1 Introduction↩︎

1.1 Double covers and Landau–Ginzburg models↩︎

Let \(X\) denote a smooth, projective variety. All varieties in this paper will be defined over \(\mathbb{C}\). The Hodge theory of a branched double cover \[\rho: \widehat{X} \to X\] is a classical topic that has been extensively studied; for instance in the foundational work of Esnault and Viehweg [1] (see also [2], [3]). Without loss of generality, we may assume that the double cover \(\widehat{X}\) is determined by a global section \(\sigma \in \Gamma(X, L^{ 2})\) of a line bundle \(L\) on a smooth variety \(X\). The branch locus is the zero locus of \(\sigma\), denoted by \(B = \{\sigma = 0\}\), and the ramification locus is given by the preimage of \(B\), \(R:=\rho^{-1}(B)\). When \(B\) is smooth (or more generally, a simple normal crossings divisor), the de Rham complex \(\mathrm{DR}({\widehat{X}}) := (\Omega^\bullet_{\widehat{X}}, \mathrm{d})\), equipped with the usual Hodge filtration \(F^\bullet\), admits a natural decomposition induced by the canonical \((\mathbb{Z}/2)\)-action on \(\widehat{X}\): \[\label{e:intro32DR32Z472} \mathrm{R}\rho_*\mathrm{DR}({\widehat{X}}) = \mathrm{DR}({\widehat{X}})^{(+)} \oplus \mathrm{DR}({\widehat{X}})^{(-)}\tag{1}\] where the superscripts \((+)\) and \((-)\) indicate the invariant and coinvariant part, respectively. The invariant part is isomorphic to the de Rham complex of \(X\), \(\mathrm{DR}(X)\), while the coinvariant part is naturally identified with the following twisted de Rham complex on the open subset \(X \setminus B\), \[\mathrm{DR}(X \setminus B, \log \sqrt{\sigma}) := (\Omega_X^\bullet(\log B) \otimes L^{-1},\mathrm{d} + \tfrac{1}{2}\mathrm{d}\log \sigma).\] We often denote its cohomology by \(H^*(\widehat{X}\setminus R)^{(-)}\) and Hodge filtration by \(F^\bullet\), which is simply defined as the stupid filtration as usual. Then the identification in 1 respects the Hodge filtration \(F^\bullet\).

An interesting feature of the setup above is that the data \((L, \sigma)\) gives rise to two distinct Landau–Ginzburg models. The first one is a pair \((\mathrm{Tot}(L^{-2}), g_{1,\sigma})\) where the potential \(g_{1,\sigma}:=\langle \sigma, -\rangle\) is the linear map induced by the section \(\sigma\). The second one is a pair \((\mathrm{Tot}(L^{-1}), g_{2,\sigma})\) where the potential \(g_{2,\sigma}=\langle \sigma, (-)^2 \rangle\) is the quadratic function induced by \(\sigma\). One of our goals is to study Hodge theory of these Landau–Ginzburg models and verify the relationship with Hodge theory of the double cover \(\widehat{X}\) (see Theorem 1).

The Hodge theory of Landau–Ginzburg models has been developed in recent years [4]–[9]. Recall that for any Landau–Ginzburg model \((Y,f: Y \to \mathbb{A}^1)\), the relevant de Rham cohomology is given by the cohomology of the twisted de Rham complex \(\mathrm{DR}(Y, f):=(\Omega_Y^\bullet, \mathrm{d} + \mathrm{d}f)\) where we simply write \(\mathrm{d}f:=\mathrm{d}f\wedge\). The Hodge filtration on cohomology was constructed by Yu [6], which is called the irregular Hodge filtration, denoted by \(F^\bullet_{\mathrm{irr}}\). This filtration is a finite \(\mathbb{Q}\)-filtration where the non-integrality of the grading reflects the multiplicities of the polar divisor of \(f\). We review some basic properties of the irregular Hodge filtration in Section 2.

According to the general philosophy of the Cayley method, the Hodge-theoretic data of the Landau–Ginzburg model \((\mathrm{Tot}(L^{-2}), g_{1,\sigma})\) reflects the Hodge theory of the branch hypersurface \(B \subset X\). More precisely, there is a filtered isomorphism \[(\mathrm{DR}_B(X), F^\bullet) \cong (\mathrm{R}\pi_*\mathrm{DR}(\mathrm{Tot}(L^{-2}), g_{1,\sigma}), F^\bullet_{\mathrm{irr}}),\qquad \pi : \mathrm{Tot}(L^{-2}) \longrightarrow X.\] where \(\mathrm{DR}_B(X)\) denotes the de Rham complex of \(X\) localized to \(B\). A natural question then arises: what kind of Hodge-theoretic information is encoded in the Landau–Ginzburg model \((\mathrm{Tot}(L^{-1}), g_{2,\sigma})\)?

In the first part of this article, we answer this question. The key observation is that the Landau–Ginzburg model \((\mathrm{Tot}(L^{-1}), g_{2,\sigma})\) is the double cover of \((\mathrm{Tot}(L^{-2}), g_{1,\sigma})\) branched along the zero section, so that there is a \((\mathbb{Z}/2)\)-decomposition similar to 1 .

Finally, one can generalize this construction to iterated or fibered double covers (cf. [11]). Fix an index set \([k]=\{1, \cdots, k\}\). Let \(\{(L_i, \sigma_i)|i\in [k]\}\) be a collection of pairs, where each \(L_i\) is a line bundle on \(X\) and \(\sigma_i \in \Gamma(X, L_i^{\otimes 2})\). Each pair defines a branched double cover \(\widehat{X}_i\) branched over \(B_i := \{\sigma_i = 0\}\) as before. For any subset \(I=\{i_1, ..., i_r\} \subset [k]\), the fiber product \[\widehat{X}_I:=\widehat{X}_{i_1} \times_X \cdots \times_X \widehat{X}_{i_r}\] is a \((\mathbb{Z}/2)^{|I|}\)-Galois cover of \(X\) branched over the divisor \(B_I := \bigcup_{j\in I} B_j\). Let \(\rho_I:\widehat{X}_I \to X\) be the covering map. Moreover, this cover can be realized as the complete intersection \[\{ y_i^2 - \sigma_i = 0 \mid i\in I\} \subset \operatorname{Tot}\left( \bigoplus_{i\in I} L_i \right)\] where \(y_i\) is the fiber coordinate of \(L_i\). On the other hand, for any subset \(J\), we consider the Landau–Ginzburg model \[\mathbb{V}_J:=\mathrm{Tot}\left(\bigoplus_{i\in J} L^{-1}_i \oplus \bigoplus_{i\notin J} L^{-2}_i \right), \quad g_{2,J}+g_{1,J^c}\] where \(g_{2,J}=\sum_{i \in J} g_{2,\sigma_i}\) and \(g_{1,J^c}=\sum_{i \notin J} g_{1,\sigma_i}\). Our first main theorem finds the Hodge theoretic relationship between this Landau–Ginzburg model and the Galois covers \(\widehat{X}_I\)’s for \(I \subseteq J\).

The summands on the right hand side of 2 are equipped with fractionally shifted versions of the usual Hodge filtration, while the left hand side carries Yu’s irregular Hodge filtration. In the case where \(I=[k]\), the right hand side of 2 has a factor of \(H^{*-k}(\widehat{X}_{[k]}\setminus R_{[k]})^{([k]^-)}(k/2)\). We show in Proposition 17 that this cohomology group is a direct summand of the cohomology of the double cover \(\widetilde{X}\rightarrow X\) branched along \(B_{[k]}\). Therefore, there is a concrete cohomological relationship between the irregular Hodge numbers of \((\mathbb{V}_{[k]},g_{2,{[k]}})\) and those of the branched double cover \(\widetilde{X}\rightarrow X\). This is a key observation for the applications described in the following section.

1.2 Applications to mirror symmetry↩︎

In mirror symmetry, one of the foundational achievements is the work of Batyrev and Borisov, who established a striking duality between certain Calabi–Yau complete intersections in toric varieties [12]. As we will rely on this construction, let us first recall it. Their setup begins with a reflexive Gorenstein cone or, more concretely, with a pair of dual reflexive polytopes equipped with nef partitions, \[(\Delta, \{\Delta_i\}_{i=1}^k) \quad \text{and} \quad (\check{\Delta}, \{\check{\Delta}_i\}_{i=1}^k)\] where \(\Delta=\Delta_1\cup\cdots\cup\Delta_k\) and \(\check{\Delta}=\check{\Delta}_1\cup\cdots\cup\check{\Delta}_k\). Let \(T(\Sigma_{\Delta})\) (resp. \(T(\Sigma_{\check{\Delta}})\) be the toric variety associated to the spanning fan \(\Sigma_\Delta\) of \(\Delta\) (resp. \(\Sigma_{\check{\Delta}}\)) of \(\Delta\) (resp. \(\check{\Delta}\)). Throughout this article, we assume that both admit maximal projective crepant smooth (MPCS, for short) resolutions. This corresponds to the existence of projective unimodular triangulations of both \(\Delta\) and \(\check{\Delta}\). Thus, we choose such resolutions and denote them by \(T_\Delta\) and \(T_{\check{\Delta}}\), respectively.

The nef partition induces the decomposition of the anticanonical line bundle: \[K^{-1}_{T_\Delta} = \bigotimes_{i=1}^k \mathcal{O}(E_{\Delta_i}),\] where each \(E_{\Delta_i}\) is the toric divisor corresponding to the chosen projective unimodular triangulation of \(\Delta_i\) and \(\mathcal{O}(E_{\Delta_i})\) is the corresponding line bundle. The lattice points of \(\check{\Delta}_i\) give a basis of torus invariant sections of \(\mathcal{O}(E_{\Delta_i})\), and we denote by \(\sigma_{i,{gen}}\) a generic section of \(\mathcal{O}(E_{\Delta_i})\). The choice of generic sections determines the Calabi–Yau complete intersection that is the common vanishing locus of these generic sections, \[X_\Delta := \bigcap_{i=1}^k \{ \sigma_{i,{gen}} = 0 \} \subset T_\Delta.\] A parallel construction yields the mirror candidate, denoted by \(X_{\check{\Delta}}\). Throughout the article, we use the notation \(\check{(-)}\) to indicate the mirror counterpart. The main result of [12] is the stringy Hodge number duality, which asserts that the stringy Hodge numbers of the mirror pair are related by \[h^{p,q}_{\mathrm{st}}(X_\Delta) = h^{d-p,q}_{\mathrm{st}}(X_{\check{\Delta}}),\] where \(d = \dim X_\Delta = \dim X_{\check{\Delta}}\). In particular, since we assume that \(T_\Delta\) and \(T_{\check{\Delta}}\) are MPCS resolutions, the stringy Hodge numbers are the same as the usual Hodge numbers.

Building on the same initial data, Hosono, Lee, Lian, and Yau introduced a new family of singular Calabi–Yau varieties [13]. Alongside a generic section \(\sigma_{i,{gen}}\) of each \(\mathcal{O}(E_{\Delta_i})\), there is a canonical toric section, denoted by \(\sigma_{i,{tor}}\). They consider the product \[\sigma_i := \sigma_{i,{gen}} \cdot \sigma_{i,{tor}},\] which becomes a section of \(\mathcal{O}(2E_{\Delta_i})\). The collection of \(\sigma_i\)’s determines a double cover \(\widetilde{T}_\Delta \to T_\Delta\), branched along the divisor \(B_\Delta=\{\sigma:= \prod_{i=1}^k \sigma_i = 0\}\). The resulting space \(\widetilde{T}_\Delta\) is a singular Calabi–Yau variety with at worst orbifold singularities (Lemma [l:CY]). A mirror candidate \(\widetilde{T}_{\check{\Delta}}\) is obtained in the same way from the dual data. In loc.cit., the following conjecture is implicit and the case when \(d=3\) is verified.

We prove Conjecture 4. To do this, we introduce other pairs of singular Calabi–Yau varieties. As introduced in Section 1.1, consider a \((\mathbb{Z}/2 )^k\)-Galois cover: \[\widehat{T}_\Delta := \bigcap_{i=1}^k \left\{ y_i^2 = \sigma_i \right\}\subset \mathrm{Tot} \left( \bigoplus_{i=1}^k \mathcal{O}(E_{\Delta_i}) \right)\] where \(y_i\) denotes the fiber coordinate in the line bundle \(\mathcal{O}(E_{\Delta_i})\). The variety \(\widehat{T}_\Delta\) is again a singular Calabi–Yau variety (Lemma [l:CY]). Our second main theorem is to establish Hodge number duality for the mirror pair \((\widehat{T}_\Delta, \widehat{T}_{\check{\Delta}})\) first, and use it to verify Conjecture 4.

The main tool in the proof is our previous work on Hodge number duality for stacky Clarke mirror pairs of Landau–Ginzburg models [14]. As described in the previous section, the Hodge numbers of branched covers can be computed from certain Landau–Ginzburg models (see Theorems 1 and 3). Our strategy is therefore to verify Hodge number duality for the corresponding Landau–Ginzburg models. For further background, we refer the reader to Section 3.

Among the various classes of singular mirror pairs that can be reinterpreted through the lens of Clarke duality—and thereby lead to Hodge number dualities for singular Calabi–Yau varieties—we highlight one particular example: the so-called toric extremal transition.

For instance, there is a classical notion of a conifold transition in the mathematics and physics literature where the setup is as follows. Let \(X\) be a Calabi–Yau manifold so that \(X\) degenerates to a Calabi–Yau variety \(X'\) with a number of isolated \(\mathrm{A}_1\) (“conifold”) singularities. If \(X'\) admits a crepant resolution, \({X}''\), then we say that \(X\) and \({X}''\) are related by conifold transition.

It is expected that, in certain circumstances, the mirrors, \(\check{X}\) and \(\check{X}''\) to \(X\) and \({X}''\) respectively, are also related by a conifold transition, where \(\check{X}''\) degenerates to a conifold Calabi–Yau variety \(\check{X}'\), and \(\check{X}\) is a crepant resolution of \(\check{X}'\). In other words, we have the following diagrams

\[\begin{tikzcd} & {X}''\ar[d,dashed] \\ X \ar[r,rightsquigarrow] & X' \end{tikzcd}\begin{tikzcd} & \check{X} \ar[d,rightsquigarrow] \\ \check{X}'' \ar[r,dashed] & \check{X}' \end{tikzcd}\] This construction was generalized to the notion of an extremal transition in the work of Morrison in the late 1990s [15]. Furthermore, Morrison provides a combinatorial description of a class of extremal transitions arising from inclusions between reflexive integral polytopes, \(\Delta_{\mathrm{II}} \subseteq \check{\Delta}_{\mathrm{I}}\), which correspond to a dual inclusion \(\Delta_{\mathrm{I}} \subseteq \check{\Delta}_{\mathrm{II}}\) of polar-dual polytopes: \[\begin{tikzcd} & X_{\Delta_{\mathrm{II}}} \ar[d] \\ X_{\check{\Delta}_{\mathrm{I}}} \ar[r,rightsquigarrow] & X'_{\Delta_{\mathrm{II}}} \end{tikzcd}\begin{tikzcd} & X_{\check{\Delta}_{\mathrm{II}}} \ar[d,rightsquigarrow] \\ X_{\Delta_{\mathrm{I}}} \ar[r] & X'_{\Delta_{\mathrm{I}}} \end{tikzcd}\] It is a consequence of [12] that \[h^{p,q}(X_{\Delta_{\mathrm{II}}}) = h^{d-p,q}(X_{\check{\Delta}_{\mathrm{II}}}),\qquad h^{p,q}(X_{\Delta_{\mathrm{I}}}) = h^{d-p,q}(X_{\check{\Delta}_{\mathrm{I}}}).\] We prove the Hodge number duality between \(X'_{\Delta_{\mathrm{I}}}\) and \(X'_{\Delta_{\mathrm{II}}}\) (see Theorem 30): For \(p,q \in \mathbb{Z}\), \[\label{e:intro-transition} h^{p,q}(X_{\Delta_{\mathrm{I}}}') = h^{d-p,q}(X_{\Delta_{\mathrm{II}}}')\tag{3}\] Note that both \(X_{\Delta_{\mathrm{I}}}'\) and \(X_{\Delta_{\mathrm{II}}}'\) are singular in general, and we use the notation \(h^{p,q}\) to denote the dimension of the \(p-\)th Hodge graded piece of Deligne’s canonical mixed Hodge structures. In particular, if we take the following polytopes with a choice of projective unimodular triangulations, \[\begin{align} \Delta_{\mathrm{II}}&:=\mathrm{Conv}(\check{\Delta}_i\times e_i \cup 0\times -e_i|i=1, \cdots, k) \subset M_\mathbb{R}\times \mathbb{R}^k, \\ \check{\Delta}_{\mathrm{I}}&:=\mathrm{Conv}(2\check{\Delta}_i\times e_i \cup 0\times -e_i|i=1, \cdots, k) \subset M_\mathbb{R}\times \mathbb{R}^k, \end{align}\] then the Hodge number duality 3 becomes the first item in Theorem 5.

For the second item in Theorem 5, we make use of a certain yoga involving several distinct stacky Clarke mirror pairs including pairs related to those that appear in Theorem 1. Moreover, the first part of Theorem 5 plays a key role in the proof of the second part. We refer the reader to Section 6 for further details.

Acknowledgements↩︎

AH was supported by a Simons Foundation Collaboration Grant for Mathematicians. SL was supported by the Institute for Basic Science (IBS-R003-D1).

2 Cyclic covers and Landau–Ginzburg models↩︎

This section develops technical results relating the irregular Hodge numbers of certain Landau–Ginzburg models to the irregular Hodge numbers of a class of \((\mathbb{Z}/2)^k\)-covers of a variety \(X\). The reader interested mostly in mirror symmetry applications might find it convenient to skip Section 2.4 upon first reading.

2.1 Landau–Ginzburg models and the irregular Hodge filtration↩︎

In this article, a Landau–Ginzburg (LG) model will denote a pair \((Y,w)\) where \(Y\) is a quasiprojective variety and \(w\) is a regular function on \(Y\). A compactified LG model will be a triple \((X,D,f)\) where \(X\) is a smooth projective variety, and \(D\) is a simple normal crossings (snc) hypersurface in \(X\), and the map \(f\) is a rational function on \(X\) whose pole divisor is contained in \(D\). We say a compactified LG model \((X,D,f)\) nondegenerate if the vanishing locus \(Z(f)\) is irreducible and \(Z(f) \cup D\) is snc. Let \(P\) denote the pole divior of \(f\). For simplicity, we also assume that \(Z(f) \cap P = \emptyset.\) Note that given any LG model for which \(Y\) and \(\{w=0\}\) are smooth, one can always find a nondegenerate compactified LG model \((X,D,f)\) for which \(Y = X\setminus D\) and \(w = f|_Y\) by Hironaka’s theorem.

2.1.1 Twisted de Rham cohomology↩︎

Given a nondegenerate compactified LG model \((X,D,f)\), we have a twisted de Rham complex \((\Omega_X^\bullet(\log D)(*P), \mathrm{d} + \mathrm{d}f )\). We denote its cohomology by \(H^*(X\setminus D,f)\). Given a snc hypersurface \(E\) for which \(E\cup D \cup Z(f)\) is also snc, we define the local twisted cohomology of \(E\), denoted by \(H^*_E(X\setminus D,f)\), to be the hypercohomology of the complex, \[\mathrm{DR}_{E}(X\setminus D,f):=\left(\dfrac{\Omega^\bullet_X(\log D + E)(*P)}{\Omega^\bullet_X(\log D)(*P)}, \mathrm{d} + \mathrm{d}f \right).\] Suppose \(E\) is irreducible. Then there is a residue morphism \[\mathrm{Res}_E: \mathrm{DR}(X\setminus (D\cup E),f) \longrightarrow \mathrm{DR}(E \setminus (D\cap E), f|_E)[1].\] This morphism is induced by the usual residue morphism (see e.g. [16]) on sheaves of \(p\)-forms. The kernel of \(\mathrm{Res}_E\) is \(\mathrm{DR}(X\setminus D,f)\) so that we have an isomorphism between \((\Omega_X^\bullet(\log D,E)(*P),\mathrm{d} + \mathrm{d}f)\) and \((\Omega^{\bullet -1}_E(\log E\cap D)(*P\cap D), \mathrm{d} +\mathrm{d}f|_E)\) when \(E\) is irreducible.

More generally, if \(E_1,\dots, E_r\) are snc hypersurfaces (not necessarily irreducible) such that \(E_1\cup \dots \cup E_r\) is also snc, then the twisted de Rham complex, localized to \(E^{[r]}:=E_1\cap \dots \cap E_r\) is defined by \[\mathrm{DR}_{E^{[r]}}(X\setminus D,f):=\left(\Omega^\bullet_X(\log D, E^{[r]}):= \dfrac{\Omega_X^\bullet(\log D + \sum_{i} E_i)(*P)}{\sum_i \Omega^\bullet(\log D + \sum_{j \neq i} E_j)(*P)}, \mathrm{d} + \mathrm{d}f\right).\] If \(r=0\), then we take \(E^{[r]}\) to mean all of \(X\), and the local twisted cohomology coincides with the twisted cohomology of \(X\). The hypercohomology of this complex is denoted by \(H^*_{E^{[r]}}(X\setminus D,f)\). It is an exercise in local coordinates to check that there is a short exact sequence of complexes, \[\label{e32:32local32cohomology32les32132-32sheaf} 0 \longrightarrow \mathrm{DR}_{E^{[r-1]}}(X\setminus D,f) \longrightarrow \mathrm{DR}_{E^{[r-1]}}(X\setminus (D\cup E_r),f) \xlongrightarrow{\mathrm{Res}_{E_r}} \mathrm{DR}_{E^{[r]}}(X\setminus D,f) \longrightarrow 0\tag{4}\] where \(E^{[r-1]} = E_1\cap \dots \cap E_{r-1}\) and \(\mathrm{Res}_{E_r}\) is the residue morphism. If we write \(E_r = R_1\cup \dots \cup R_m\) for smooth, irreducible hypersurfaces \(R_i\) and define \(R^I= \bigcap_{i\in I} R_i\), \(B^I = R^I \cap D \cap \bigcup_{j \notin I} R_{j}\), there is a residue resolution \[\label{e32:32residue32resolution} \begin{array}{ll} 0 \longrightarrow \Omega_X^\bullet(\log D, E^{[r]})(*P) & \longrightarrow \bigoplus_{|I| =1}\Omega_{R^I}^{\bullet-1}(\log B_I, E^{[r]}\cap R^I)(*P) \\ & \longrightarrow \bigoplus_{|I| = 2}\Omega_{R^I}^{\bullet-2}(\log B_I, E^{[r]}\cap R^I)(*P) \longrightarrow \dots \end{array}\tag{5}\] The morphisms in 5 are alternating sums of the residue morphisms whose precise definition is standard (see e.g. [14]), but we will suppress here because it only plays a minor role in our computations.

2.1.2 Irregular Hodge filtration↩︎

There is a filtration on the twisted de Rham complex, called the irregular Hodge filtration, which was first defined by Yu [6] as follows. Let \(\lambda \in \mathbb{Q}\). \[F_{\mathrm{irr}}^\lambda \mathcal{O}_X(*P) = \begin{cases} 0 & \text{ if } \lambda < 0 \\ \mathcal{O}_X(\lfloor \lambda P\rfloor) & \text{ if } \lambda \geq 0 \end{cases},\quad F_{\mathrm{irr}}^\lambda\Omega_X^p(\log D)(*P) = \Omega_X^p(\log D)\otimes F_{\mathrm{irr}}^{p-\lambda}\mathcal{O}_X(*P).\] Exactness of the tensor product with the line bundle \(\mathcal{O}_X(\lfloor \lambda P\rfloor)\) implies that this filtration also induces an irregular Hodge filtration \(\mathrm{DR}_{E^{[r]}}(X\setminus D,f)\). Note that the irregular Hodge filtration is a \(\mathbb{Q}\)-filtration. The induced filtration on \(H^p_{E^{[r]}}(X\setminus D,f)\) is defined as usual by letting \[\label{e32:32definition32of32the32irregular32filtration32on32cohomology} F_\mathrm{irr}^\lambda H_{E^{[r]}}^p(\Omega^\bullet_X(\log D)(*P)) = \mathrm{im}\left(H_{E^{[r]}}^p(F^\lambda_\mathrm{irr}\Omega^\bullet_X(\log D)(*P)) \rightarrow H_{E^{[r]}}^p(\Omega^\bullet_X(\log D)(*P) ) \right).\tag{6}\] The irregular filtration is exhaustive. In fact, \(F^0_\mathrm{irr}H^p_{E^{[r]}}(\Omega_X^\bullet(\log D)(*P)) = H^p_{E^{[r]}}(\Omega_X^\bullet(\log D)(*P))\) [6] and \(F_\mathrm{irr}^\lambda \Omega_X^\bullet(\log D)(*P) = 0\) if \(\lambda > \dim X\). Furthermore, if \[\begin{align} F^{>\lambda}_\mathrm{irr}H^p_{E^{[r]}}(X\setminus D,f) & = \bigcup_{\lambda' >\lambda} F^{\lambda'}_\mathrm{irr}H^p_{E^{[r]}}(X\setminus D,f), \\ {\mathrm{gr}}^\lambda_{F_\mathrm{irr}}H^p_{E^{[r]}}(\Omega_X^\bullet(\log D)(*P))& = \dfrac{{F}^{\lambda}_\mathrm{irr}H^p_{E^{[r]}}(\Omega_X^\bullet(\log D)(*P))}{F_\mathrm{irr}^{>\lambda}H^p_{E^{[r]}}(\Omega_X^\bullet(\log D)(*P))}, \end{align}\] then \({\mathrm{gr}}_{F_\mathrm{irr}}^\lambda H^p_{E^{[r]}}(\Omega_X^\bullet(\log D)(*P)) = 0\) for all but finitely many \(\lambda \in \mathbb{Q}_{\geq 0}\). By results of Yu [6], the twisted cohomology and irregular Hodge filtration of \((X,D,f)\) only depend on \(Y = X\setminus D\) and \(w = f|_Y\).

An important result in [5] is that the morphism in 6 is injective for all \(\lambda\) and as a consequence, \[\label{e32:32ESY} {\mathrm{gr}}_{F_\mathrm{irr}}^\lambda H_{E^{[r]}}^{p}(\Omega^\bullet_X(\log D)(*P)) \cong H_{E^{[r]}}^{p}({\mathrm{gr}}_{F_\mathrm{irr}}^\lambda \Omega^\bullet_X(\log D)(*P)).\tag{7}\] There are filtered long exact sequences whenever \(E\) is an snc hypersurface in \(X\), coming from 4 , \[\label{e32:32local32cohomology32les} \dots \longrightarrow H_{E^{[r-1]}}^p(X\setminus D,f) \longrightarrow H^p_{E^{[r-1]}}(X\setminus (D\cup E_r),f) \longrightarrow H_{E^{[r]}}^p(X\setminus D,f) \longrightarrow \dots\tag{8}\] The following fact can be deduced directly from the results of [5] (in particular 7 ).

There is also a Künneth formula for the irregular Hodge filtration, which appears in work of Chen and Yu [17] and Sabbah and Yu [10], which we state in slightly greater generality than [17] and significantly less generality than [10].

The Kontsevich sheaves of \((X,D,f)\) are defined to be: \[\Omega_f^p : = \{ \omega \in \Omega_X^p(\log D) \mid \mathrm{d}f \wedge \omega \in \Omega_{X}^{p+1}(\log D)\}.\] Equipping \(\Omega_f^\bullet\) with the differential \(\mathrm{d} + \mathrm{d}f\), we obtain the Kontsevich complex, and equipping \(\Omega_f^\bullet\) with the differential \(\mathrm{d}\) we obtain the Kontsevich–de Rham complex.

There is an inclusion of complexes \[\label{e:32inlcusion32kontsevich} (\Omega_f^\bullet, \mathrm{d} +\mathrm{d}f ,\sigma^\bullet) \hookrightarrow (\mathrm{DR}(X\setminus D,f),F^{\mathrm{Yu},\bullet}_0).\tag{9}\] Here \(\sigma^\bullet\) denotes the stupid filtration and where, for any \(\alpha \in \mathbb{Q}\), \(F^{\mathrm{Yu},p}_\alpha\) is the \(\mathbb{Z}\)-filtration defined so that \(F^{\mathrm{Yu},p}_\alpha = F_\mathrm{irr}^{p+\alpha}\) for \(p\in \mathbb{Z}\). The inclusion 9 a quasi-isomorphism of filtered complexes ([5]). In particular, if \(\mathrm{gr}_{F_\mathrm{irr}}^\lambda H^*(X\setminus D, f)=0\) for \(\lambda \in \mathbb{Q} \setminus \mathbb{Z}\) (as will be assumed in Proposition 10 below) the irregular filtration on \(H^*(X\setminus D,f)\) can be identified with the filtration induced by \(\sigma^\bullet\) on \(\mathbb{H}^*(\Omega_f^\bullet, \mathrm{d} + \mathrm{d}f)\).

According to [5], we have equalities, \[\dim \mathbb{H}^n(X,(\Omega_f^\bullet, \mathrm{d} +\mathrm{d}f)) = \dim \mathbb{H}^n(X,(\Omega_f^\bullet, \mathrm{d})) = \sum_{p+q = n} \dim H^p(X,\Omega_f^q).\] It is known, for instance by work of Hien [18], that \(\dim H^*(X\setminus D,f) = \dim H^*(X\setminus D,f^{-1}(t);\mathbb{C})\) for a generic value \(t\). The next result is presumably well known but does not seem to be stated anywhere in the literature. It says that if \(t\) is a generic point in \(\mathbb{A}^1\), the usual Hodge numbers of the pair \((X\setminus D,f^{-1}(t))\) can be computed using the irregular Hodge filtration under certain conditions.

We present a particular case, which we will discuss in more detail in the next section. Suppose \(\sigma\) is a global section of a vector bundle \(\pi: \mathcal{E}\rightarrow X\) and let \(Z_\sigma\) denote the vanishing locus of \(\sigma\) in \(X\). There is an induced regular map \(g=g_\sigma : \mathbb{V}:=\mathrm{Tot}(\mathcal{E}^\vee) \rightarrow \mathbb{A}^1\), and we may view \((\mathbb{V},g)\) as a LG model. Completely explicitly, if we trivialize \(\mathbb{V}\) on an open set \(U\), \(\varphi : \mathbb{C}^k \times U \rightarrow \mathbb{V}|_U\) and we may write \[\label{e32:32local32representation32for32potential} \varphi^*g = t_1f_1(\underline{x}) + \dots + t_k f_k(\underline{x})\tag{13}\] where \(t_1,\dots, t_k\) are coordinates on \(\mathbb{C}^k\) and \((f_1, \dots, f_k)\) is the pullback of \(\sigma\). Results closely related to Proposition 11 below appear several places (e.g. [6], [19], [20]) but we are not aware of a statement in the literature where the precise form used below can be found.

By Poincaré–Lefschetz duality (e.g. [16] or [21]) there is an isomorphism of mixed Hodge structures between \(H^*_{Z_\sigma}(X)\) and \(H^{*-2c}(Z_\sigma)\) where \(c = \mathrm{codim}_\mathbb{C}\, Z_\sigma\). Therefore, Proposition 11 identifies the twisted cohomology of \((\mathbb{V},g)\) with the usual cohomology of \(Z_\sigma\) after an appropriate Tate twist.

2.1.3 Affine bundles↩︎

The following result is a rather straightforward generalization of known results for the cohomology of projective varieties. Let \(\pi:\mathbb{P}\rightarrow X\) denote a \(\mathbb{P}^1\) bundle, which is the projectivization of the total space of a line bundle \(\mathbb{L} = \mathrm{Tot}(L^{-1})\). Let \(\mathbb{X}\) denote the hypersurface in \(\mathbb{P}\) so that \(\mathbb{L} = \mathbb{P}\setminus \mathbb{X}\). Let \(E_1,\dots, E_r\) denote hypersurfaces in \(X\) as above and let \(\mathbb{E}_i:=\pi^{-1}E\). If \(D\) is a hypersurface in \(X\), let \(\mathbb{D}:=\pi^{-1}D\), and if \(f\) is a regular function on \(X\setminus D\), let \(\mathsf{f} := \pi^*f\).

2.2 Statement of Theorem 14 and discussion↩︎

Let \(X\) be a smooth projective variety and suppose \(B_1,\dots, B_k\) are snc hypersurfaces chosen so that \(B = \bigcup_{i=1}^k B_i\) is also snc. We assume that there are line bundles \(L_1,\dots, L_k\) so that \(\mathcal{O}_X(B_i) \cong L_i^{\otimes 2}\) for all \(i\). To this data, there are two classes of geometric objects that are related to one another.

Cyclic covers↩︎

For each subset \(I \subseteq [k]\), we have a \((\mathbb{Z}/2)^{|I|}\)-cover \(\widehat{X}_I\) of \(X\) with branch divisors \(\bigcup_{i \in I} B_i\). Precisely, if \(\widehat{X}_i\) is the double cover of \(X\) branched at \({B}_i\), and \(I = \{i_1,\dots,i_j\} \subseteq [k]\) \[\widehat{X}_I = \widehat{X}_{i_1} \times_X \dots \times_X \widehat{X}_{i_j}.\] We let \({R}_I\) denote the ramification divisor of \(\rho_I : \widehat{X}_I\rightarrow X\), which is the preimage of \(B_I:=\bigcup_{i \in I}B_i\) in \(\widehat{X}_I\). We also let \(B^J = \cap_{j\in J} B_j\) and \({Z}^{I} = \rho_I^{-1}B^{I^c}\). The local cohomology \(H^*_{{Z}^{I}}(\widehat{X}_I\setminus {R}_I)\) admits a linear \((\mathbb{Z}/2)^{|I|}\) action which respects the Hodge filtration because the \(\mathbb{Z}/2\) actions in question are by automorphisms preserving \(Z^I\). We use the notation \(H^*_{{Z}^{I}}(\widehat{X}_I\setminus {R}_I)^{(I)}\) to denote the intersection of the \((-1)\)-eigenspaces of each of the \(|I|\) different \(\mathbb{Z}/2\) actions inherited from the automorphisms \(\tau_{i_j} : \widehat{X}_{i_j} \rightarrow \widehat{X}_{i_j}\) with \(i_j \in I\). \(H_{Z^{I}}^*(\widehat{X}_I\setminus {R}_I)^{(I)}\) is a direct summand of the mixed Hodge structure \(H_{Z^I}^*(\widehat{X}_I\setminus {R}_I)\). When \(I\) is a single element set, we often simplify notation to \((-)\) instead of \((I)\).

LG models↩︎

For each subset \(J \subseteq [k]\) we denote \[\mathbb{V}_J = \mathrm{Tot}\left( \bigoplus_{i \in J} L_i^{-1} \oplus \bigoplus_{i \notin J} L_i^{-2}\right).\] Let \(\pi_J : \mathbb{V}_J \rightarrow X\) denote the canonical morphism. For each \(i\), there are tautological sections \(s_i\) for \(\pi_J^*L_i^{-1}\) (if \(i \in J\)) and \(t_i\) of \(\pi_J^*L_i^{-2}\) (if \(i \notin J\)). There are sections \(\sigma_i\) of \(L_i^{\otimes 2}\) so that \(\{\sigma_i = 0\} = B_i\). When \(i \in J\), \(g_{2,i} := s_i^2\cdot \pi_J^*\sigma_i\) is a regular function on \(\mathbb{V}_J\) and when \(i \notin J\), \(g_{1,i} := t_i\cdot \pi_J^*\sigma_i\) is a regular function on \(\mathbb{V}_J\). Let \(g_{2,J} := \sum_{i \in J}g_{2,i}\) and \(g_{1,J^c} := \sum_{i \notin J} g_{1,i}\).

The main result of this section relates the cohomology of these two objects. For a \(\mathbb{Q}\)-filtered vector space \((V,F^\bullet)\), and \(\alpha \in \mathbb{Q}\) we let \((V(\alpha),F^\bullet)\) denote the vector filtered vector space \(F^\lambda(V(\alpha)) = F^{\lambda-\alpha}V\). This notation is meant to be reminiscent of the Tate twist in usual Hodge theory.

The remainder of Section 2 is devoted to proving Theorem 14. The remainder of Section 2.2 is meant to explain the geometric meaning of Theorem 14 to the reader.

In Theorem 14, we may either view objects on the right hand side as LG models equipped with potential \(f =0\) or, by Example 1, the objects on the right hand side may be viewed as cohomology groups equipped with their canonical Hodge filtrations.

The following remarks and examples describe the content of Theorem 14 in a few particular cases.

The following statement is an obvious consequence of Theorem 14, which we include because the underlying principle is important in Section 6.

Suppose notation is as above. There is also a double cover \(\widetilde{X}_{[k]}\) of \(X\) whose branch locus is \(B_{[k]} = \bigcup_{i=1}^k B_i\). Let \(\widetilde{R}_{[k]}\) be the preimage of \(B_{[k]}\) in \(\widetilde{X}_{[k]}\).

As a consequence, we see that \(H^*(\widetilde{X}_{[k]})^{(-)}(k/2)\) is a filtered direct summand of \(H^*(\mathbb{V}_{[k]},g_{2,[k]})\). This observation is important in Section 6 because it allows us to recast part of the cohomology of the double cover \(\widetilde{X}_{[k]}\) in terms of the cohomology of the LG model \((\mathbb{V}_{[k]},g_{2,[k]})\).

2.3 Proof of Theorem 14, assuming Theorem 18↩︎

Let \((X,D,f)\) denote a nondegenerate compactified LG model and assume that \(E^{[r]} \subseteq X\) is the intersection of a collection of snc hypersurfaces, \(E_1,\dots, E_r\). There are three auxiliary LG models that we study in relation to \((X,D,f)\):

• Suppose \(\rho : \widehat{X}\rightarrow X\) is a double cover with normal crossings branch divisor \(B\). Let \(\widehat{D}\) denote \(\rho^{-1}D\) and \(\widehat{f} = \rho^{*}f\). Let \(\widehat{E}^{[r]}\) denote \(\rho^{-1}E^{[r]}\) and let \({R} := \rho^{-1}B(\cong B)\). We are interested in understanding the nondegenerate compactified LG model \((\widehat{X},\widehat{D},\widehat{f})\).

Suppose there exists a line bundle \(L\) so that \(L^2 \cong \mathcal{O}_X(B)\).

• Let \(\mathbb{L}_{1} = \mathrm{Tot}(L^{-1})\) and let \(\pi_{1} :\mathbb{L}_{1}\rightarrow X\) denote the usual projection morphism. Let \(\mathbb{D}_{1} = \pi_{1}^{-1}D, \mathbb{E}^{[r]}_{1} = \pi_{1}^{-1}E^{[r]}\), and let \(s\) denote the tautological section of \(\pi_1^*L\). Then \(s^2 \in \Gamma(\mathbb{L}_1,\pi_1^*L^{-2})\) and \(\pi_1^*\sigma \in \Gamma(\mathbb{L}_2,\pi_1^*L^2)\). Let \(\mathsf{g}_{2} = s^2\cdot (\pi_1^*\sigma), \mathsf{f}_{1} = \pi^{*}_{1}f\), which are regular functions on \(\mathbb{L}_1\setminus \mathbb{D}_1\). We consider the LG model \((\mathbb{L}_{1}\setminus \mathbb{D}_{1}, \mathsf{g}_{2} + \mathsf{f}_{1})\).

• Let \(\mathbb{L}_{2} = \mathrm{Tot}(L^{-2})\) and let \(\pi_{2} :\mathbb{L}_{2}\rightarrow X\) denote the usual projection morphism. Let \(\mathbb{D}_{2} = \pi_{1}^{-1}D, \mathbb{E}^{[r]}_{2} = \pi_{2}^{-1}E^{[r]}\) and let \(t\) denote the tautological section of \(\pi_2^*L^{-2}\). Then \(t \in \Gamma(\mathbb{L}_2,\pi_2^*L^{-2})\) and \(\pi_2^*\sigma \in \Gamma(\mathbb{L}_2,\pi_1^*L^2)\). Let \(\mathsf{g}_{1} = t\cdot (\pi^*\sigma), \mathsf{f}_{2} = \pi^{*}_{2}f\), which are regular functions on \(\mathbb{L}_2 \setminus \mathbb{D}_2\). We consider the LG model \((\mathbb{L}_{2}\setminus \mathbb{D}_{2}, \mathsf{g}_{1} + \mathsf{f}_{2})\).

Section 2.4 will be devoted to proving the following theorem which is a global variation on results proved by Sabbah and Yu [10] and Fresán, Sabbah, and Yu [20] respectively.

Theorem 14 follows from Theorem 18. Let us start with a brief remark to make the relation between Theorem 14 and Theorem 18 clear; if \(r = 1\) then \(I = \emptyset\) or \(\{1\}\). In either case \(g_{2,\emptyset} = 0, g_{1,\emptyset^c} = \mathsf{g}_{1}\) and \(\mathbb{V}_{\emptyset} = \mathbb{L}_{2}\) or \(g_{2,\{1\}} = \mathsf{g}_{2}, g_{1,\{1\}^c} = 0\) and \(\mathbb{V}_{\{1\}} = \mathbb{L}_{1}\). Applying 18 and 17 respectively, with \(\mathbb{E}^{[r]} = \mathbb{D} = \emptyset\) and \(\mathsf{f} = 0\) yields, \[H^*(\mathbb{V}_{\emptyset}, g_{1,\emptyset^c}) = H^{*-1}_{B}(X),\qquad H^*(\mathbb{V}_{\{1\}},g_{2,\{1\}}) = H^{*-1}(\widehat{X}\setminus R)^{(-)}(1/2) \oplus H^{*}_{B}(X)\] as desired. This recovers the first part of Example 3 immediately.

Taking Theorem 18 for granted, we prove Theorem 14.

2.4 Proof of Theorem 18↩︎

The entirety of the section is devoted to proving Theorem 18. There are two basic morphisms that play a significant role in this proof. The first will be explained now, and the second will be explained later.

Observe that \(\mathbb{L}_{1}:=\mathrm{Tot}(L^{-1})\) is a double cover of \(\mathbb{L}_{2}:=\mathrm{Tot}({L}^{-2})\) ramified along the zero section. Precisely, if \(U \times \mathbb{C}\) is a trivializing chart on \(\mathbb{L}_1\), then the covering map takes \((\underline{x},t) \in U\times \mathbb{C}\subseteq \mathbb{L}_1\) to \((\underline{x},t^2) \in \mathbb{L}_2\). We let \(\rho : \mathbb{L}_{1} \rightarrow \mathbb{L}_{2}\) denote this morphism, and we let \(\pi_{2}:\mathbb{L}_{2} \rightarrow X\) denote the canonical morphism. Recall that \(\mathsf{g}_{1} = t\cdot \pi_{2}^*\sigma\) where \(t\) denotes the canonical section of \(\pi_{2}^*L^{-2}\). The following diagram commutes: \[\begin{tikzcd} \mathbb{L}_{1}\ar[rd,"\pi_{1}",swap] \ar[r,"\rho"] & \mathbb{L}_{2} \ar[d,"\pi_{2}"] \\ & X \end{tikzcd}\] We notice that \(\rho^*\mathsf{g}_{1} = \mathsf{g}_{2}\) and if \(\mathbb{E}_{2}^{[r]} = \pi_{2}^{-1}E^{[r]}\) then \(\rho^{-1}\mathbb{E}_{2}^{[r]} = \mathbb{E}_{1}^{[r]}\). There is a \(\mathbb{Z}/2\) action on \(H_{\mathbb{E}_{1}^{[r]}}^*(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} +\mathsf{f}_{1})\) which preserves the irregular Hodge filtration. Therefore, we have a filtered direct sum decomposition into eigenspaces, \[H^*_{\mathbb{E}_{1}^{[r]}}(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} +\mathsf{f}_{1}) = H^*_{\mathbb{E}_{1}^{[r]}}(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} +\mathsf{f}_{1})^{(+)} \oplus H^*_{\mathbb{E}_{1}^{[r]}}(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} +\mathsf{f}_{1})^{(-)}.\] We may identify the \((+)\) eigenspace using the following result which is a direct generalization of a classical result about the Hodge numbers of cyclic coverings.

Therefore, we see that \[H^*_{\mathbb{E}_{1}^{[r]}}(\mathbb{L}_{1}\setminus \mathbb{D}_1,\mathsf{g}_{2} +\mathsf{f}_{1}) = H_{\mathbb{E}^{[r]}_{2}}^*(\mathbb{L}_{2} \setminus \mathbb{D}_{2},\mathsf{g}_{1} +\mathsf{f}_{2}) \oplus H_{\mathbb{E}_{1}^{[r]}}^*(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} +\mathsf{f}_{1})^{(-)}.\] Proposition 20 below is surely well known to experts but it does not seem to be proven in the generality that we need (see [6], [19], [20] for similar results when \(f =0\), \(\mathbb{B}\) is smooth, or \(X\) is affine, respectively). It is a consequence of the following Lemma.

Note that 18 is a consequence of the following proposition.

Now we use this to decompose the cohomology of the LG model \((\mathbb{L}_{1}\setminus \mathbb{D}_{1}, \mathsf{g}_{2} + \mathsf{f}_{2})\). There is a direct sum decomposition \[H_{\mathbb{E}_{1}^{[r]}}^*(\mathbb{L}_{1}\setminus \mathbb{D} _{1},\mathsf{g}_{2} +\mathsf{f}_{1}) \cong H_{\mathbb{E}_{1}^{[r]}}^*(\mathbb{L}_{1} \setminus \mathbb{D}_{1},\mathsf{g}_{2}+\mathsf{f}_{1})^{(+)} \oplus H_{\mathbb{E}_{1}^{[r]}}^*(\mathbb{L}_{1} \setminus \mathbb{D}_{1},\mathsf{g}_{2}+\mathsf{f}_{1})^{(-)}.\] By Proposition 19 (using the double covering map \(\mathbb{L}_1 \setminus \mathbb{D}_1 \rightarrow \mathbb{L}_2 \setminus \mathbb{D}_2\)) and Proposition 20 this becomes \[\label{e32:32eigenspace32decomposition} H_{\mathbb{E}_{1}^{[r]}}^*(\mathbb{L}_{1}\setminus \mathbb{D} _{1},\mathsf{g}_{2} +\mathsf{f}_{1}) \cong H^*_{B \cap E^{[r]}}(X\setminus D,f) \oplus H_{\mathbb{E}_{1}^{[r]}}^*(\mathbb{L}_{1} \setminus \mathbb{D}_{1},\mathsf{g}_{2}+\mathsf{f}_{1})^{(-)}.\tag{31}\] We provide an alternative description of the second summand of the right hand side of 31 . There is a commutative diagram, 32 below, whose top horizontal arrow is an isomorphism by Proposition 20 and the right vertical morphism is injective by Proposition 19. \[\label{e32:32commutative32diagram32from32pullback} \begin{tikzcd} H_{\mathbb{E}^{[r]}_{2} \cap \mathbb{B}_{2}}^{*-1}({\mathbb{L}}_{2}\setminus {\mathbb{D}}_{2},\mathsf{g}_1 + \mathsf{f}_{2}) \ar[r,"\cong"]\ar[d,"\rho^*"]& H_{\mathbb{E}^{[r]}_{2}}^*(\mathbb{L}_{2}\setminus \mathbb{D}_{2},\mathsf{g}_{1} + \mathsf{f}_{2}) \ar[d,hookrightarrow,"p"] \\ H^{*-1}_{\mathbb{E}_{1}^{[r]}\cap \mathbb{B}_1}(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} + \mathsf{f}_{1}) \ar[r,"i_*"] & H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} + \mathsf{f}_{1}). \end{tikzcd}\tag{32}\] By commutativity, \(\rho^*\) is injective, so by Lemma 1, it must be an isomorphism. Therefore, the image of \(i_*\) in 32 is identified with the image of \(p\) which, by Proposition 19, is \(H^*_{\mathbb{E}_{1}^{[r]}}(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} +\mathsf{f}_{1})^{(+)}\). From 8 we have a long exact sequence \[\begin{align} \label{e32:32les32that32splits} \dots \longrightarrow H^{*-1}_{\mathbb{E}^{[r]}_{1}\cap \mathbb{B}_{1}}(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} + \mathsf{f}_{1}) &\xrightarrow{\,\, i_*\,\,} H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus \mathbb{D}_{1}, \mathsf{g}_{2} + \mathsf{f}_{1}) \\ &\longrightarrow H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus (\mathbb{D}_{1}\cup \mathbb{B}_{1}),\mathsf{g}_{2} + \mathsf{f}_{1}) \longrightarrow \dots \end{align}\tag{33}\] Since \(i_*\) is injective the long exact sequence 33 splits into a collection of short exact sequences: \[\begin{align} \label{ckqodyhv} 0 \longrightarrow H^{*-1}_{\mathbb{E}^{[r]}_{1}\cap \mathbb{B}_{1}}(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2} + \mathsf{f}_{1}) &\xrightarrow{\,\, i_*\,\,} H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus \mathbb{D}_{1}, \mathsf{g}_{2} + \mathsf{f}_{1}) \\ &\longrightarrow H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus (\mathbb{D}_{1}\cup \mathbb{B}_{1}),\mathsf{g}_{2} + \mathsf{f}_{1}) \longrightarrow 0 \end{align}\tag{34}\] As above, we have a filtered splitting, \[H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus \mathbb{D}_{1}, \mathsf{g}_{2} + \mathsf{f}_{1}) \cong H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus \mathbb{D}_{1}, \mathsf{g}_{2} + \mathsf{f}_{1})^{(-)} \oplus H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus \mathbb{D}_{1}, \mathsf{g}_{2} + \mathsf{f}_{1})^{(+)}.\] Since the image of \(i_*\) is \(H_{\mathbb{E}_{1}^{[r]}}^*( \mathbb{L}_{1}\setminus \mathbb{D}_{1}, \mathsf{g}_{2} + \mathsf{f}_{1})^{(+)}\), there is a filtered isomorphism \[\label{e32:32remove32divisor} H_{\mathbb{E}_{1}^{[r]}}^{*}( \mathbb{L}_{1}\setminus (\mathbb{D}_{1}\cup \mathbb{B}_{1}),\mathsf{g}_{2} + \mathsf{f}_{1}) \cong H_{\mathbb{E}_{1}^{[r]}}^*(\mathbb{L}_{1}\setminus \mathbb{D}_{1},\mathsf{g}_{2}+\mathsf{f}_{1})^{(-)}.\tag{35}\]

We will now construct a morphisms that will allow us to identify the left hand side of 35 with \(H^{*-1}_{\widehat{E}^{[r]}}(\widehat{X}\setminus \widehat{D}, \widehat{f})^{(-)}(1/2)\). This will complete the proof. Given the ramified double cover \(\eta: \widehat{X} \rightarrow X\), there is a cartesian diagram \[\begin{tikzcd} \widehat{\mathbb{L}}_{1} \ar[r,"\eta"] \ar[d,"\widehat{\pi}_{1}"] & \mathbb{L}_{1} \ar[d,"\pi_{1}"] \\ \widehat{X} \ar[r,"\rho"] & X \end{tikzcd}\] where \(\widehat{\mathbb{L}}_1\) is \(\mathrm{Tot}(\rho^*L)\). Let \(\widehat{\mathbb{E}}_{1}^{[r]} = \rho^{-1}\mathbb{E}_{1}^{[r]}\) and let \({\mathbb{R}}_{1} = \rho^{-1}\mathbb{B}_{1}\). By Proposition 19 there is a filtered isomorphism \[H^*_{\widehat{\mathbb{E}}_{1}^{[r]}}(\widehat{\mathbb{L}}_{1}\setminus (\widehat{\mathbb{D}}_{1}\cup {\mathbb{R}}_{1}),\widehat{\mathsf{g}}_{2}+\mathsf{f}_{1})^{(+)}\cong H^*_{\mathbb{E}_{1}^{[r]}}(\mathbb{L}_{1}\setminus (\mathbb{D}_{1}\cup \mathbb{B}_{1}),\mathsf{g}_{2} + \mathsf{f}_{1}).\]

Let \(\tau: \widehat{X}\rightarrow \widehat{X}\) be the automorphism so that \(\widehat{X}/\tau \cong X\). The final step in the proof of Theorem 14 is to show that \[\label{e32:32final32isomorphism} H^*_{\widehat{\mathbb{E}}_{1}^{[r]}}(\widehat{\mathbb{L}}_{1}\setminus (\widehat{\mathbb{D}}_{1}\cup {\mathbb{R}}_{1}),\widehat{\mathsf{g}}_{2}+\mathsf{f}_{1})^{(+)}\cong H^{*-1}(\widehat{X}\setminus (\widehat{D}\cup R),f)^{(-)}(1/2).\tag{36}\] This is deduced from the following geometric calculation and the Künneth formula. Details appear below.

We apply the Künneth formula for the irregular filtered twisted cohomology (Theorem 9) to deduce 36 , which will complete the proof of Theorem 18. The Künneth formula can be interpreted as usual by taking the external tensor product of twisted de Rham cohomology classes. Specializing Example 2 we see that the classes \([\alpha\cdot \mathrm{d}q]\), \(\alpha\in\mathbb{C}\) give \({\mathrm{gr}}^{1/2}_{F_\mathrm{irr}}H^1(\mathbb{A}^1_q,q^2) \cong H^1(\mathbb{A}^1_q,q^2)\), so the invariant cohomology classes with respect to \(\widehat{\tau}\) are \([\omega] \boxtimes [\alpha \cdot \mathrm{d}q] \in H_{\widehat{E}^{[r]}}^*(\widehat{X}\setminus \widehat{D},\widehat{f}) \boxtimes H^{1}(\mathbb{A}^1,q^2)\) where \([\omega] \in H_{\widehat{E}^{[r]}}^*(\widehat{X}\setminus \widehat{D},\widehat{f})^{(-)}\). Therefore, \[\begin{align} H_{\widehat{\mathbb{E}}_{1}^{[r]}}^*(\widehat{\mathbb{L}}_{1} \setminus (\widehat{\mathbb{D}}_{1}\cup {\mathbb{B}}_{1}), \mathsf{g}_2 + \mathsf{f}_{1})^{(+)} & \cong (H_{\widehat{E}^{[r]}}^{*-1}(\widehat{X}\setminus \widehat{D},\widehat{f}) \boxtimes H^{1}(\mathbb{A}^1,q^2))^{(+)} \\ & \cong H_{\widehat{E}^{[r]}}^{*-1}(\widehat{X}\setminus \widehat{D},\widehat{f})^{(-)}(1/2). \end{align}\] This completes the proof of Theorem 18.

3 Hodge number duality for Clarke mirror pairs↩︎

In this section, we recall background on toric varieties, Clarke duality, and one of the main results of our previous work [14]. We first recollect some background about toric varieties to set up notation.

Let \(N\) and \(M\) be dual lattices of rank \(d\) with the natural bilinear pairing \(\langle -,- \rangle:N \times M \to \mathbb{Z}\). We write \(N_\mathbb{K}:=N \otimes_\mathbb{Z} \mathbb{K}\) and \(M_\mathbb{K}=M \otimes_\mathbb{Z} \mathbb{K}\) for \(\mathbb{K}=\mathbb{Q}, \mathbb{R}, \mathbb{C}\). A fan \(\Sigma \in N_\mathbb{R}\) is a collection of strongly convex polyhedral cones such that each face of a cone in \(\Sigma\) is also a cone in \(\Sigma\), and the intersection of two cones in \(\Sigma\) is a face of each cone.

Let \(\Sigma[1]=\{\rho_i\mid i=1, \dots, n\}\) be the set of primitive generators of \(\Sigma\). Consider the monomial ideal of \(\mathbb{C}[x_1, \cdots, x_n]\), \(J_\Sigma:=\langle \prod_{\rho_i \subsetneq c} x_i \mid c \in \Sigma \rangle\) and the induced quasi-affine variety \(\mathbb{C}^n \setminus V(J_\Sigma)\). We also have the morphism of the lattices \(\beta:\mathbb{Z}^n \to N\) that sends the standard basis \(e_i\) to \(\rho_i\), which induces the morphism of tori \(T_{\beta}:(\mathbb{C}^*)^n \to (\mathbb{C}^*)^d\). We let \(G_{\beta}\) be its kernel.

Any toric variety is stratified by tori \(T_c\) of dimension \((d-\dim c)\) corresponding to the cone \(c \in \Sigma\). We let \(T(\Sigma)_c\) denote the closure of \(T_c\) in \(T(\Sigma)\). In particular, each one dimensional cone \(c\), or its ray generator \(\rho\) determines the torus-invariant divisor, which we also denote by \(E_\rho = T(\Sigma)_c\).

Below, we list some properties of the fans and their relations to the associated toric variety:

1. A cone \(c\) is called unimodular if the primitive integral ray generators of \(c\) form a basis of \(N\). A fan \(\Sigma\) is unimodular if every maximal cone is unimodular. Then, the associated toric variety \(T(\Sigma)\) is a manifold.

2. A cone \(c\) is called simplicial if the primitive integral ray generators of \(c\) form a basis of \(N\). A fan \(\Sigma\) is simplicial if every maximal cone is simplicial. Then, the associated toric variety \(T(\Sigma)\) is an orbifold.

3. A cone \(c\) in \(N_\mathbb{R}\) is called Gorenstein if there is some \(m_c\in M\) so that the integral collection of points \(c \cap \{n \mid \langle n, m_c \rangle = 1\}\) generates the cone \(c\). A fan \(\Sigma\) is Gorenstein if all cones are Gorenstein. Then, the associated toric variety \(T(\Sigma)\) is Gorenstein.

4. A fan \(\Sigma\) is quasiprojective if there is a convex function on \(\mathrm{Supp}(\Sigma)\) which is linear on each cone of \(\Sigma\) and takes integral values on \(N \cap \mathrm{Supp}(\Sigma)\). Then, the associated toric variety \(T(\Sigma)\) is quasiprojective.

Suppose that we have a finite collection of integral points \(A\) in \(M\). We consider a Laurent polynomial \(w \in \mathbb{C}[M]\) of the form \[w=\sum_{m \in A}u_m\underline{x}^m\] for some \(u_m \in \mathbb{C}^*\). For any toric variety \(T(\Sigma)\), one can see that \(w\) becomes a regular function on \(T(\Sigma)\) if \(\langle m, \rho \rangle \geq 0\) for every primitive integral ray generator \(\rho \in \Sigma[1]\) and \(m \in A\).

For later use, we also define the notion of toric Deligne–Mumford stack introduced in [22]. A stacky fan is a simplicial fan \(\Sigma\) equipped with additional data: a positive integer \(\beta_\rho\) assigned to each primitive integral ray generator \(\rho \in \Sigma[1]\). We denote this as \({\boldsymbol{\Sigma}} = (\Sigma, \beta)\), where \(\Sigma\) is referred to as the underlying fan of \({\boldsymbol{\Sigma}}\). The assignment \(\beta\) can be viewed as a lattice morphism \(\beta:\mathbb{Z}^n \to N\) that sends the standard basis \(e_i\) to \(\beta_\rho \cdot \rho\). Similar to the global construction of toric varieties, it induces the morphism of tori \(T_{\beta}:(\mathbb{C}^*)^n \to (\mathbb{C}^*)^d\). Since \(\beta\) has finite cokernel, \(T_{\beta}\) is surjective. Let \(G_{\beta}\) be its kernel.

The underlying toric variety \(T(\Sigma)\) is the coarse moduli space for \(T({\boldsymbol{\Sigma}})\) so that we have a canonical morphism \(\pi_{\boldsymbol{\Sigma}}:T({\boldsymbol{\Sigma}}) \to T(\Sigma)\). Any simplicial fan \(\Sigma\) has a canonical stacky fan structure where \(\beta_\rho = 1\) for all \(\rho \in \Sigma[1]\). For a stacky fan \({\boldsymbol{\Sigma}}\), let \({\boldsymbol{\Sigma}}[1]\) denote the set \(\{\beta_\rho \rho \mid \rho \in \Sigma[1]\}\). We often write \(c \in {\boldsymbol{\Sigma}}\) which means that we take ray generators of the cone \(c\) to be extended ones \(\{\beta_\rho \rho \mid \rho \in c\}\).

Some of the properties of a fan naturally extend to a stacky fan as properties of the underlying fan: \({\boldsymbol{\Sigma}}\) is called unimodular, simplicial, or quasiprojective if its underlying fan \(\Sigma\) has these properties. An exception is the Gorenstein property, where we say \({\boldsymbol{\Sigma}}\) is Gorenstein if every cone \(c\) in \({\boldsymbol{\Sigma}}\) is Gorenstein.

Given a Clarke dual pair of stacky fans, the regularity condition ensures that a Laurent polynomial \[w({\boldsymbol{\Sigma}}) = 1 + \sum_{n \in {\boldsymbol{\Sigma}}[1]} u_n \underline{x}^n \in \mathbb{C}[N]\] where \(u_n \in \mathbb{C}^*\) are chosen generically, defines a regular function on \(T(\check{\Sigma})\). By composing it with the canonical morphism \(\pi_{\mathbf{\Sigma}}\), we obtain a regular function on \(T(\check{\boldsymbol{\Sigma}})\) which we also write as \(w({\boldsymbol{\Sigma}})\) by abuse of notation. Thus, we have an induced pair of LG models \[(T({\boldsymbol{\Sigma}}),w(\check{\boldsymbol{\Sigma}})),\qquad (T(\check{\boldsymbol{\Sigma}}), w({\boldsymbol{\Sigma}}))\] which is called a Clarke mirror pair of stacky LG models. In [14], we have justified this terminology, particularly the term “mirror", by verifying an irregular version of Hodge number duality (see Theorem 25). To state this, we briefly recall the notion of orbifold cohomology [24].

Let \(Y\) be a toric DM stack (or an orbifold). The orbifold cohomology of \(Y\) is defined as the usual cohomology of the inertia stack of \(Y\) with a particular twist. Each component of the inertia stack is referred to as a twisted sector, and the twist assigned to each component is called its age. To simplify the discussion, we only present a description for the toric case, which suffices for the purposes of this article.

Let \({\boldsymbol{\Sigma}}\subset N\) be a simplicial stacky fan and \(T({\boldsymbol{\Sigma}})\) be the associated toric DM stack. For each cone \({c}\) in \(\boldsymbol{\Sigma}\), we let \[\mathrm{Box}^\circ({c}) := \left\{ \left. \sum_{\rho\in {c}[1]} a_\rho \rho \,\, \right| \,\, a_\rho \in (0,1) \right\}\cap N.\]

We note that the age grading \(\iota(g)\) is rational in general. When \(\mathbf{\Sigma}\) has a trivial stacky structure, equivalently \(\mathbf{\Sigma}=\Sigma\), then \(\iota(g) \in \mathbb{Z}\) if and only if \(\Sigma\) is Gorenstein.

Analogously, for a LG model \((T({\boldsymbol{\Sigma}}),w)\), we can naturally define the orbifold twisted cohomology and the orbifold irregular Hodge filtration. We denote the orbifold twisted cohomology by \(H^*_{\mathrm{orb}}(T({\boldsymbol{\Sigma}}), w)\), and the associated graded pieces of the irregular Hodge filtration is given by \[H_\mathrm{orb}^{\lambda, \mu}(T(\mathbf{\Sigma}), w)=\bigoplus_{c \in {\boldsymbol{\Sigma}}} \bigoplus_{g \in \mathrm{Box}^\circ({c})} H^{\lambda-\iota(g),\mu-\iota(g)}(T({\boldsymbol{\Sigma}})_c, w)\] for \(\lambda, \mu \in \mathbb{Q}\). When the rational gradings happen to be integral, we adopt the more familiar notation \((p,q)\) in place of \((\lambda, \mu)\) to denote Hodge components. Therefore, the corresponding orbifold irregular Hodge numbers² are \[h^{\lambda,\mu}_{\mathrm{orb}}(T(\mathbf{\Sigma}), w)=\sum_{c \in {\boldsymbol{\Sigma}}}\sum_{g \in \mathrm{Box}^\circ(c)}h^{\lambda-\iota(g), \mu-\iota(g)}(T({\boldsymbol{\Sigma}})_c, w).\]

4 Mirror symmetry for Galois cover Calabi–Yau varieties↩︎

We introduce two different singular Calabi–Yau pairs associated to the nef partition data, which will be proved to satisfy Hodge number duality.

Let \(\Delta \subset M_{\mathbb{R}}\) be a reflexive polytope and let \(\Sigma_\Delta\) denote its spanning fan. A nef partition of \(\Delta\) is a partition of the set of vertices of \(\Delta\) into subsets \(S_1,\dots, S_k\) so that for each \(i\), there is a \(\Sigma_\Delta\) linear convex function \(\psi_i\) so that \(\psi_i(n) = 1\) if \(n \in S_i\) and \(\psi_i(n) = 0\) if \(n \in S_j, j\neq i\). We let \(\Delta_i = \mathrm{conv}(S_i \cup 0_N)\). To indicate that \(\Delta_1,\dots, \Delta_k\) form a nef partition of \(\Delta,\) we will write \(\Delta = \Delta_1\cup \dots \cup \Delta_k\). Given a nef partition, denote \[\check{\Delta}_i = \{ m \in M_\mathbb{R} \mid m(n) \geq -1, \,\, \forall \,\, n \in \Delta_i, m(n) \geq 0 \,\, \forall \,\, n \in \Delta_j, j\neq i\}.\] Let \(\check{\Delta} = \mathrm{conv}(\check{\Delta}_1,\dots, \check{\Delta}_k)\). Then \(\check{\Delta}_1,\dots, \check{\Delta}_k\) forms a nef partition of \(\check{\Delta}\). We have \(\Delta^\circ = \check{\Delta}_1 + \dots + \check{\Delta}_k\) and \(\check{\Delta}^\circ = \Delta_1 + \dots + \Delta_k\) where \(+\) indicates Minkowski sum of polytopes³.

As mentioned in the introduction, we always assume that the associated toric varieties \(T(\Sigma_\Delta)\) and \(T(\Sigma_{\check{\Delta}})\) admit MPCS resolutions, and denote them by \(T_\Delta\) and \(T_{\check{\Delta}}\), respectively. Note that this is equivalent to choosing projective unimodular triangulations of \(\Delta\) and \(\check{\Delta}\). Such triangulations can fail to exist if \(\dim \Delta \geq 4\). We make this assumption to ensure that we only have to deal with singular varieties, rather than singular DM stacks.

For each \(i\), let \(E_{\Delta_i}\) be the toric divisor on \(T_{\Delta}\) corresponding to the chosen projective unimodular triangulation of \(\Delta_i\) and \(\mathcal{O}(E_{\Delta_i})\) be the corresponding line bundle. The nef partition \(\Delta=\Delta_1\cup\cdots\cup\Delta_k\) induces the decomposition of the anticanonical line bundle \(K^{-1}_{T_\Delta}=\bigotimes_{i=1}^k \mathcal{O}(E_{\Delta_i})\). Also, each integral point of \(\check{\Delta}_i\) provides a section of \(\mathcal{O}(E_{\Delta_i})\), and sections corresponding to integral points in \(\check{\Delta}_i\) form a basis of \(H^0(T_{\Delta},\mathcal{O}(E_{\Delta_i})\). In particular, we will use the notation \(\sigma_{i,gen}\) to denote a generic section and \(\sigma_{i, tor}\) will denote the section corresponding to the point \(0\). The vanishing locus of \(\sigma_{i,tor}\) is \(E_{\Delta_i}\), and the vanishing locus of \(\sigma_{i,gen}\) is a smooth hypersurface in \(T_{\Delta}\). Then the product of these two sections \(\sigma_i:=\sigma_{i,gen}\sigma_{i,tor}\) becomes a section of \(\mathcal{O}(2E_{\Delta_i})\). We also write \(D_i=\{\sigma_i=0\}, D_{i,gen}=\{\sigma_{i,gen}=0\}, D_{i,tor}=\{\sigma_{i,tor}=0\}\) for the vanishing loci and \(D=D_1\cup\cdots\cup D_k\). Then by construction, these divisors are at worst simple normal crossings.

To this data, we associate two branched covers of \(T_\Delta\):

1. \(\pi:\widehat{T}_\Delta \to T_\Delta\), where \(\widehat{T}_\Delta\) is the \((\mathbb{Z}/2)^k\)-Galois cover of \(T_{\Delta}\).

2. \(\pi:\widetilde{T}_\Delta \to T_\Delta\), where \(\widetilde{T}_\Delta\) is the branched double cover of \(T_{\Delta}\) whose branch locus is \(D\).

Applying the parallel construction on the mirror side, we obtain two singular Calabi–Yau pairs \[(\widehat{T}_\Delta, \widehat{T}_{\check{\Delta}}), \qquad (\widetilde{T}_{\Delta}, \widetilde{T}_{\check{\Delta}}).\] We will prove the following result in Section 5. Since both \(\widehat{T}_\Delta\) and \(\widehat{T}_{\check{\Delta}}\) have at worst orbifold singularities, their cohomology groups carry pure Hodge structures. We emphasize that the Hodge numbers referred to in Theorem 26 are usual (not orbifold) Hodge numbers. The fact that singularities are at worst orbifold follows from the fact that \(\widehat{T}_\Delta\) is an iterated double cover branched along orbifold normal crossings divisors at each step.

The second pair \((\widetilde{T}_{\Delta}, \widetilde{T}_{\check{\Delta}})\) was first introduced in [13], and we refer to such pairs as HLLY mirror pairs. It was conjectured that the Hodge number duality holds so that it is indeed a mirror pair. The case where \(d=3\) was proven in op. cit.

We will prove the Hodge number duality for HLLY mirror pairs in Section 6, providing justification for Conjecture 27.

5 Toric extremal transition↩︎

We describe a general framework of toric extremal transitions [15] and the Hodge number duality results. Using this framework, we will prove Theorem 26.

Suppose we have a pair of reflexive polytopes \({\Delta}_{\mathrm{II}} \subseteq \check{\Delta}_{\mathrm{I}}\) of the same dimension \(d\). Dually, we also have a pair of reflexive polytopes \({\Delta}_{\mathrm{I}} \subseteq \check{\Delta}_{\mathrm{II}}\). Let \({T}_{\Delta_\mathrm{I}}\) be an MPCP resolution of the toric Fano variety \(T(\Sigma_{\Delta_{\mathrm{I}}})\) attached to the polytope \({\Delta}_{\mathrm{I}}\), and \({T}_{\check{\Delta}_{\mathrm{II}}}\) be an MPCP resolution of the Fano toric variety \(T(\Sigma_{\check{\Delta}_{\mathrm{II}}})\) attached to \(\check{\Delta}_{\mathrm{II}}\). The classic geometric picture [15] is that attached to this data we have two families of singular Calabi–Yau varieties in \({T}_{\Delta_\mathrm{I}}\) and \({T}_{\check{\Delta}_\mathrm{II}}\). Viewing integral points of \(\check{\Delta}_{\mathrm{I}}\) as monomial sections of \(K^{-1}_{{T}_{\Delta_{\mathrm{I}}}}\), we see that integral points of \({\Delta}_{\mathrm{II}}\) determine a family of hypersurfaces in \({T}_{\Delta_\mathrm{I}}\) as well. Furthermore, for a generic choice of such a section, the corresponding hypersurface, which we may denote \(X_{{\Delta}_{\mathrm{II}}}'\), is usually singular. We may view \(X_{{\Delta}_{\mathrm{II}}}'\) as a degeneration of a very general anticanonical hypersurface \(X_{\check{\Delta}_{\mathrm{I}}}\) in \({T}_{\Delta_\mathrm{I}}\). Each \(X'_{{\Delta}_{\mathrm{II}}}\) is birational to an anticanonical hypersurface \(X_{{\Delta}_{\mathrm{II}}}\) in \({T}_{\check{\Delta}_\mathrm{II}}\). The pair of operations consisting of degeneration of \(X_{\check{\Delta}_{\mathrm{I}}}\rightsquigarrow X_{{\Delta}_{\mathrm{II}}}'\) along with the birational map \(X_{{\Delta}_{\mathrm{II}}} \dashrightarrow X_{{\Delta}_{\mathrm{II}}}'\) is called an extremal transition. Applying polar duality, we obtain a dual extremal transition, from which we obtain the following diagrams. \[\label{e32:32geometric32transition} \begin{tikzcd} & X_{{\Delta}_{\mathrm{II}}} \ar[d,dashed] \\ X_{\check{\Delta}_{\mathrm{I}}} \ar[r,rightsquigarrow] & X'_{{\Delta}_{\mathrm{II}}} \end{tikzcd}\begin{tikzcd} & X_{\check{\Delta}_{\mathrm{II}}} \ar[d,rightsquigarrow] \\ X_{{\Delta}_{\mathrm{I}}} \ar[r,dashed] & X'_{{\Delta}_{\mathrm{I}}} \end{tikzcd}\tag{37}\]

We have Hodge number duality between pairs \(X_{\check{\Delta}_{\mathrm{I}}}\) and \(X_{\Delta_{\mathrm{I}}}\), and between \(X_{\Delta_{\mathrm{II}}}\) and \(X_{\check{\Delta}_{\mathrm{II}}}\). The first goal of this section is to prove that a similar duality holds between \(X_{{\Delta}_{\mathrm{II}}}'(\subset T_{\Delta_\mathrm{I}})\) and \(X_{{\Delta}_{\mathrm{I}}}'(\subset T_{\Delta_\mathrm{II}})\).

In the following statement the varieties \(X'_{\Delta_\mathrm{I}}\) and \(X'_{\Delta_{\mathrm{II}}}\) are singular and their cohomology admits a mixed Hodge structure which is not necessarily pure. We let \(h^{p,q}(X) = {\mathrm{gr}}_F^p H^{p+q}(X)\). In the case where \(H^{p+q}(X)\) admits a pure Hodge structure, these are just the usual Hodge numbers.

The following example demonstrates that when \(k=1\), the HLLY mirror construction can be interpreted as a duality between singular varieties sitting in a diagram of the form 37 , therefore, Theorem 28 follows from Theorem 30 when \(k=1\).

There is a direct generalization of Theorem 30 to the case of complete intersections but we do not spell this out in general here. Instead, we will prove Theorem 26 which can be considered as a complete intersection generalization of Theorem 30 in a very particular case.

6 A proof of HLLY conjecture↩︎

We prove Theorem 28 (HLLY conjecture) by combining Theorem 26 and some other Clarke dualities.

As before, let’s start with a pair of reflexive dual polytopes \((\Delta,\check{\Delta})\) with a nef partition \(\Delta=\Delta_1\cup\cdots\cup\Delta_k\) and its dual nef partition \(\check{\Delta}=\check{\Delta}_1\cup\cdots\cup\check{\Delta}_k\). We choose a projective unimodular triangulation of each \(\Delta_i\) and \(\check{\Delta}_i\) whose integral points are denoted by \(A_i\) and \(\check{A}_i\), respectively. We keep the same notation as in the previous sections.

For \(a_i \in \mathbb{Z}_{\geq 0}\), we define a stacky fan \(\Sigma_{\{a_iA_i|i=1, \dots, k\}} \subset M \times \mathbb{Z}^k\) whose ray generators are of the form \(\{\rho \times a_ie_i|\rho \in A_i \}\). For simplicity, once a subset \(J \subseteq \{1, \cdots, k\}\) is taken, we write \(\Sigma_{2A_{J}, A_{J^c}}=\Sigma_{\{2A_i |i \in J\} \cup \{A_i | i \notin J\}}\). For example, when \(J = \emptyset\), then \(\Sigma_{2A_{J}, A_{J^c}}\) is the fan for the total space of the vector bundle \(\mathcal{O}(-E_{\Delta_1}) \oplus \cdots \oplus \mathcal{O}(-E_{\Delta_k})\) over \(T_{\Delta}\). The general case is described below. By applying the parallel construction for \(\check{A}_i\)’s, we obtain several Clarke dual pairs.

To such a Clarke pair \((\Sigma_{2A_{J}, A_{J^c}}, \Sigma_{\check{A}_{J}, 2\check{A}_{J^c}})\), the associated LG models are given as follows: \[(T(\Sigma_{2A_{J}, A_{J^c}}), w(\Sigma_{\check{A}_{J}, 2\check{A}_{J^c}})), \qquad (T(\Sigma_{\check{A}_{J}, 2\check{A}_{J^c}}), w(\Sigma_{{2A}_{J}, {A}_{J^c}}))\] where

• \(T(\Sigma_{2A_{J}, A_{J^c}})=\sqrt{\mathrm{Tot}(-2E_{\Delta_J}-E_{\Delta_{J^c}})/0_{J}}\) and \(w(\Sigma_{\check{A}_{J}, 2\check{A}_{J^c}})=g_{1,J}+g_{2,J^c}\).

• \(T(\Sigma_{\check{A}_{J}, 2\check{A}_{J^c}})=\sqrt{\mathrm{Tot}(-2E_{\check{\Delta}_{J^c}}-E_{\check{\Delta}_{J}})/0_{J^c}}\) and \(w(\Sigma_{2{A}_{J}, {A}_{J^c}})=\check{g}_{1,J^c}+\check{g}_{2,J}\).

We simplify notation by dropping \(\mathcal{O}\) when referring to the total space of the corresponding line bundle. For example, \[\mathrm{Tot}(-2E_{\Delta_J}-E_{\Delta_{J^c}}):=\mathrm{Tot}\left(\bigoplus_{j \in J}\mathcal{O}(-2E_{\Delta_j})\oplus \bigoplus_{j \notin J} \mathcal{O}(-E_{\Delta_j})\right).\] Also, if \(J = \{j_1,\dots, j_m\}\), then \(\sqrt{\mathrm{Tot}(-2E_{\Delta_J} - E_{\Delta_{J^c}})/0_{J}}\) is the fiber product \[\sqrt{\mathrm{Tot}(-2E_{j_1})/0_{E_{j_1}}} \times_{T_\Delta} \dots \times_{T_\Delta}\sqrt{\mathrm{Tot}(-2E_{j_m})/0_{E_{j_m}}}\times_{T_\Delta} \mathrm{Tot}(-E_{\Delta_{J^c}})\] where \(0_{E_i}\) denotes the zero section of the projection map. An analogous description applies to \(\sqrt{\mathrm{Tot}(-2E_{\check{\Delta}_{J^c}}-E_{\check{\Delta}_{J}})/0_{J^c}}\).

By Theorem 14 and Theorem 24, the orbifold cohomology \[H_{\mathrm{orb}}^*\left(\sqrt{\mathrm{Tot}(-2E_{\Delta_J}-E_{\Delta_{J^c}})/0_{J}}, g_{1,J}+g_{2,J^c}\right)\] can be decomposed based on both age grading and the grading in Corollary 2 by coinvariant degree. Namely, this orbifold cohomology is decomposed as follows: \[\begin{align} &\bigoplus_{I^{(0)} \subseteq J} H^{*-|I^{(0)}|}\left(\mathrm{Tot}(-2E_{\Delta_{J\setminus I^{(0)}}}-E_{\Delta_{J^c}}), g_{1,J\setminus I^{(0)}}+g_{2,J^c}\right)\left(\frac{|I^{(0)}|}{2}\right)=\\ & \bigoplus_{I^{(0)} \subseteq J, I^{(-)} \subseteq J^c}H^{*-|I^{(0)}|}\left(\mathrm{Tot}(-2E_{\Delta_{J\setminus I^{(0)} \cup J^c \setminus I^{(-)}}}-E_{\Delta_{I^{(-)}}}), g_{1,J\setminus I^{(0)} \cup J^c \setminus I^{(-)}}+g_{2, I^{(-)}}\right)^{(I^{(-)})}\left(\frac{|I^{(0)}|}{2}\right)\\ & =\bigoplus_{I^{(0)} \subseteq J, I^{(-)} \subseteq J^c}H^{*-|I^{(0)}|}\left(\mathrm{Tot}(-2E_{\Delta_{I^{(+)}}}-E_{\Delta_{I^{(-)}}}), g_{1,I^{(+)}}+g_{2, I^{(-)}}\right)^{(I^{(-)})}\left(\frac{|I^{(0)}|}{2}\right). \end{align}\] where \(I^{(0)}\subseteq J\) consists of the indices corresponding to the twisted sector (=number of shifting), and \(I^{(-)}\) parametrizes the indices of the coinvariant part and \(I^{(+)}:=\{1, \cdots, k\} \setminus I^{(-)} \sqcup I^{(0)}\) that parametrizes the number of invariant parts. See also Remark 15 for the superscript notation \((I^{(-)})\). We define \(B^{\lambda, \mu}_{I^{(0)}, I^{(-)}, I^{(+)}}\) to be the dimension of the direct summand indexed by \((I^{(0)}, I^{(-)})\) in the above cohomology group. Namely \[\label{e:B32def} B^{\lambda, \mu}_{I^{(-)}, I^{(0)}, I^{(+)}}:=\dim H^{\lambda-\frac{|I^{(0)}|}{2}, \mu-\frac{|I^{(0)}|}{2}}\left(\mathrm{Tot}(-2E_{\Delta_{I^{(+)}}}-E_{\Delta_{I^{(-)}}}), g_{1,I^{(+)}}+g_{2, I^{(-)}}\right)^{(I^{(-)})}.\tag{38}\] We will use the notation \(B_{I^{(-)},I^{(0)},I^{(+)}}\) to denote the element of \(\mathbb{Z}^{\tfrac{1}{2}\mathbb{Z} \times \tfrac{1}{2}\mathbb{Z}}\) whose \((\lambda,\mu)\) component is \(B_{I^{(-)},I^{(0)},I^{(+)}}^{\lambda,\mu}\). Later, we will use a shifting operation on such objects, defined by letting \(B_{I^{(-)},I^{(0)},I^{(+)}}(a)\) be such that \[\label{e:32shifting32degrees} B_{I^{(-)},I^{(0)},I^{(+)}}^{\lambda ,\mu}(a) = B_{I^{(-)},I^{(0)},I^{(+)}}^{\lambda -a,\mu-a}\tag{39}\] for any \(a \in \tfrac{1}{2}\mathbb{Z}\).

Due to Theorem 25, we have the identity of Hodge numbers: For \(\lambda, \mu \in \mathbb{Q}\), \[h_{\mathrm{orb}}^{\lambda, \mu}\left(\sqrt{\mathrm{Tot}(-2E_{\Delta_J}-E_{\Delta_{J^c}})/0_{J}}, g_{1,J}+g_{2,J^c}\right)=h_{\mathrm{orb}}^{d+k-\lambda, \mu}\left(\sqrt{\mathrm{Tot}(-2E_{\check{\Delta}_{J^c}}-E_{\check{\Delta}_{J}})/0_{J^c}}, \check{g}_{1,J}+\check{g}_{2,J^c}\right).\] The degree shifting of each summand in the both is determined by \(|I^{(-)} \sqcup I^{(0)}|\). Therefore, to acquire symmetry between \(I^{(-)}\) and \(I^{(0)}\), we consider another Clarke dual pair \((\Sigma_{A_J, 2A_{J^c}}, \Sigma_{2\check{A}_{J}, \check{A}_{J^c} })\) that is obtained by simply changing the role of \(J\) and \(J^c\) from the previous pair. Then we get a similar identity of Hodge numbers: For \(\lambda, \mu \in \mathbb{Q}\), \[h_{\mathrm{orb}}^{\lambda, \mu}\left(\sqrt{\mathrm{Tot}(-2E_{\Delta_{J^c}}-E_{\Delta_{J}})/0_{J^c}}, g_{1,J^c}+g_{2,J}\right)=h_{\mathrm{orb}}^{d+k-\lambda, \mu}\left(\sqrt{\mathrm{Tot}(-2E_{\check{\Delta}_{J}}-E_{\check{\Delta}_{J^c}})/0_{J}}, \check{g}_{1,J^c}+\check{g}_{2,J}\right).\] Now we combine these two Hodge number dualities to deduce the weaker one. In other words, we pair up the orbifold cohomology \[\label{e:grping} H_{\mathrm{orb}}^*\left(\sqrt{\mathrm{Tot}(-2E_{\Delta_J}-E_{\Delta_{J^c}})/0_{J}}, g_{1,J}+g_{2,J^c}\right)\bigoplus H_{\mathrm{orb}}^*\left(\sqrt{\mathrm{Tot}(-2E_{\Delta_{J^c}}-E_{\Delta_{J}})/0_{J^c}}, {g}_{1,J^c}+{g}_{2,J}\right).\tag{41}\] The mirror counterpart can be simply obtained by replacing \(\Delta\) and \(g\) by \(\check{\Delta}\) and \(\check{g}\), respectively. Also, the degree shifting of each summand in the both is determined by \(|I^{(-)} \sqcup I^{(0)}|\). Then the weaker version of Hodge number duality can be written as \[\label{e:mir} \sum_{I^{(0)}\subseteq J, I^{(-)}\subseteq J^c} B^{\lambda, \mu}_{I^{(-)}, I^{(0)}, I^{(+)}}+B^{\lambda, \mu}_{I^{(0)}, I^{(-)}, I^{(+)}}= \sum_{I^{(0)}\subseteq J, I^{(-)}\subseteq J^c} \check{B}^{d+k-\lambda, \mu}_{I^{(-)}, I^{(0)}, I^{(+)}}+\check{B}^{d+k-\lambda, \mu}_{I^{(0)}, I^{(-)}, I^{(+)}}\tag{42}\] where \(\check{B}^{d+k-\lambda, \mu}_{I^{(-)}, I^{(0)}, I^{(+)}}\)’s are defined in the same way as 38 . Since the degree of the mirror dual part is canonically determined once the Hodge grading \((\lambda, \mu)\) of \(B_{I^{(-)}, I^{(0)}, I^{(+)}}\) is specified, we simply say that the sum \[\sum_{I^{(0)}\subseteq J, I^{(-)}\subseteq J^c} B_{I^{(-)}, I^{(0)}, I^{(+)}}+B_{I^{(0)}, I^{(-)}, I^{(+)}}\] satisfies the mirror relation because 42 holds for all \((\lambda, \mu)\). We will say that the mirror relation is satisfied for any finite sum of \(B_{I^{(-)},I^{(0)},I^{(+)}}\) if the analogue of 42 holds.

Note that \[B^{\lambda,\mu}_{[k],\emptyset,\emptyset} = h^{\lambda-k/2,\mu-k/2}(\widetilde{T}_\Delta)^{(-)},\qquad B^{\lambda,\mu}_{\emptyset,[k],\emptyset} = h^{\lambda-k/2,\mu-k/2}(T_\Delta).\] So, by Proposition 17, Theorem 28 reduces to showing that \[\label{e32:32satisfies32the32mirror32relation} B_{[k],\emptyset,\emptyset} + B_{\emptyset,[k],\emptyset} \text{ satisfies the mirror relation.}\tag{43}\] The reader should also observe that, following Proposition 17 and 40 , we have \[\sum_{I^{(0)}\cup I^{(-)} = [k]} B^{\lambda,\mu}_{I^{(-)},I^{(0)},\emptyset} = h^{\lambda - k,\mu-k}(\widehat{T}_\Delta).\] Therefore, Proposition 17 and Theorem 26 tell us that: \[\sum_{I^{(0)}\cup I^{(-)} = [k]} B_{I^{(-)},I^{(0)},\emptyset} \text{ satisfies the mirror relation.}\]

Note that if \(k-|I^{(+)}| \in 2\mathbb{Z}\) then \(B^{\lambda,\mu}_{I^{(-)}, I^{(0)}, I^{(+)}} = 0\) unless \(\lambda,\mu \in \mathbb{Z}\) and similarly if \(k - |I^{(+)}| \in 2\mathbb{Z}+1\) then \(B^{\lambda,\mu}_{I^{(-)}, I^{(0)}, I^{(+)}} = 0\) unless \(\lambda,\mu \in (\tfrac{1}{2}) + \mathbb{Z}\). Since we are largely interested in the case where \(|I^{(+)}| = 0\), we can distinguish between the cases where \(|I^{(+)}|\) is even or odd. We focus only on the case where \(|I^{(+)}|\) is even, as this is the equality relevant to 43 and hence the proof of Conjecture 27. Also, note that the mirror relation 42 is preserved under scalar multiplication and addition.

Before getting into the proof of Theorem 28, we examine the cases \(k=2\) and \(k=3\) to illustrate how this description is used in the proof of Conjecture 27.

For the cases when \(k \geq 4\), we do not need to keep track of all the indices of the summands for each \(J\). Instead, we group all \(J\)’s with the same size together. For \(|J| = p\) with \(0 \leq p \leq \lfloor \frac{k}{2} \rfloor\), we group the summands \(B\)’s and denote \(B_{a,b,c}=\sum_{|I^{(-)}|=a, |I^{(0)}|=b, |I^{(+)}|=c} B_{I^{(-)}, I^{(0)}, I^{(+)}}\) for any \(a,b,c \geq 0\).

Using these binomial identities, we prove the following theorem.

Recall that an element \(i\) in \(I_1\) or \(I_2\) corresponds to a set of integral points \(A_i\). For \(j=1,2\), we introduce a singleton \(\sum I_j\) that corresponds to the union of \(A_i\)’s over all \(i \in I_j\). For example, \(B_{\sum I_2, \sum I_1, \emptyset}+B_{\sum I_1, \sum I_2, \emptyset}\) corresponds to the coarser nef partition \(\Delta=\Delta_1' \cup \Delta_2'\) and \(\check{\Delta}=\check{\Delta}_1' \cup \check{\Delta}_2'\) where \(\Delta'_1=\cup_{i \in I_1} \Delta_i\),\(\Delta'_2=\cup_{i \in I_2}\Delta_i\), \(\check{\Delta}'_1=\cup_{i \in I_1} \check{\Delta}_i\) and \(\check{\Delta}'_2=\cup_{i \in I_2}\check{\Delta}_i\), with the previous choice of projective unimodular triangulations. A crucial observation, which can be deduced easily, e.g. from 40 , is that \[\label{e32:32contraction32relation} B_{I^{(-)},I^{(0)},I^{(+)}} = B_{I^{(-)},\sum I^{(0)},I^{(+)}}((|I^{(0)}|-1)/2).\tag{49}\] Here, the \((-)\) notation is as in 39 .

The proof of Theorem 33 then reduces to proving that the mirror relation holds for \(B_{N+1,0,0} + B_{0,N+1,0}\) and \(B_{N,1,0} + B_{1,N,0}\) for all \(N\). This can be deduced from Lemma 5 as argued below.

References↩︎

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1. This step uses the full force of 19 . A direct proof of the weaker statement in Theorem 14 would require more subtle induction at this point.↩︎

2. In [14], we used the letter \(f\) to denote Hodge numbers in order to distinguish them from the Deligne–Hodge numbers. However, since we primarily work with smooth and proper varieties in this article, such a distinction is unnecessary, and we will use the letter \(h\) throughout the article.↩︎

3. In [13], Hosono, Lee, Lian, and Yau let \(\Sigma_\Delta\) denote the normal fan of \(\Delta\), or equivalently the spanning fan of \(\Delta^\circ\), rather than the spanning fan of \(\Delta\). The notation used in this paper is consistent with that of our earlier work [14], and was chosen for its compatibility with the Clarke mirror construction.↩︎

4. The total space of any split vector bundle over a toric variety is itself toric [26].↩︎