rho\rangle \ge -a_{\rho i} \text{ for all } \rho \in \Sigma(1)\}. \end{align*} $$

Note that $m \in P_{D_i}$ if and only if $\langle m + e_i^*, u_\rho + a_{\rho i}e_i\rangle \ge 0$ for all $\rho \in \Sigma (1)$ . Thus $m \in P_{D_i}$ if and only if $m + e_i^* \in |\Sigma _{-D_1,\dots ,-D_r}|^\vee $ . Consider the (codimension r) hyperplane ${H_i := \{ (m, \delta _{1i}, \dots , \delta _{ri}) \ | \ m \in M_{\operatorname {\mathbb {R}}}\}}$ .Footnote ¹ Thus we get that the global sections of each divisor

$$ \begin{align*}f_i = \sum_{m \in P_{D_i} \cap M} c_m \prod_{\rho \in \Sigma(1)} x_\rho^{\langle m, u_\rho\rangle+a_{\rho i}} \in \Gamma(X_{\Sigma}, \operatorname{\mathcal{O}}(D_i))\end{align*} $$

correspond to the global functions on $\operatorname {tot} \bigoplus _{i=1}^r \operatorname {\mathcal {O}}(-D_i)$

(3.4) $$ \begin{align}\begin{aligned} s_i &= \sum_{(m, b_1, \dots, b_r) \in H_i \cap (M\times \operatorname{\mathbb{Z}}^r)} c_m \prod_{ \bar\rho \in \Sigma_{-D_1,\dots,-D_r}(1)} x_{\bar\rho}^{\langle m+ \sum_{i=1}^r b_i e_i, u_{\bar\rho}\rangle} \\ &= u_i \sum_{(m, b_1, \dots, b_r) \in H_i\cap (M \times \operatorname{\mathbb{Z}}^r)} c_m \prod_{ \rho \in \Sigma(1)} x_{\bar\rho}^{\langle m, u_\rho\rangle+a_{\rho i}} , \end{aligned}\end{align} $$

where we denote the coordinate associated to the ray $\epsilon _i$ by $u_i$ and that corresponding to $\bar{\rho} $ by $x_{\bar{\rho}} $ . By abuse of notation, we will write $s_i = u_if_i$ , as it is equivalent if one conflates $x_\rho $ with $x_{\bar \rho }$ . Write

$$ \begin{align*}[\mathcal{Z}/G] = [Z(f_1, \dots, f_r) / S_{\Sigma}] \subseteq [U_\Sigma / S_{\Sigma}]\end{align*} $$

and $W = \sum _{i=1}^r u_if_i$ . We recall the R-charge action of $\operatorname {\mathbb {G}_m}$ acting on a vector bundle by fiberwise dilation (see Notation 2.3) and consider the projection character

$$ \begin{align*}\chi: S_{\Sigma} \times \operatorname{\mathbb{G}_m} \to \operatorname{\mathbb{G}_m}. \end{align*} $$

Note that

$$ \begin{align*}W \in \Gamma(U_{\Sigma_{-D_1,\dots,-D_r}}, \operatorname{\mathcal{O}}_{U_{\Sigma_{-D_1,\dots,-D_r}}}(\chi))^{S_{\Sigma_{-D_1,\dots,-D_r}} \times \operatorname{\mathbb{G}_m}}. \end{align*} $$

We remark this is equivalent to W being semi-invariant with respect to the character $\chi $ [Reference Favero and KellyFK19, Definition 4.3].

Corollary 3.4. There exists an equivalence of categories

$$ \begin{align*}\Omega: \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal{Z}/G]) \stackrel{\sim} \longrightarrow \operatorname{D}^{\operatorname{abs}}[ U_{\Sigma_{-D_1,\dots,-D_r}}, S_{\Sigma_{-D_1,\dots,-D_r}(1)} \times \operatorname{\mathbb{G}_m}, W], \end{align*} $$

where the $\operatorname {\mathbb {G}_m}$ acts with weights 0 on the coordinates $x_\rho $ and 1 on the $u_i$ .

In light of the above corollary, the gauged Landau-Ginzburg model associated to the complete intersection $[\mathcal {Z}/G]$ is

$$\begin{align*}(U_{\Sigma_{-D_1,\dots,-D_r}}, S_{\Sigma_{-D_1,\dots,-D_r}(1)} \times \operatorname{\mathbb{G}_m}, W). \end{align*}$$

3.3 Partial compactifications and crepant categorical resolutions

We continue with the setup of § 3.2 and keep the notations $\Sigma , D_i, W, H_i, f_i, s_i$ as above. In light of Remark 3.5, we assume $\Sigma $ is simplicial and thus so is $\Sigma _{-D_1,\dots ,-D_r}$ . Given the global function $W: \operatorname {tot} \bigoplus _{i=1}^r \operatorname {\mathcal {O}}(-D_i) \to \mathbb {A}^1$ , we can define

$$ \begin{align*}\Xi_{i, W} := \{ \bar m = (m, \delta_{1i}, \dots, \delta_{ri}) \in H_i \ | \ c_m \ne 0\}. \end{align*} $$

Note $\Xi _{i,W} \subseteq P_{D_i}+e_i^*$ . Take $\Xi _W = \bigcup _{i=1} \Xi _{i,W}$ . Note that $\sigma _W := \operatorname {Cone}(\Xi _W) \subseteq |\Sigma _{-D_1,\dots ,-D_r}|^\vee $ , hence $\sigma _W^\vee \supseteq |\Sigma _{-D_1,\dots ,-D_r}|$ .

Take a (strictly convex) rational polyhedral cone $\Sigma ^{\prime }$ so that $ |\Sigma _{-D_1,\dots ,-D_r}| \subseteq \Sigma ^{\prime } \subseteq \sigma _{W}^\vee $ . We have the following result.

This lemma is a corollary of Lemma 4.5 proven in Subsection 4.1 using convex geometry, so we postpone its proof.

Using Corollary 4.23 of [Reference Favero and KellyFK18], we have a stack isomorphism

$$ \begin{align*}\varphi: [U_{\Sigma_{-D_1, \dots, -D_r}} \times \operatorname{\mathbb{G}_m}^{\Psi(1) \setminus \Sigma_{-D_1, \dots, -D_r}(1)} / S_{\Psi}] \stackrel{\sim}{\longrightarrow} [U_{\Sigma_{-D_1, \dots, -D_r}} / S_{\Sigma_{-D_1, \dots, -D_r}}],\end{align*} $$

which induces an equivalence of categories on the associated absolute derived categories. We then consider the $S_\Psi $ -equivariant open immersion

(3.5) $$ \begin{align} i: U_{\Sigma_{-D_1, \dots, -D_r}} \times \operatorname{\mathbb{G}_m}^{\Psi(1) \setminus \Sigma_{-D_1, \dots, -D_r}(1)} \hookrightarrow U_{\Psi}. \end{align} $$

As shorthand, we write $U_{\Sigma , D_i}' := U_{\Sigma _{-D_1, \dots , -D_r}} \times \operatorname {\mathbb {G}_m}^{\Psi (1) \setminus \Sigma _{-D_1, \dots , -D_r}(1)}$ . Note that the function $W: \operatorname {tot} \bigoplus _{i=1}^r \operatorname {\mathcal {O}}(-D_i)\to \mathbb {A}^1$ can be viewed as an $S_{\Sigma _{-D_1, \dots , -D_r}}$ -invariant function $W: U_{\Sigma _{-D_1, \dots , -D_r}} \to \mathbb {A}^1$ . Extend the R-charge trivially to the new variables. Since W is a linear combination of toric monomials in the dual cone to $|\Psi |$ , the function is then extended to

(3.6) $$ \begin{align} \bar W \in \Gamma( U_{\Psi}, \operatorname{\mathcal{O}}_{U_\Psi}(\chi))^{S_{\Psi} \times \operatorname{\mathbb{G}_m}} \end{align} $$

where $\chi :S_{\Psi } \times \operatorname {\mathbb {G}_m} \to \operatorname {\mathbb {G}_m}$ is again the projection. Indeed, the extension $\bar W$ is a section of $\operatorname {\mathcal {O}}_{U_\Psi }(\chi )$ because each toric monomial $x^{\bar m}$ , once written in coordinates, is of the form $u_i \cdot \prod _{\bar \rho \in \Psi (1) \setminus \{\epsilon _i\} } x_{\bar \rho }^{\langle \bar m, u_{\bar \rho }\rangle }$ for some i, where the $u_i$ are the bundle coordinates as in § 3.2. The following is essentially an alternative stating of Theorem 3.7 of [Reference Favero and KellyFK18] in the toric setting.

Theorem 3.7. Suppose $\operatorname {D}^{\operatorname {abs}}[U_{\Psi }, G_{\Psi } \times \operatorname {\mathbb {G}_m}, \bar W]$ is homologically smooth and proper. Then we have the following crepant categorical resolution

(3.7) $$ \begin{align}\begin{aligned} i_* \circ \varphi^* \circ\Omega: \operatorname{Perf} [\mathcal Z/G] &\longrightarrow \operatorname{D}^{\operatorname{abs}}[U_{\Psi}, G_{\Psi} \times \operatorname{\mathbb{G}_m}, \bar W]; \\ \Omega^{-1}\circ \varphi_* \circ i^*: \operatorname{D}^{\operatorname{abs}}[U_{\Psi}, G_{\Psi} \times \operatorname{\mathbb{G}_m}, \bar W] &\longrightarrow \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal Z/G]); \end{aligned}\end{align} $$

Proof. First, we note that using Lemma 2.8, the image under $\varphi _* \circ \Omega $ of $\operatorname {Perf} [\mathcal Z/G] $ is supported on the hyperplane $Z(u_1, \dots , u_r)\subseteq U_{\Sigma , D_i}' $ . Note that the partial compactification given by the open immersion i given in (3.5) does not intersect the hyperplane, thus the image under $i_*$ will be in the subcategory $\operatorname {D}^{\operatorname {abs}}[U_{\Psi }, G_{\Psi } \times \operatorname {\mathbb {G}_m}, \bar W]_{\operatorname {rel} U_{\Sigma , D_i}' }$ .

The claim then follows from the fact that

(3.8) $$ \begin{align}\begin{aligned} \Omega^{-1} \circ \varphi_*& \circ i^* \circ i_*\circ\varphi^* \circ\Omega \\&= \Omega^{-1} \circ\varphi_* \circ \operatorname{Id}_{\operatorname{D}^{\operatorname{abs}}[ U_{\Sigma_{-D_1,\dots,-D_r}} \times \operatorname{\mathbb{G}_m}^{\Psi(1) \setminus \Sigma_{-D_1, \dots, -D_r}(1)} , S_{\Sigma_{\Psi}} \times \operatorname{\mathbb{G}_m}, W]|_{\operatorname{rel } U_\Psi}} \circ \varphi^* \circ \Omega \\ &= \operatorname{Id}_{\operatorname{Perf} [\mathcal Z/G]}. \end{aligned}\end{align} $$

and Definition 2.6.

We then obtain the following corollary when $[\mathcal {Z}/G]$ is smooth.

Furthermore, we have the following corollary we use later.

4.1 Semiprojectivity of partial compactifications

In this subsection, we discuss the existence of semiprojective partial compactifications by using regular triangulations and convex geometry. This subsection may at first seem like a digression, but it proves we have a partial compactification that is a GIT quotient. However, since the results of the subsection are true outside of the above context, we rename $N \times \operatorname {\mathbb {Z}}^r$ to N temporarily until Corollary 4.6. Said corollary is a strengthening of Lemma 3.6 and the main result for our purposes in this subsection.

There are two standard notions of regular triangulation in the literature. We start by reviewing them. Take a finite subset of distinct elements $\nu = \{v_1, \dots , v_r\} \in N_{\mathbb {Q}}$ lying on an integral affine hyperplane $H\subseteq N_{\operatorname {\mathbb {R}}}$ with $0 \notin H$ . This gives the lattice polytope $Q_\nu = \operatorname {Conv}(v_1, \dots , v_r)\subseteq H$ . We will assume that $Q_\nu $ has full dimension in H and that the cone $C_\nu = \operatorname {Cone}(\nu )$ has full dimension in $N_{\operatorname {\mathbb {R}}}$ and is strongly convex. The below is a minor variant on the definition of triangulation from that in [Reference Cox, Little and SchenckCLS11, §15.2], which we follow.

One can define a special class of triangulations, regular triangulations, as follows. Given nonnegative weights $\omega = (w_1, \dots , w_r) \in \mathbb {Q}^{r}_{\ge 0}$ (or, equivalently a weight function ${w: \nu \to \operatorname {\mathbb {Q}}_{\ge 0}}$ ) to obtain a cone

$$ \begin{align*}C_{\nu, \omega} := \operatorname{Cone}( (v_1, w_1) \dots, (v_r, w_r)) \subseteq N_{\operatorname{\mathbb{R}}} \times \operatorname{\mathbb{R}}. \end{align*} $$

The lower hull of $C_{\nu , \omega }$ consists of all facets of the cone $C_{\nu , \omega }$ whose inner normal has a positive last coordinate. Projecting the facets in the lower hull and their faces gives a fan $\Sigma _{\omega }$ in $N_{\operatorname {\mathbb {R}}}$ such that $|\Sigma _\omega | = C_{\nu }$ and $\Sigma _{\omega } \subseteq \{\operatorname {Cone}(v_i) \ | \ 1 \le i \le r\}$ . The fan $\Sigma _\omega $ naturally provides a polyhedral subdivision of the convex hull $Q_\nu $ . The following is a variant of Definition 15.2.8 of [Reference Cox, Little and SchenckCLS11] (requiring rational weights $\omega \subseteq \mathbb {Q}^{r}_{\ge 0}$ rather than $\operatorname {\mathbb {R}}^r_{\ge 0}$ ) for when this polyhedral subdivision is a triangulation.

There is an alternate definition of regular triangulation by Hausel and Sturmfels for cones [Reference Hausel and SturmfelsHS02]. We modify their definitions by adding the word ‘conical’ to avoid confusion with the above.

Definition 4.3. A conical triangulation of $\nu $ is a simplicial fan $\Sigma $ whose rays have generators in $\nu \subseteq N$ . A T-Cartier divisor on $\Sigma $ is a continuous function

$$ \begin{align*}\Phi: C_{\nu} \to \operatorname{\mathbb{R}}\end{align*} $$

which is linear on each cone of $\Sigma $ and takes integer values on $N\cap C_{\nu }$ . The conical triangulation $\Sigma $ is called regular if there exists a T-Cartier divisor $\Phi $ which is ample, i.e., the function $\Phi $ is convex and restricts to a different linear function on each maximal cone of $\Sigma $ .

Proof. For any $q \in Q_{\nu }$ , there is a unique $w_q \in \mathbb {R}_{>0}$ such that $(q,w_q)$ is in the lower hull of $C_{\nu , \omega }$ . Note that if $q \in Q_{\nu } \cap N$ then $w_q \in \mathbb {Q}_{>0}$ as it is a rational linear combination of $w_1, \dots , w_r$ . Write $C_{\nu } = \{ a q \ | \ q \in Q_{\nu }, a \in \mathbb {R}_{\ge 0}\}$ . Then define a map $\Psi : C_{\nu } \to \mathbb {R}$

$$ \begin{align*}\Phi: C_{\nu} \to \mathbb{R}_{\ge 0}, \qquad \Psi(aq) = aw_q. \end{align*} $$

Using the fact that the weights $w_q$ are built from the lower hull of $C_{\nu ,\omega }$ , one can check this is continuous, convex, linear on each cone of $\Sigma _w$ , and restricts to a different linear function on each maximal cone of $\Sigma _w$ . One can then use Gordan’s lemma on each maximal cone of $\Sigma _w$ to find a constant D so that the function

$$ \begin{align*}\Phi_D: C_{\nu} \to \mathbb{R}_{\ge 0}, \qquad \Psi(aq) =D aw_q. \end{align*} $$

satisfies the properties above and also takes integer values on $N\cap C_\nu $ .

Proof. We first prove the existence of a regular polyhedral subdivision $\mathcal {S}$ of L containing $\mathcal {T}_0$ in the sense of the lemma, and then we prove that this regular subdivision can be refined into a regular triangulation containing $\mathcal {T}_0$ , thus proving the Lemma.

If $L_1 \subset L_0$ , the lemma is trivial. We proceed by induction. Suppose $L_1\setminus L_0 =\{v\}$ . Denote the standard basis of $M_{\operatorname {\mathbb {R}}}\times \operatorname {\mathbb {R}}$ by $e_1, e_2,\dots , e_{d+1}$ . By definition, the regular triangulation $\mathcal {T}_0$ of $L_0$ is obtained by projecting the lower facets of a polyhedron $\mathcal {Q}_0=\operatorname {Conv}(\{x+w_0(x) e_{d+1}\vert x\in L_0\})\subseteq M_{\operatorname {\mathbb {R}}}\times \operatorname {\mathbb {R}}$ for some weight function $w_0:L_0\rightarrow \operatorname {\mathbb {Q}}_{\ge 0}$ . To obtain the regular subdivision $\mathcal {S}$ of $L=L_0\cup \{v\}$ which contains $\mathcal {T}_0$ , we extend $w_0$ to a weight function $w_1:L\rightarrow \operatorname {\mathbb {R}}^+$ and project the lower facets of the polyhedron $\mathcal {Q}_1=\operatorname {Conv}(\{x+w_1(x)e_{d+1}\vert x\in L\})\subseteq M_{\operatorname {\mathbb {R}}}\times \operatorname {\mathbb {R}}$ , noting that $\mathcal {Q}_0\subseteq \mathcal {Q}_1$ . We distinguish two cases: $v\in \operatorname {Conv}(L_0)$ and $v\not \in \operatorname {Conv}(L_0)$ .

Case 1: v ∈ Conv(L ₀). Define the weight function $w_1:L\rightarrow \operatorname {\mathbb {Q}}_{\ge 0}$ by setting $w_1(v)=1+\max _{x\in L_0}(w_0(x))$ and $w_1(x)=w_0(x)$ otherwise. Since $v \in \operatorname {Conv}(L_0)$ , there exists weights $\lambda _x \in [0,1]$ for all $x \in L_0$ such that $\sum _{x \in L_0} \lambda _x = 1$ and $\sum _{x \in L_0} \lambda _x x = v$ . Thus, for any such collection of weights,

$$ \begin{align*}w_1(v)> \max_{x\in L_0}(w_0(x)) \ge \sum_{x\in L_0} \lambda_x w_0(x) \end{align*} $$

hence $(v,w_1(v))$ is not in any lower facet of the polyhedron $\mathcal {Q}_1$ . Thus $\mathcal {S} = \mathcal {T}_0$ is a regular triangulation.

Case 2: v∉Conv(L ₀). First we construct a weight function $w_1$ . Fix a lower facet F of $\mathcal {Q}_0$ with inner pointing normal $u_F$ . Write $F=\mathcal {Q}_0\cap A_F$ where $A_F$ is the boundary of the supporting halfspace $H_F=\{x\in N_{\operatorname {\mathbb {R}}}\times \operatorname {\mathbb {R}}\vert \langle u_F, x\rangle \geq c_F\}$ at the facet F, so $A_F = \{x\in N_{\operatorname {\mathbb {R}}}\times \operatorname {\mathbb {R}}\vert \langle u_F, x \rangle =c_F\}$ . Write $\mu _F=\langle u_F, e_{d+1}\rangle>0$ and $\overline {u}_F=u_F-\mu _F e_{d+1}^*$ , where $e_{d+1}^*$ is the dual basis vector to $e_{d+1}$ and $\overline {u}_F$ (respectively $\mu _Fe_{d+1}^*$ ) is the projection of $u_F$ onto the first d coordinates (last coordinate). Define $w_1: L_1 \cup L_0 \to \operatorname {\mathbb {Q}}_{\ge 0}$ to be

$$\begin{align*}w_1(x) = \begin{cases} w_0(x) & \text{ if } x \in L_0; \\ 1+\max_{x\in L_0}\frac{c_F-\langle \overline{u}_F, v\rangle}{\mu_F} & \text{ if } x = v. \end{cases} \end{align*}$$

Consider the polyhedron $\mathcal {Q}_1=\operatorname {Conv}(\{x+w_1(x)e_{d+1}\vert x\in L\})$ . Note

$$\begin{align*}\bigcup_{T \text{ a lower facet of }\mathcal{Q}_0} T\subseteq \mathcal{Q}_0\subseteq \mathcal{Q}_1. \end{align*}$$

Fix a lower facet $F=\mathcal {Q}_0\cap A_F$ of $\mathcal {Q}_0$ . We claim that F is a lower facet of $\mathcal {Q}_1$ , i.e. $\mathcal {Q}_1\cap A_F=F$ and $\mathcal {Q}_1\subseteq H_F$ .

Suppose that $\mathcal {Q}_1\not \subseteq H_F$ , i.e. there is a point $q\in \mathcal {Q}_1$ such that $q\not \in H_F$ . Then $c_F> \langle u_F, q\rangle $ and there are non-negative real numbers $(\lambda _x)_{x\in L_0},\lambda _v$ where $1=\lambda _v+\sum _{x\in L_0}\lambda _x$ such that

$$\begin{align*}q=\lambda_v(v+w_1(v) e_{d+1})+\sum_{x\in L_0}\lambda_x(x+w_1(x)e_{d+1}). \end{align*}$$

We obtain

(4.1) $$ \begin{align}\begin{aligned} c_F&>\left\langle u_F, \lambda_v(v+w_1(v)e_{d+1})+\sum_{x\in L_0}\lambda_x(x+w_1(x)e_{d+1})\right\rangle\\ & =\lambda_v(\langle \overline{u}_F,v\rangle +w_1(v)\mu_F)+\left\langle u_F, \sum_{x\in L_0}\lambda_x(x+w_1(x)e_{d+1})\right\rangle\\ & > \lambda_v\left( \langle \overline{u}_F,v\rangle+\frac{c_F-\langle \overline{u}_F,v\rangle}{\mu_F}\mu_F\right)+\sum_{x\in L_0}\lambda_xc_F\\ & =c_F. \end{aligned} \end{align} $$

This is a contradiction, and so $\mathcal {Q}_1\subseteq H_F$ .

Suppose, for the point of contradiction, there is a point $q\in (\mathcal {Q}_1\setminus \mathcal {Q}_0)\cap A_F$ . Then there is a collection of non-negative real numbers $\{\lambda _x\vert x\in L_0\}\cup \{\lambda _v\}$ such that $\lambda _v+\sum _{x\in L_0}\lambda _x=1$ with $\lambda _v>0$ and $\lambda _v(v+w_1(v)e_{d+1})+\sum _{x\in L_0}\lambda _x(x+w_1(x)e_{d+1})=q$ and

$$ \begin{align*} c_F&=\langle u_F, q\rangle \\ & =\langle u_F, \lambda_v(v+w_1(v)e_{d+1})\rangle +\langle u_F, \sum_{x\in L_0}\lambda_x(x+w_1(x)e_{d+1})\rangle. \end{align*} $$

For each $x\in L_0$ , we have $(x+w_1(x)e_{d+1})\in \mathcal {Q}_0$ so $\langle u_F, x+w_1(x)e_{d+1} \rangle \geq c_F$ . Thus we obtain

$$ \begin{align*} c_F& \geq \langle u_F, \lambda_v(v+w_1(v)e_{d+1})\rangle +c_F\sum_{x\in L_0}\lambda_x\\ &= \lambda_v\langle u_F, v+w_1(v)e_{d+1}\rangle+(1-\lambda_v)c_F;\\ \lambda_vc_F & \geq \lambda_v\langle \overline{u}_F + \mu_F e_{d+1}^*, v+w_1(v)e_{d+1}\rangle;\\ c_F-\langle \overline{u}_F, v\rangle &\geq \langle \mu_F e_{d+1}^*, w_1(v)e_{d+1}\rangle> \frac{c_F-\langle \overline{u}_F, v\rangle}{\mu_F}\mu_F=c_F-\langle \overline{u}_F, v\rangle; \end{align*} $$

a contradiction. Hence, $(\mathcal {Q}_1\setminus \mathcal {Q}_0)\cap A_F=\emptyset $ . Consequently, as $\mathcal {Q}_0\subseteq \mathcal {Q}_1$ , we have $\mathcal {Q}_1\cap A_F=\mathcal {Q}_0\cap A_F$ as required.

In summary, we have shown that all lower facets of $\mathcal {Q}_0$ are also lower facets of $\mathcal {Q}_1$ . Thus,

$$\begin{align*}\mathcal{T}_0\subseteq \mathcal{S}. \end{align*}$$

Suppose $\mathcal {S}$ is not a triangulation. Then there exist a lower facet $F = \mathcal {Q}_1 \cap A_F$ of $\mathcal {Q}_1$ that is not a simplex. Recall that $\mathcal {T}_0$ is a triangulation, so $F \notin \mathcal {Q}_0$ and $v \in F$ . Write

$$ \begin{align*}F = \operatorname{Conv}(v+ w_1(v)e_{d+1}, (x_j + w_1(x_j) e_{d+1})_{j \in J}) \end{align*} $$

for some $J \subseteq L_0$ . Assume J is maximal in the sense that if $x + w_1(x) e_{d+1} \in F$ for some $x \in L_0$ then $x \in J$ . Note that $\dim \operatorname {Conv}(L_0) = \dim F \le |J| -1$ , as F is not a simplex. This would require there to be a facet $F' \subseteq \mathcal {Q}_0$ given by $F' = \operatorname {Conv} (x_j + w_1(x_j) e_{d+1})_{j \in J}$ such that $ F' = \mathcal {Q}_0 \cap A_F$ and $\dim F = \dim F'$ . Since $\mathcal {T}_0$ is a triangulation, $|J|-1 =\dim F' = \dim F= \dim \operatorname {Conv}(L_0)$ . This implies that the minimal affine linear subspace containing $ (x_j + w_1(x_j) e_{d+1})_{j \in J}$ must contain $v+ w_1(v)e_{d+1}$ , which contradicts the fact that $\mathcal {Q}_0 \subset \mathcal {Q}_1$ (e.g., Equation (4.1)).

The induction step now follows easily by iterating the above for each element of $L_1$ .

The following is a strengthened version of Lemma 3.6. Recall the (strictly convex) rational polyhedral cone $\Sigma ^{\prime }$ so that $ |\Sigma _{-D_1,\dots ,-D_r}| \subseteq \Sigma ^{\prime } \subseteq \sigma _{W}^\vee $ .

4.2 Variation of GIT Quotients and Gorenstein Cones

Corollary 4.6 shows there exists a simplicial fan $\Psi $ with support $\Sigma ^{\prime }$ so that $X_{\Psi }$ is semiprojective and $\Psi $ contains $\Sigma _{-D_1, \dots , -D_r}$ as a subfan. Recall that we have the global $G_{\Psi (1)}$ -invariant function $\bar W$ , defined as in (3.6), yielding the gauged LG model

$$ \begin{align*}(U_\Psi, G_{\Psi(1)} \times \operatorname{\mathbb{G}_m}, \bar W) \end{align*} $$

Since $X_{\Psi }$ is semiprojective, $\Psi $ corresponds to a chamber of a secondary fan corresponding to the point collection given by intersecting a hyperplane with the rays in $\Psi (1)$ (see, e.g., Exercise 15.1.8 of [Reference Cox, Little and SchenckCLS11]). Thus the quotient stack $[U_{\Psi } / G_{\Psi }]$ is a GIT quotient and we can vary the choice of linearization to find other GIT quotients, and provide relationships between their corresponding factorization categories using the technology developed in [Reference Ballard, Favero and KatzarkovBFK19, Reference Halpern-LeistnerHL15]. Such a change will give a birational transformation, yielding the ‘flop’ portion of the exoflop. One can use this technology directly and the above in order to generate results on relations in various cases.

The story simplifies when one restricts to Calabi-Yau complete intersections. In this case, the relationships between the factorization categories of different GIT quotients were streamlined in [Reference Favero and KellyFK18] using Gorenstein cones. The following definitions for various variants of Gorenstein cones will be necessary for some of the following statements.

Definition 4.7. Consider a cone $\sigma $ in $N_{\operatorname {\mathbb {R}}}$ . We say $\sigma $ is

1. (1) Gorenstein with respect to 𝔪 _σ if there exists an element $\mathfrak m_\sigma \in M$ so that the cone $\sigma $ is generated over $\operatorname {\mathbb {Q}}$ by finitely many lattice points in $\{n \in N \ | \ \langle \mathfrak m_\sigma , n \rangle =1 \}$ .

2. (2) ℚ-Gorenstein with respect to 𝔪 _σ if there exists an element $\mathfrak m_{\sigma } \in M_{\operatorname {\mathbb {Q}}}$ so that the cone $\sigma $ is generated over $\operatorname {\mathbb {Q}}$ by finitely many lattice points in $\{n \in N \ | \ \langle \mathfrak m_\sigma , n \rangle =1 \}$ .

If $\sigma $ has primitive lattice generators $v_1, \dots , v_k \in N$ , the support $\Delta _\sigma $ of $\sigma $ is the polytope $\operatorname {Conv}(\{v_1, \dots , v_k\})$ in the hyperplane

$$ \begin{align*}H_{\mathfrak m_\sigma}(1) := \{ x \in N_{\operatorname{\mathbb{R}}} \ | \ \langle \mathfrak m_{\sigma}, x\rangle=1\}.\end{align*} $$

We say a Gorenstein cone $\sigma $ is reflexive Gorenstein of index r if its dual cone $\sigma ^\vee $ is Gorenstein with respect to an element $\mathfrak n_{\sigma ^\vee } \in N$ and $\langle \mathfrak m_{\sigma }, \mathfrak n_{\sigma ^\vee }\rangle = r$ .

Note that if $\sigma $ is $\operatorname {\mathbb {Q}}$ -Gorenstein and dimension $\dim N_{\operatorname {\mathbb {R}}} = d$ , then $m_{\sigma }$ is unique.

Let $\sigma \subseteq N_{\operatorname {\mathbb {R}}}$ be a $\operatorname {\mathbb {Q}}$ -Gorenstein cone and $\nu \subseteq \sigma \cap N$ be a finite, geometric collection of lattice points which contains the (primitive) ray generators of $\sigma $ . Partition the set $\nu $ into two subsets

(4.2) $$ \begin{align}\begin{aligned} \nu_{=1} = \{v \in \nu \ | \ \langle m_{\sigma}, v\rangle = 1\} \text{ and }\\ \nu_{\ne 1} = \{ v \in \nu \ | \ \langle m_{\sigma}, v \rangle \ne 1\}. \end{aligned}\end{align} $$

Note that since $\sigma $ is $\operatorname {\mathbb {Q}}$ -Gorenstein, the ray generators of $\sigma $ are contained in $\nu _{=1}$ .

We consider the fan $\Psi $ above. Suppose the cone $|\Psi |$ is $\operatorname {\mathbb {Q}}$ -Gorenstein. Then we have the following result.

Using Theorem 4.11, we are left to study the category

$$ \begin{align*}\operatorname{D}^{\operatorname{abs}}[U_{\tilde \Sigma} \times \operatorname{\mathbb{G}_m}^{\nu \setminus \tilde\Sigma(1)}, G_{\Psi} \times \operatorname{\mathbb{G}_m}, \bar W]. \end{align*} $$

With some additional assumptions, we can prove that it is geometric. The following is half of [Reference Favero and KellyFK18, Corollary 5.15].

Proposition 4.12. Take a fan $\tilde \Sigma $ as in Theorem 4.11. Suppose there exist rays ${\rho ^{\prime }_1, \dots , \rho ^{\prime }_r \in \tilde \Sigma (1)}$ with primitive generators $e_{\rho ^{\prime }_1}, \dots , e_{\rho ^{\prime }_r} \in N \times \operatorname {\mathbb {Z}}^r$ so that

1. (1) the induced projection

   $$ \begin{align*}\pi: N\times \operatorname{\mathbb{Z}}^r \to (N\times \operatorname{\mathbb{Z}}^r) / (\oplus_{i=1}^t \operatorname{\mathbb{Z}} \cdot e_{\rho^{\prime}_i}) \end{align*} $$
   induces the toric morphism $\pi : X_{\Sigma ^{\prime }_{-D_1', \dots , -D_r'}} \to X_{\Sigma ^{\prime }}$ and this toric morphism is a rank r vector bundle whose sheaf of sections is $\oplus _{i=1}^r \mathcal {O}_{X_{\Sigma ^{\prime }}}(-D_i')$ , and

2. (2) $e_{\rho ^{\prime }_1}+ \dots + e_{\rho ^{\prime }_r}= e_1 + \dots + e_r \in \operatorname {\mathbb {N}} \times \operatorname {\mathbb {Z}}^r$ .

Write the function $\bar W$ as

$$ \begin{align*}\bar W = \sum_{\bar m \in \Xi_{i,W}} c_m \prod_{\rho \in \tilde\Sigma(1)} x_\rho^{\langle \bar m, u_{\rho}\rangle}. \end{align*} $$

Then, there exists a partition $\Xi _{i,W} = H_1' \cup \dots \cup H_r'$ so that we can write

$$ \begin{align*}\bar W = u_1'g_1 + \dots + u_r' g_r, \text{ where } g_i = \sum_{\bar m \in H_i'} \prod_{\rho \in \tilde\Sigma(1)\setminus\{\rho^{\prime}_1, \dots, \rho_r'\}} x_\rho^{\langle \bar m, u_\rho\rangle} \end{align*} $$

where $g_i \in \Gamma (X_{\Sigma ^{\prime }}, \mathcal {O}_{X_{\Sigma ^{\prime }}}(D_i'))$ . Then we have the quotient stack

(4.3) $$ \begin{align} [\mathcal Z'/G']:= [Z(g_1, \dots, g_r) / S_{\tilde\Sigma(1)}] \subseteq [U_{\tilde \Sigma}/S_{\Sigma}] \end{align} $$

where

$$ \begin{align*}\operatorname{D}^{\operatorname{abs}}[U_{\tilde \Sigma} \times \operatorname{\mathbb{G}_m}^{\nu \setminus \tilde\Sigma(1)}, G_{\Psi} \times \operatorname{\mathbb{G}_m}, \bar W] \cong \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal Z'/G']). \end{align*} $$

Note that if $[\mathcal Z'/G']$ is smooth, then $\operatorname {D}^{\operatorname {b}}(\operatorname {coh }[\mathcal Z'/G'])$ is homologically smooth and proper. Also, if $[\mathcal Z'/G']$ is a Calabi-Yau orbifold, $\operatorname {D}^{\operatorname {b}}(\operatorname {coh }[\mathcal Z'/G'])$ is Calabi-Yau. These observations are useful for satisfying the hypotheses of Theorem 3.7 and Corollary 3.10, respectively.

4.3 Categorical ramifications of the exoflop

In Section 3, we established the following diagram for when one partially compactifies the gauged Landau-Ginzburg model corresponding to a complete intersection in a toric variety:

Here, $\Omega $ is the equivalence in Corollary 3.4 and $i_* \circ \varphi ^*$

1. (i) with $\varphi _*\circ i^*$ forms a crepant categorical resolution if $\operatorname {D}^{\operatorname {abs}}[U_{\Psi }, G_{\Psi } \times \operatorname {\mathbb {G}_m}, \bar W]$ is homologically smooth and proper (Theorem 3.7);

2. (ii) is a fully faithful functor if $\operatorname {D}^{\operatorname {abs}}[U_{\Psi }, G_{\Psi } \times \operatorname {\mathbb {G}_m}, \bar W]$ is homologically smooth and proper and $\operatorname {D}^{\operatorname {b}}(\operatorname {coh }[\mathcal {Z}/G]) $ is a smooth Deligne-Mumford stack (Corollary 3.9); and

3. (iii) is an equivalence if $\operatorname {D}^{\operatorname {abs}}[U_{\Psi }, G_{\Psi } \times \operatorname {\mathbb {G}_m}, \bar W]$ is homologically smooth, proper, connected, and Calabi-Yau, and $\operatorname {D}^{\operatorname {b}}(\operatorname {coh }[\mathcal {Z}/G]) $ is a smooth Deligne-Mumford stack (Corollary 3.10).

In this section, we established a chain of relations:

Putting together, we have the following corollary that summarizes the exoflop’s power:

There are many conditions at play in Corollary 4.13. However, to illustrate its power, we describe in the next section combinatorial sufficient conditions using reflexive completely split Gorenstein cones.

5.1 Gorenstein Cones and CICYs

The following is a general result about almost Gorenstein cones.

Proof. Being reflexive Gorenstein with respect to $\mathfrak {m}$ implies that $\sigma $ is almost Gorenstein with respect to $\mathfrak {m}$ . Firstly, we note that $\sigma $ and $\sigma ^\vee $ both being completely split reflexive Gorenstein is equivalent to $\sigma $ being associated to a nef-partition by Corollary 3.7 of [Reference Batyrev and NillBN08]. That is, there exists some $e_1', \dots , e_r' \in N \times \operatorname {\mathbb {Z}}^r$ that form part of a $\operatorname {\mathbb {Z}}$ -basis for $N\times \operatorname {\mathbb {Z}}^r$ so that there exists a nef partition $\Delta _1+\dots +\Delta _r=\Delta $ with unique interior point $0$ in $N' := N\times \operatorname {\mathbb {Z}}^r / (\oplus _{i=1}^r \operatorname {\mathbb {Z}} e_i')$ and $\sigma =\operatorname {Cone}(\Delta _1+ e_1', \dots , \Delta _r + e_r')$ .

Denote by $V_i$ the vertex set of $\Delta _i + e_i'$ . Note $V_i\cap V_j=\emptyset $ . Let $V=\bigcup _{i=1}^r V_i$ . We note $\sigma =\operatorname {Cone}(V)$ , and by associating to each $p\in V$ the ray $\rho _p$ with primitive generator p, we have $\sigma (1)=\{\rho _p\ \vert \ p\in V\}$ . We want to show that $\sigma =|\Sigma ^{\prime }_{-D_1',\dots ,-D_r'}|$ for some vector bundle over a simplicial $\Sigma ^{\prime }$ . We prove this by direct construction.

Write $\pi : N_{\operatorname {\mathbb {R}}} \times \operatorname {\mathbb {R}}^r \to N_{\operatorname {\mathbb {R}}}'$ for the projection and $\overline {\rho }_p := \pi (\rho _p)$ . We have that the cone over $\pi (V) \setminus \{0\}$ has support $N_{\operatorname {\mathbb {R}}}'$ as $e_1'+\dots +e_r'$ is in the relative interior of $\sigma $ . Thus, there exists a complete fan in $N_{\operatorname {\mathbb {R}}}'$ with rays $ \{\overline {\rho }_p \ | \ p \in V, p \ne e_i' \text { for all } i\}$ . One can then simplicially subdivide to obtain a fan $\Sigma ^{\prime }$ . Let

$$ \begin{align*} D^{\prime}_i:=\sum_{\substack{ p\in V_i\\ p \ne e_i'\text{ for all } i}}D_{\overline{\rho}_p}. \end{align*} $$

Note that, by Corollary 3.17 of [Reference Batyrev and NillBN08], the images of $V_i$ and $V_j$ in $N'$ intersect only at the origin, thus each $D_{\rho _p}$ appears as a nontrivial summand in a unique divisor $D_i'$ . Following the standard toric vector bundle construction [Reference Cox, Little and SchenckCLS11, §7.3], we find that the vector bundle $\bigoplus \operatorname {\mathcal {O}}_{X_{\Sigma ^{\prime }}}(-D^{\prime }_i)$ has a fan $\Sigma ^{\prime }_{-D^{\prime }_1,\dots ,D^{\prime }_r}$ with rays $\{\rho _p\ \vert \ p\in V\}$ , i.e. $\Sigma ^{\prime }_{-D^{\prime }_1,\dots ,D^{\prime }_r}(1)=\sigma (1)$ , implying that $\sigma =|\Sigma ^{\prime }_{-D^{\prime }_1,\dots ,-D^{\prime }_r}|$ . As desired, we therefore have constructed directly a fan $\Sigma ^{\prime }_{-D^{\prime }_1,\dots ,-D_r'}$ that fulfills the conditions of Assumption 5.1.

We look to apply Corollary 5.2 while using the above Proposition. To do so, it is sufficient to check if the dual cone $\sigma _W^\vee $ to $\sigma _W$ defined in (5.3) is completely split reflexive Gorenstein, and that the LG model corresponds to a smooth complete intersection. In the rest of the section, we find combinatorial criteria and genericity hypotheses where both are satisfied.

The next result uses Lemma 5.4 and Bertini’s theorem to allow us to apply Corollary 5.2, hence providing a crepant categorical resolution as desired. First, let us set up some notation.

Notation 5.6. Let $\Xi $ be saturated and $\Psi $ a fan so that $\Xi \subseteq |\Psi |^\vee $ . We write $\mathcal {F}_{\Xi }$ for the family of polynomials $W: U_{\Psi } \to \mathbb {A}^1$ of the form

$$ \begin{align*}W = \sum_{m \in \Xi} c_m \prod_{\rho\in \Psi(1)} x^{\langle m, u_\rho\rangle }. \end{align*} $$

Corollary 5.7. Let $\Xi _W$ and $\sigma _W$ be as defined in (5.3) and suppose $[\mathcal Z / G]$ is positive dimensional. Suppose $\Xi _W$ is saturated, $\sigma _W^\vee $ is completely split reflexive Gorenstein of index r and that $W\in \mathcal F_\Xi $ is sufficiently generic. Then there is a crepant categorical resolution of $[\mathcal Z/G]$

$$ \begin{align*} F&:\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal Z'/G']) \to \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal Z/G]), \\ G&:\operatorname{Perf} [\mathcal Z/G] \to \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal Z'/G']) \end{align*} $$

by $[\mathcal Z'/G']$ as in (4.3). Moreover, if $[\mathcal {Z} / G]$ is smooth and $\dim [\mathcal {Z} / G]>0$ , then there is a derived equivalence between $[\mathcal {Z}/G]$ and $[\mathcal {Z}'/G']$ .

Proof. By Lemma 5.4, the cone $\sigma _W^\vee $ fulfills Assumption 5.1. To apply Corollary 5.2 and obtain the desired categorical resolution, it remains to show that the complete intersection $[\mathcal Z'/G']$ in $\mathcal X_{\Sigma ^{\prime }}$ is indeed smooth. In the proof of Proposition 5.3, we have shown that the lattice polytopes giving $\sigma _W^\vee $ its Cayley structure in fact give a nef-partition of $\Delta _{-K_{\Sigma ^{\prime }}}$ . Hence, the divisors $D^{\prime }_i$ corresponding to the nef-partition and giving the vector bundle $\bigoplus _{i=1}^r \operatorname {\mathcal {O}}_{X_{\Sigma ^{\prime }}}(-D^{\prime }_i)$ are nef. In particular, by Proposition 6.3.12 in [Reference Cox, Little and SchenckCLS11] these divisors are basepoint free. Recall that any section $g\in \Gamma (\bigoplus \operatorname {\mathcal {O}}_{X_{\Sigma ^{\prime }}}(-D_i'))$ can be expressed via a sum of monomials

$$\begin{align*}\sum_{m\in H_{\mathfrak{n}}(1)\cap (M\times\operatorname{\mathbb{Z}}^r)\cap |\Psi|^\vee\cap M}c_mx^m\end{align*}$$

for some coefficients $c_m$ . By Bertini’s Theorem, a generic section $(g_1,\dots ,g_r)$ of the vector bundle will give a smooth complete intersection $[\mathcal Z'/G']:=Z(g_i)\subseteq \mathcal {X}_{\Sigma ^{\prime }}$ . The set of such generic sections is open and dense in the linear system spanned by the divisors $D_i'$ . As $\Xi _W$ is saturated, the open and dense set of generic sections must thus intersect the family corresponding to sections of the form $\sum _{m\in \Xi _W}c_mx^m$ , i.e. there is an element W in $\mathcal F_W$ such that the corresponding complete intersection $[\mathcal Z'/G']\subseteq \mathcal {X}_{\Sigma ^{\prime }}$ is smooth. Corollary 5.2 then gives the crepant categorical resolution as desired. We remark that these categories are connected when the Calabi-Yau orbifolds are positive dimensional and thus the equivalence holds if the additional hypotheses are satisfied.

The Corollary 5.7 gives us combinatorial conditions we can check to generate categorical resolutions as in Corollary 5.2. Since this is useful, we give below another formulation of it that may be more user-friendly for applications.

One can rewrite Corollary 5.7 in terms of the support polytopes in the following way.

Corollary 5.9. Suppose $\Xi $ is saturated and that $\tilde {\Delta }=\operatorname {Conv}(\Xi _W)$ and its dual $\tilde {\Delta }^\vee $ are both integrally closed Gorenstein polytopes (of index r). Then, for a generic polynomial W in the family $\mathcal F_\Xi $ , there is a crepant categorical resolution of $[\mathcal Z/G]$

$$ \begin{align*} F&:\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal Z'/G']) \to \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal Z/G]), \\ G&:\operatorname{Perf} [\mathcal Z/G] \to \operatorname{D}^{\operatorname{b}}(\operatorname{coh }[\mathcal Z'/G']) \end{align*} $$

by $[\mathcal Z'/G']$ as in (4.3).

6.1 Aspinwall’s example

We first explain an explicit example. We choose to repeat Aspinwall’s primary example in his paper [Reference AspinwallAsp15, §2.3, 3.1-5]. In some sense this is a simple case in comparison to the general case of what can happen above as the partial compactification in the exoflop is as simple as possible, but it provides intuition on why an exoflop can give rise to a categorical resolution.

Consider a quartic

$$ \begin{align*}f = f_4(x_1, x_2, x_3) + x_0 f_3(x_1,x_2,x_3)+ x_0^2 f_2(x_1,x_2,x_3) \end{align*} $$

where $f_k(x_1, x_2, x_3)$ are homogeneous equations of degree k. We assume that the $f_k$ are generic enough to avoid additional singularities. We have that $f \in \Gamma (\operatorname {\mathbb {P}}^3, \mathcal {O}_{\operatorname {\mathbb {P}}^3}(4))$ , and $Z(f)\subseteq \operatorname {\mathbb {P}}^3$ is a singular quartic surface. Indeed, its singular locus is the point $(1:0:0:0)$ .

The vector bundle $\operatorname {tot} \operatorname {\mathcal {O}}_{\operatorname {\mathbb {P}}^3}(-4)$ can be written as a quotient stack $[(\operatorname {\mathbb {A}}^4 \setminus \{0\}) \times \operatorname {\mathbb {A}}^1 / \operatorname {\mathbb {G}_m}]$ , where $\operatorname {\mathbb {G}_m}$ acts with weights $(1,1,1,1,-4)$ . Write u for the variable corresponding to the last coordinate. There is a $\operatorname {\mathbb {G}_m}$ -invariant global function

$$ \begin{align*}W=uf: (\operatorname{\mathbb{A}}^4 \setminus\{0\}) \times \operatorname{\mathbb{A}}^1 \to \operatorname{\mathbb{A}}^1. \end{align*} $$

One computes that the critical locus has two irreducible components, when $u=f=0$ and $Z(x_1,x_2, x_3)$ . The former component, when viewed in the stack $[(\operatorname {\mathbb {A}}^4 \setminus \{0\}) \times \operatorname {\mathbb {A}}^1 / \operatorname {\mathbb {G}_m}]$ is proper and isomorphic to $Z(f)$ in the zero section $u=0$ . The latter is the $\operatorname {\mathbb {A}}^1$ corresponding to the fiber over the point $(1:0:0:0)$ .

We construct a partial compactification of the quotient stack $[(\operatorname {\mathbb {A}}^4 \setminus \{0\}) \times \operatorname {\mathbb {A}}^1 / \operatorname {\mathbb {G}_m}]$ and extend W. To do so, there is a stack isomorphism

$$ \begin{align*}\varphi: [(\operatorname{\mathbb{A}}^4 \setminus\{0\}) \times \operatorname{\mathbb{A}}^1 / \operatorname{\mathbb{G}_m}] \to [(\operatorname{\mathbb{A}}^4 \setminus\{0\}) \times \operatorname{\mathbb{A}}^1 \times \operatorname{\mathbb{G}_m} / (\operatorname{\mathbb{G}_m})^2] \end{align*} $$

where the two $\operatorname {\mathbb {G}_m}$ act by weights $(1,1,1,1,-4,0)$ and $(1,0,0,0,-2,1)$ . We will use the variable y for this new coordinate. Consider the $(\operatorname {\mathbb {G}_m})^2$ -equivariant open immersion

$$ \begin{align*}i: (\operatorname{\mathbb{A}}^4 \setminus\{0\}) \times \operatorname{\mathbb{A}}^1 \times \operatorname{\mathbb{G}_m} \hookrightarrow \operatorname{\mathbb{A}}^6 \setminus Z(yx_0, yx_1, yx_2, yx_3, ux_0). \end{align*} $$

Here, we can extend W by taking the $(\operatorname {\mathbb {G}_m})^2$ -invariant function

$$ \begin{align*}\bar W= u(\bar f), \text{ where } \bar f = y^2f_4(x_1, x_2, x_3) + yx_0 f_3(x_1,x_2,x_3)+ x_0^2 f_2(x_1,x_2,x_3) \end{align*} $$

Since the additional strata added by the open immersion are away from the zero section of the line bundle, the first component of the critical locus of $\bar W$ is the same as W. However, the second is compactified to be a weighted projective line.

Remark 6.1. The above construction is simple in toric geometry. Take the standard fan for $\operatorname {\mathbb {P}}^3$ . The vector bundle is the toric variety associated to the fan obtained by the star subdivision at the ray generated by the lattice point $(0,0,0,1)$ of the cone

$$ \begin{align*}\operatorname{Cone}((1,0,0,1),(0,1,0,1),(0,0,1,1), (-1,-1,-1,1)).\end{align*} $$

The partial compactification is found by adding the cone

$$ \begin{align*}\operatorname{Cone}((-1,0,0,1),(0,1,0,1),(0,0,1,1), (-1,-1,-1,1))\end{align*} $$

to the fan. The maximal special linear system allowed to take this partial compactification corresponds to

$$ \begin{align*} \Xi = (M \times \operatorname{\mathbb{Z}}) \cap \operatorname{Conv}(&(-1,-1,-1,1), (-1,3,-1,1), (-1,-1,3,1), \\ &(1,-1,-1,1), (1,1,-1,1),(1,-1,1,1)) \end{align*} $$

Taking a generic enough potential using $\Xi $ is equivalent to choosing $f_4, f_3$ , and $f_2$ above generic enough to avoid singularities.

There is a toric flop (given by GIT) corresponding to the following birational map

$$ \begin{align*}\psi: \operatorname{\mathbb{A}}^6 \setminus Z(yx_0, yx_1, yx_2, yx_3, ux_0) \dashrightarrow \operatorname{\mathbb{A}}^6 \setminus Z(x_0x_1, x_0x_2, x_0x_3, yx_1, yx_2, yx_3) \end{align*} $$

One can check that $[\operatorname {\mathbb {A}}^6 \setminus Z(x_0x_1, x_0x_2, x_0x_3, yx_1, yx_2, yx_3) / (\operatorname {\mathbb {G}_m})^2]$ is the total space of the anticanonical bundle of the blow up $\operatorname {Bl}_{(1:0:0:0)}\operatorname {\mathbb {P}}^3$ of $\operatorname {\mathbb {P}}^3$ at $(1:0:0:0)$ , and that ${\bar f \in \Gamma (\operatorname {Bl}_{(1:0:0:0)}\operatorname {\mathbb {P}}^3, -K_{\operatorname {Bl}_{(1:0:0:0)}\operatorname {\mathbb {P}}^3})}$ . Since f was chosen sufficiently generically, $Z(\bar f)$ is smooth. One then obtains that $Z(\bar f)\subseteq \operatorname {Bl}_{(1:0:0:0)}\operatorname {\mathbb {P}}^3 $ is a categorical resolution of $Z(f)\subseteq \operatorname {\mathbb {P}}^3$ (as an example of Corollary 5.2).

6.2 Derived equivalences with varying bundle structures

As seen above, a standard (toric) resolution of singularities can appear from an exoflop, but there are some derived equivalences that are found where a birational equivalence is not obvious. In this section we exhibit the convex geometry that leads to such a derived equivalence. This involves when the toric vector bundle structures differ (that is, there are different sets of minimal generators that sum to $\mathfrak {n}$ , as seen in Assumption 5.1). We will consider three different Calabi-Yau complete intersections (we have chosen a small dimensional case to attempt to not cloud the example with too much unnecessary toric geometry).

For this example, we work in $N= \operatorname {\mathbb {Z}}^5$ and take the rays $\rho _1, \dots , \rho _{12}$ with minimal generators

$$ \begin{align*} &u_{\rho_1}= (2,0,-1,0,1), \hspace{0.45em} &u_{\rho_2} = (0,2,-1,0,1), \hspace{0.45em} &u_{\rho_3}=(-1,-1,2,1,0), \hspace{0.45em} &u_{\rho_4} = (-1,-1,0,1,0), \\ &u_{\rho_5} = (1,-1,0,1,0), \hspace{0.45em} &u_{\rho_6} = (-1,1,0,1,0), \hspace{0.45em} &u_{\rho_7}=(0,0,1,0,1), \hspace{0.45em} &u_{\rho_8} = (0,0,-1,0,1), \\ &u_{\rho_9} = (0,-1,0,1,0), \hspace{0.45em} &u_{\rho_{10}} = (0,1,0,0,1), \hspace{0.45em} &u_{\rho_{11}} = (0,0,0,1,0), \hspace{0.45em} &u_{\rho_{12}}=(0,0,0,0,1). \end{align*} $$

Consider the following three cones:

$$ \begin{align*} & \sigma = \operatorname{Cone}(\rho_1,\dots,\rho_8),\\ & \sigma_1=\operatorname{Cone}(\rho_1,\rho_2,\rho_3,\rho_4,\rho_{11},\rho_{12}),\\ & \sigma_2=\operatorname{Cone}(\rho_1,\rho_2,\rho_3,\rho_4, \rho_9, \rho_{10}). \end{align*} $$

We refer the reader to Figure 1 to get some geometric intuition in the convex geometry. Note that $\rho _9, \rho _{10}, \rho _{11}, \rho _{12} \in \sigma $ , hence $\sigma _1,\sigma _2\subseteq \sigma $ . Moreover, all 3 cones are completely split Gorenstein cones of index 2 with respect to $\mathfrak {m} = (0,0,0,1,1)$ .

Figure 1

A depiction of the minimal generators $u_i := u_{\rho _i}$ of the cone $\sigma \subseteq \operatorname {\mathbb {R}}^5$ when projected to the first three coordinates. Here, the minimal generators in black have their fourth coordinate $1$ and those in magenta have fifth coordinate $1$ . The purple vertex is $(0,0,0)$ and the only point where the tetra intersect after projecting down to the first three coordinates. The cone $\sigma $ is a cone over the Cayley product of these two tetrahedra.

Figure 1 Long description

At the center is a purple vertex labeled u sub 7, marking the intersection of two tetrahedra. The black tetrahedron, positioned lower left, has vertices labeled u sub 3, u sub 4, u sub 5, and u sub 6, with additional black points u sub 9 on edge u sub 5 to u sub 7. The magenta tetrahedron, positioned upper right, has vertices labeled u sub 1, u sub 2, u sub 7, and u sub 8, with additional magenta points u sub 10 on edge u sub 2 to u sub 7 and u sub 8 on edge u sub 1 to u sub 7. All edges are drawn between corresponding vertices, forming two overlapping tetrahedra. The black tetrahedron's vertices have fourth coordinate one, while the magenta tetrahedron's vertices have fifth coordinate one. The intersection at u sub 7 is the only shared point after projection to the first three coordinates.

Note that the cone $\sigma ^\vee $ is a completely split Gorenstein cone of index 2 with respect to $\mathfrak {n} = (0,0,0,1,1)$ . We take $\Xi _W = H_{\mathfrak {n}}(1)\cap M \cap \sigma ^\vee $ , which one can compute to be

$$ \begin{align*}\Xi_W = \{(1,0,0,1,0), (0,1,0,1,0), (0,0,1,0,1), (-1,-1,-1,0,1), (0,0,0,1,0), (0,0,0,0,1)\}. \end{align*} $$

We name each of these above lattice points $m_1, \dots , m_6 \in \Xi _W$ (in the above order), and can write the following global function on any fan $\Sigma $ with $\Sigma (1) = \{ \rho _1, \dots , \rho _{12}\}$ .

$$ \begin{align*} W = \sum_{i=1}^6 c_i \prod_{\rho\in \sigma(1)} x^{\langle m_i, u_{\rho_i}\rangle } &= c_1 x_1^2 x_5^2x_9x_{11} + c_2 x_2^2 x_6^2 x_{10}x_{11} + c_3 x_3^2 x_7^2 x_{10} x_{12} \\ &\qquad +c_4 x_4^2 x_8^2 x_9x_{12} + c_5 x_3x_4x_5x_6x_9 x_{11} + c_6 x_1x_2x_7x_8x_{10}x_{12}. \end{align*} $$

We can use W to define complete intersections in different toric varieties. Since $\sigma , \sigma _1, \sigma _2$ all are completely split Gorenstein cones of index 2, there exist fans $\Sigma , \Sigma _1,$ and $\Sigma _2$ that are total spaces of rank two vector bundles over dimension 3 toric varieties.

For $\Sigma _1$ , we star subdivide with respect to $\rho _{11}$ and $\rho _{12}$ as $u_{\rho _{11}}+u_{\rho _{12}}= \mathfrak {n}$ . In this case, we can reduce the potential to

$$ \begin{align*}W' = (c_1 x_1^2 + c_2 x_2^2 + c_5x_3x_4)x_{11} + (c_3 x_3^2 + c_4 x_4^2 + c_6 x_1x_2)x_{12}. \end{align*} $$

and one can compute that this corresponds to a complete intersection

$$ \begin{align*}\mathcal{Z}' := Z(c_1 x_1^2 + c_2 x_2^2 + c_5x_3x_4, c_3 x_3^2 + c_4 x_4^2 + c_6 x_1x_2) \subseteq [\mathbb{P}^3 / (\operatorname{\mathbb{Z}}/4\operatorname{\mathbb{Z}})], \end{align*} $$

where a generator g of the $\operatorname {\mathbb {Z}}/4\operatorname {\mathbb {Z}}$ acts on $\operatorname {\mathbb {P}}^3$ by $g\cdot (x_1:x_2:x_3:x_4) = (x_1:-x_2:ix_3:-ix_4)$ .

On the other hand, for $\Sigma _2$ , we star subdivide with respect to $\rho _9$ and $\rho _{10}$ as $u_{\rho _{9}}+u_{\rho _{10}}= \mathfrak {n}$ . The potential reduces to

$$ \begin{align*}W"= (c_1 x_1^2 +c_4 x_4^2+ c_5 x_3x_4)x_9 + (c_2 x_2^2 + c_3 x_3^2 + c_6 x_1x_2)x_{10} \end{align*} $$

and this corresponds to the complete intersection

$$ \begin{align*}\mathcal{Z}" = Z(c_1 x_1^2 +c_4 x_4^2+ c_5 x_3x_4, c_2 x_2^2 + c_3 x_3^2 + c_6 x_1x_2) \subseteq [\mathbb{P}^3 / (\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}})], \end{align*} $$

where the generator g of $\operatorname {\mathbb {Z}}/2\operatorname {\mathbb {Z}}$ acts on $\operatorname {\mathbb {P}}^3$ by $g\cdot (x_1:x_2:x_3:x_4) = (-x_1:-x_2:x_3:x_4)$ .

If the $c_i$ are generic, one can check that both $\mathcal {Z}'$ and $\mathcal {Z}''$ are smooth Calabi-Yau orbifolds. Lastly, there is a $\Sigma $ with support $\sigma $ that is a rank 2 vector bundle and one can use W to define a Calabi-Yau complete intersection $[\mathcal {Z}/G]$ in it. By Corollary 5.7, there is a derived equivalence between $[\mathcal {Z}/G]$ and both $\mathcal {Z}'$ and $\mathcal {Z}''$ , hence $\operatorname {D}^{\operatorname {b}}(\operatorname {coh }\mathcal {Z}') \cong \operatorname {D}^{\operatorname {b}}(\operatorname {coh }\mathcal {Z}'')$ .

Note the ‘shuffling’ of monomials that happens between the two potentials $W'$ and $W''$ , which happens when different rays are used as the rays corresponding to the bundle coordinates. This can be used to make interesting equivalences between CICYs. In this example, it can be shown that $\mathcal Z'$ and $\mathcal Z''$ are birational, but it is not clear if the derived equivalent Calabi-Yau complete intersections are always birational for higher-dimensional examples.

6.3 A higher-dimensional generalization of the Libgober-Teitelbaum family

In [Reference Libgober and TeitelbaumLT93], Libgober and Teitelbaum proposed a mirror to a highly symmetric Calabi-Yau complete intersection given by two cubics in $\operatorname {\mathbb {P}}^5$ . In [Reference MalterMal24], it was proven to be derived equivalent to the Batyrev-Borisov mirror to two cubics in $\operatorname {\mathbb {P}}^5$ . In this subsection, we look at the most natural generalization to the Libgober-Teitelbaum family, and show it has a crepant categorical resolution using the Batyrev-Borisov mirror to the complete intersection of two degree n polynomials in $\operatorname {\mathbb {P}}^{2n-1}$ . We fix $n\in \operatorname {\mathbb {Z}}$ , $n\ge 2$ , throughout what follows.

Define two polynomials

$$ \begin{align*} Q_{1,\lambda}&=x_1^n+x_2^n+\dots+x_n^n-\lambda x_{n+1}\dots x_{2n}, \\ Q_{2,\lambda}&=x_{n+1}^n+x_{n+2}^n+\dots+x_{2n}^n-\lambda x_1\dots x_n. \end{align*} $$

Their complete intersection $Z_{\lambda }:=Z(Q_{1,\lambda }, Q_{2,\lambda })\subseteq \operatorname {\mathbb {P}}^{2n-1}$ is a smooth Calabi-Yau complete intersection in $n=3$ for $\lambda $ such that $\lambda ^6\ne 0,3^6$ and it is a singular complete intersection otherwise. It is also highly symmetric.

Denote by $\zeta _n$ a primitive n-th root of unity. Consider $\alpha _i,\beta _i\in \operatorname {\mathbb {Z}}\pmod {n}$ ( $1\le i\le n-1$ ) and $\delta \in \operatorname {\mathbb {Z}}\pmod {n^2}$ such that

$$\begin{align*}\zeta_{n^2}^\delta=\zeta_n^{\beta_1+\dots+\beta_{n-1}}=\zeta_n^{\alpha_1+\dots+\alpha_{n-1}}. \end{align*}$$

Consider the following subgroup $G_n$ of $PGL(2n-1,\operatorname {\mathbb {C}})$ , given by automorphisms of the form

$$\begin{align*}g_{\underline{\alpha},\underline{\beta},\delta}=\left\{\begin{array}{@{}ll} \operatorname{diag}\left( \zeta_{n^2}^\delta,\zeta_{n^2}^\delta\zeta_n^{\alpha_1},\dots,\zeta_{n^2}^\delta\zeta_n^{\alpha_{n-1}},\zeta_{n^2}^{-\delta}\zeta_n^{\beta_1},\dots,\zeta_{n^2}^{-\delta}\zeta_n^{\beta_{n-1}},\zeta_{n^2}^{-\delta}\right) & \text{if } n \text{ odd,} \\ \operatorname{diag}\left( \zeta_{2n^2}^\delta,\zeta_{2n^2}^\delta\zeta_n^{\alpha_1},\dots,\zeta_{2n^2}^\delta\zeta_n^{\alpha_{n-1}},\zeta_{2n^2}^{-\delta}\zeta_n^{\beta_1},\dots,\zeta_{2n^2}^{-\delta}\zeta_n^{\beta_{n-1}},\zeta_{2n^2}^{-\delta}\right) & \text{if } n \text{ even.} \end{array}\right. \end{align*}$$

One can check that the action of $G_n$ on $\operatorname {\mathbb {P}}^{2n-1}$ acts invariantly on the variety $Z_{\lambda }$ , hence we can define the orbifold $[\mathcal {Z}_n/G_n]:=Z(Q_{1,n,\lambda },Q_{2,n,\lambda })\subseteq [\operatorname {\mathbb {P}}^{2n-1}/G_n]$ . For $n=3$ , this is the Libgober-Teitelbaum mirror to the complete intersection of two cubics in $\operatorname {\mathbb {P}}^5$ , which is proven to be derived equivalent to members of the Batyrev-Borisov mirror family in [Reference MalterMal24]. However, for $n\geq 4$ , the techniques for proving derived equivalence to the corresponding Batyrev-Borisov construction used in loc. cit. do not work any further, as $[\mathcal {Z}_n/G_n]$ is singular. Using the results of § 5, we will demonstrate instead that the natural result of applying the Batyrev-Borisov construction yields categorical resolutions to the family $[\mathcal {Z}_n/G_n]$ .

We start by noting that since $[\mathcal {Z}_n/G_n]$ is a complete intersection of the two $Q_{i,\lambda }$ in $\mathcal {X}_n=[\operatorname {\mathbb {P}}^{2n-1}/G_n]$ , there is a corresponding gauged LG model with superpotential

$$\begin{align*}W=u_1Q_{1,\lambda}+u_2Q_{2,\lambda}, \end{align*}$$

This will be a global function for the total space $\operatorname {tot}(\operatorname {\mathcal {O}}_{\mathcal {X}_n}(-D_1)\oplus \operatorname {\mathcal {O}}_{\mathcal {X}_n}(-D_2))$ of a rank two vector bundle where $Q_{i,\lambda } \in \Gamma (\mathcal {X}_n, \operatorname {\mathcal {O}}_{\mathcal {X}_n} (D_i))$ . One can construct the toric variety for this vector bundle, which will be $N_{\operatorname {\mathbb {R}}}$ where $N=\operatorname {\mathbb {Z}}^{2n+1}$ . Give this lattice the standard $\operatorname {\mathbb {Z}}$ -basis $e_1, \dots , e_{2n+1}$ . We now will define the rays $\rho _1, \dots , \rho _{2n}, \tau _1, \tau _2$ of the fan $\Sigma _{\mathcal {X}_n}$ by giving its primitive generators. To do so, we first write $\delta _1 := \sum _{i=1}^n e_i$ and $\delta _2 := \sum _{i=n+1}^{2n-1} e_i$ . Then the primitive generators for the rays are

(6.1) $$ \begin{align}\begin{aligned} u_{\rho_i} &= ne_i - \delta_2 + e_{2n+1}, \text{ for } 1 \le i \le n;\\ u_{\rho_i} &= -\delta_1 + n e_i + e_{2n}, \text{ for } n+1 \le i \le 2n-1;\\ u_{\rho_{2n}} &= -\delta_1 + e_{2n};\\ u_{\tau_1} &= e_{2n}; \\ u_{\tau_2} &= e_{2n+1}. \end{aligned}\end{align} $$

We associate the coordinates $x_1, \dots , x_{2n}$ to $\rho _1, \dots , \rho _{2n}$ and $u_1, u_2$ to $\tau _1, \tau _2$ . One can see that $\operatorname {tot}(\operatorname {\mathcal {O}}_{\mathcal {X}_n}(-D_1)\oplus \operatorname {\mathcal {O}}_{\mathcal {X}_n}(-D_2))$ is the toric variety corresponding to the fan obtained by star subdividing the cone $\operatorname {Cone}(\rho _1, \dots , \rho _{2n}, \tau _1, \tau _2)$ along $\tau _1$ and $\tau _2$ .

We now move to the potential W. Write $e_i^*\in M$ for the dual basis vector to $e_i$ . Define $\delta _1^* := \sum _{i=1}^n e_i^*$ and $\delta _2^* := \sum _{i=n+1}^{2n-1} e_i^*$ . Next, one can compute that $W = \sum _{i=1}^{2n} x^{m_i} - \lambda x^{m_{2n+1}} - \lambda x^{m_{2n+2}}$ , where

(6.2) $$ \begin{align}\begin{aligned} m_i &= e_i^* + e_{2n}^*, \text{ for } 1 \le i \le n, \\ m_i &= e_i^* + e_{2n+1}^*, \text{ for } n+1 \le i \le 2n-1, \\ m_{2n} &= (- \textstyle\sum_{i=1}^{2n-1} e_i^*) + e_{2n+1}^*, \\ m_{2n+1} &= e_{2n}^*, \\ m_{2n+2} &= e_{2n+1}^*. \end{aligned}\end{align} $$

Hence $\Xi _W = \{m_1, \dots , m_{2n+2}\}$ and $\sigma _W = \operatorname {Cone}(\Xi _W)$ . The dual cone $\sigma _W^\vee $ is then the cone over the $4n+2$ points $u_{\rho _1}, \dots , u_{\rho _{4n}}, u_{\tau _1}, u_{\tau _2}$ , where $u_{\rho _1}, \dots , u_{\rho _{2n}}, u_{\tau _1}, u_{\tau _2}$ are as in (6.1) and

(6.3) $$ \begin{align}\begin{aligned} u_{\rho_{2n+i}} &= - \delta_1 + ne_i + e_{2n}, \text{ for } 1 \le i \le n, \\ u_{\rho_{3n+i}} &= - \delta_2 + ne_{n+i} + e_{2n+1}, \text{ for } 1 \le i \le n-1, \\ u_{\rho_{4n}} &= -\delta_2 + e_{2n+1}. \end{aligned}\end{align} $$

The cone $\sigma _W^\vee $ is completely split reflexive Gorenstein of index 2. Thus, Lemma 5.4 implies that $\sigma _W^\vee $ fulfills Assumption 5.1.

Denote by $\Sigma ^{\prime }_{-D_1',-D_2'}$ the fan whose support is $\sigma _W^\vee $ in Assumption 5.1. Write $x_i$ for the variables associated to the new rays $\rho _i$ . The extended superpotential $\bar W$ on the partial compactification takes the form

$$ \begin{align*} \bar W &=u_1(x_1^nx_{2n+1}^n+\dots+x_n^{n}x_{3n}^n-\lambda x_{n+1}\cdots x_{3n}) \ + \\ &\qquad u_2(x_{n+1}^nx_{3n+1}^n+\dots+x_{2n}^nx_{4n}^n-\lambda x_1\cdots x_n\cdot x_{3n+1}\cdots x_{4n}). \end{align*} $$

Then one obtains a complete intersection $[\mathcal {Z}^{\prime }_n / G_n'] \subseteq [U_{\Sigma ^{\prime }} / G_{\Sigma ^{\prime }}]$ .

Lastly, one can verify that the complete intersection $[\mathcal {Z}^{\prime }_n/G_n']$ as in (4.3) is smooth when $\lambda ^{2n} \ne 0,n^{2n}$ , hence Corollary 5.2 implies that we have a categorical resolution of $[\mathcal {Z}_n/G_n]$ via $[\mathcal {Z}^{\prime }_n/G_n']$ .

6.4 Mirror Constructions

Take a $\operatorname {\mathbb {Q}}$ -Fano toric variety corresponding to a polytope $\Delta $ . In [Reference Artebani, Comparin and GuilbotACG16], Artebani, Comparin and Guilbot prove that general hypersurfaces associated to a special linear system corresponding to canonical subpolytopes $\Delta '$ of the anticanonical polytope $\Delta $ are Calabi-Yau. In particular, given a $\operatorname {\mathbb {Q}}$ -Fano toric variety with anticanonical polytope $\Delta _2$ , one can take a special linear system corresponding to a canonical polytope $\Delta _1$ and consider its corresponding family of Calabi-Yau varieties. Here $(\Delta _1, \Delta _2)$ form a good pair if both $\Delta _1$ and $\Delta _2^*$ are canonical. Consequently, $(\Delta ^*_2, \Delta ^*_1)$ also form a good pair, forming a duality. This generalizes both mirror constructions of Batyrev-Borisov and Berglund-Hübsch-Krawitz. In the former case, when $\Delta $ is reflexive, then $(\Delta , \Delta )$ is a good pair and one recovers Batyrev duality. There has been recent work by Rossi that generalizes this work and suggests an iterative process of doing this, by introducing what is known as f-duality [Reference RossiRos22, Reference RossiRos23]. In this case, however, the type of singularities that can arise is unclear.

In [Reference Doran, Favero and KellyDFK18, Theorem 1.2], the authors proved that given good pairs $(\Delta _1, \Delta _2)$ and $(\Delta _1', \Delta _2)$ members of their dual families were derived equivalent. This is expected, as when one has multiple constructions for the mirror of two symplectomorphic manifolds, these ‘multiple’ mirrors must be derived equivalent according to the Homological Mirror Symmetry Conjecture. As $\Delta _1$ and $\Delta _1'$ define two different families of hypersurfaces in the same toric variety, a generalization of Moser’s theorem would imply a symplectomorphism.

One expects a similar situation in the complete intersection case. To our knowledge, there is no sufficient criterion for polytopes defining complete intersections that are Calabi-Yau that is weaker than using the standard nef partition of reflexive polytopes for codimension higher than one. Consider nef partitions of reflexive polytopes ${\Delta _1 = \Delta _{1,1} + \dots + \Delta _{1,r}}$ , and $\Delta _2 = \Delta _{2,1} + \dots + \Delta _{2,r}$ in $M_{\operatorname {\mathbb {R}}}$ where one has the inclusion of Cayley polytopes

$$ \begin{align*} \Delta_{1,1} * \cdots * \Delta_{1,r} \subseteq \Delta_{2,1} * \dots * \Delta_{2,r}. \end{align*} $$

Since $\{\Delta _{i,j}\kern-1pt\}_{j=1}^r$ is a nef partition, it corresponds to vector bundles $\mathcal {V} \kern1.3pt{=}\kern1.3pt \bigoplus _{j=1}^r\kern-1pt \operatorname {\mathcal {O}}_{X_{\Sigma _{\Delta _i}}}\kern-1pt(-D_{i,j}\kern-1pt)$ over the toric varieties $X_{\Sigma _{\Delta _i}}$ , where $\Sigma _{\Delta _i}$ is the normal fan to $\Delta _i$ and $D_{i,j}$ is the divisor associated to the polytope $\Delta _{i,j}$ . The lattice points in $\Delta _{1,1} * \cdots * \Delta _{1,r}$ correspond to global functions of the vector bundle. Each global function corresponds to a (stacky) complete intersection $\mathcal {Z}_i$ . We then have by Corollary 5.7 a derived equivalence between $\mathcal {Z}_1$ and $\mathcal {Z}_2$ .

If so, then, when one can use such a new mirror construction or Batyrev-Borisov, then we expect there to be a derived equivalence between these new Calabi-Yau mirrors and those constructed by Batyrev-Borisov that can be proven using exoflops.