2.1 The weight filtration

We recall basic concepts about the mixed Hodge structures [Reference Deligne10][Reference Deligne9] following the exposition in [Reference Peters and Steenbrink25]. Let U be a smooth quasi-projective variety over ${\mathbb {C}}$ and $(X,D)$ be a good compactification of U. Recall that a pair $(X,D)$ is called a good compactification of U if X is a smooth and compact variety and D is a simple normal crossing divisor. Let $j:U \to X$ be a natural inclusion. Consider the logarithmic de Rham complex

$$ \begin{align*} \Omega^{\bullet}_X(\log D) \subset j_*\Omega^{\bullet}_U. \end{align*} $$

Locally at $p \in D$ with an open neighborhood $V \subset X$ with coordinates $(z_1, \cdots , z_n)$ in which D is given by $z_1\cdots z_k=0$ , one can see

$$ \begin{align*} \begin{aligned} &\Omega^1_X(\log D)_p=\mathcal O_{X,p}\frac{dz_1}{z_1} \oplus \cdots \oplus \mathcal O_{X,p}\frac{dz_k}{z_k} \oplus \mathcal O_{X,p}dz_{k+1} \oplus \cdots \oplus \mathcal O_{X,p}dz_n\\ &\Omega^r_X(\log D)_p = \bigwedge^r\Omega^1_X(\log D)_p. \end{aligned} \end{align*} $$

There are two filtrations on the logarithmic de Rham complex $(\Omega ^{\bullet }_X(\log D),d)$ :

1. 1. (Hodge filtration) A decreasing filtration $F^{\bullet }$ on $\Omega ^{\bullet }_X(\log D)$ defined by

   $$ \begin{align*} F^p\Omega^{\bullet}_X(\log D):= \Omega^{\geq p}_X(\log D). \end{align*} $$

2. 2. (Weight filtration) An increasing filtration $W_\bullet $ on $\Omega ^{\bullet }_X(\log D)$ defined by

   $$\begin{align*}W_m\Omega^r_X(\log D):= \begin{cases} 0 & m<0\\ \Omega^r_X(\log D) & m \geq r\\ \Omega^{r-m}_X\wedge \Omega^m_X(\log D) & 0 \leq m \leq r. \end{cases} \end{align*}$$

The key properties of these two filtrations are the degenerations of the associated spectral sequences. More precisely, we have

For a given mixed Hodge structure $V=(V_{\mathbb {C}}, W_\bullet , F^{\bullet })$ and $m \in {\mathbb {Z}}$ , we define the m-th Tate twist of V by setting $V(m):=(V_{\mathbb {C}}(m), W(m)_\bullet , F(m)^{\bullet })$ , where $V_{\mathbb {C}}(m):=(2\pi i)^mV_{\mathbb {C}}$ and

$$ \begin{align*} W(m)_k:=W_{k+2m} \quad F(m)^p:=F^{m+p} \end{align*} $$

for all k and p.

In order to compute the mixed Hodge structures, we introduce the geometric description of the $E_1$ -page of the spectral sequence. Let D be a simple normal crossing divisor with N irreducible components $D_1, \dots , D_N$ . For any index set $I \subset \{1, \dots , N\}$ , we write $D_I=\cap _{i \in I}D_i$ for the intersection. We set $D(k)$ to be the disjoint union of k-tuple intersections of the components of D and $D(0)$ to be X. Also, for $I=(i_1, \cdots , i_m)$ and $J=(i_1, \cdots , \hat {i_j}, \cdots , i_m)$ , there are inclusion maps

$$ \begin{align*} & \iota^I_J:D_I \hookrightarrow D_J\\ \iota^m_j=\bigoplus_{|I|=m}& \iota_J^I:D(m) \hookrightarrow D(m-1) \end{align*} $$

which induce canonical Gysin morphisms on the level of cohomology. Therefore, we have

(2.1) $$ \begin{align} \gamma_m=\bigoplus^m_{j=1}(-1)^{j-1}(\iota^m_j)_!:H^{k-m}(D(m))(-m) \rightarrow H^{k-m+2}(D(m-1))(-m+1), \end{align} $$

where $(-)_!$ is the Gysin morphism. We call this sign convention the Mayer-Vietoris sign rule which is unique up to $\pm 1$ . Under the residue map, this gives a geometric description of the differential $d_1:E_1^{-m, k+m} \to E_1^{-m+1, k+m}$ of the $E_1$ -page of the spectral sequence for the weight filtration as follows:

Proposition 2.4 [Reference Peters and Steenbrink25, Proposition 4.7].

The following diagram is commutative

(2.2)

where $res_m$ is the residue map for all $m \geq 0$ .

Note that all the morphisms in the diagram (2.2) are compatible with Hodge filtration $F^{\bullet }$ . This description provides several computational tools as well as functorial properties of the mixed Hodge structures under geometric morphisms. For more details, we refer the reader to [Reference Peters and Steenbrink25].

One can extend the above construction to the case when U is singular. This can be done by taking a simplicial or cubical resolution of the singular variety U and associated good compactifications. We will not review this construction but describe one particular case which we will deal with.

Example 2.5. Let D be a simple normal crossing variety. Consider the long exact sequences

(2.3) $$ \begin{align} 0 \to \bigoplus_{|I|=1}H^j(D_I)\xrightarrow{d_0} \bigoplus_{|I|=2}H^j(D_I)\xrightarrow{d_1} \bigoplus_{|I|=3}H^j(D_I)\xrightarrow{d_2} \cdots, \end{align} $$

where $d_i$ is the alternating sum of the restriction map. Then we have

$$ \begin{align*} \text{Gr}^W_{j}H^{i+j}(D)=\frac{\ker d_i}{\operatorname{\mathrm{Im}} d_{i+1}}. \end{align*} $$

In fact, the sequence (2.3) is the $E_1$ -page of the spectral sequence for the weight filtration $W_\bullet $ . It is also compatible with the Hodge filtrations on each term to yield the Hodge filtration on $H^*(D)$ .

2.2 The monodromy weight filtration

Let X be a smooth complex manifold and $\Delta $ be the unit disk. We consider a holomorphic map $f:X \to \Delta $ that is smooth over the punctured disk $\Delta ^*:=\Delta \setminus \{0\}$ . We also assume that $E:=f^{-1}(0)$ is a simple normal crossing divisor. Let $E_i$ be the components of E and write

$$ \begin{align*} E_I=\bigcap_{i \in I} E_i, \qquad E(m)=\coprod _{|I|=m}E_I \end{align*} $$

as before. We present the de Rham theoretic description of the monodromy weight filtration on a generic fiber $X_t:=f^{-1}(t)$ .

Define the relative de Rham complex on X with logarithmic poles along E:

$$ \begin{align*} \Omega^{\bullet}_{X/\Delta}(\log E):=\Omega_X^{\bullet}(\log E)/f^*(\Omega^1_\Delta (\log 0)) \wedge \Omega_X^{\bullet-1}(\log E). \end{align*} $$

By definition, this fits into the short exact sequence

$$ \begin{align*} 0 \to f^*(\Omega^1_\Delta (\log 0)) \wedge \Omega_X^{\bullet-1}(\log E) \to \Omega_X^{\bullet}(\log E) \to \Omega^{\bullet}_{X/\Delta}(\log E) \to 0. \end{align*} $$

By taking $-\otimes \mathcal O_E$ on the above sequence, we will have

$$ \begin{align*} 0\to \Omega^{\bullet}_{X/\Delta}(\log E)\otimes \mathcal O_E [-1] \xrightarrow{\wedge dt/t} \Omega_X^{\bullet}(\log E) \otimes \mathcal O_E \to \Omega^{\bullet}_{X/\Delta}(\log E)\otimes \mathcal O_E \to 0. \end{align*} $$

The connecting homomorphism induces the residue at 0 of the logarithmic extension of the Gauss–Manin connection:

$$ \begin{align*} res_0(\nabla):\mathbb H^q(E, \Omega^{\bullet}_{X/\Delta}(\log E) \otimes \mathcal O_E) \to \mathbb H^q(E, \Omega^{\bullet}_{X/\Delta}(\log E) \otimes \mathcal O_E). \end{align*} $$

Note that the cohomology of the induced complex on E, $\Omega ^{\bullet }_{X/\Delta }(\log E) \otimes \mathcal O_E$ becomes isomorphic to the cohomology group $H^*(X_t)$ . Also, the morphism $res_0(\nabla )$ recovers the monodromy action on $H^k(X_t)$ .

Define the increasing filtration $W_{\bullet }$ on $\Omega _{X/\Delta }(\log E)\otimes \mathcal O_E$ by

$$ \begin{align*} W_k\Omega^{\bullet}_{X/\Delta}(\log E)\otimes \mathcal O_E:=\operatorname{\mathrm{Im}}\left(W_k\Omega^{\bullet}_{X}(\log E) \to \Omega^{\bullet}_{X/\Delta}(\log E)\otimes \mathcal O_E\right) \end{align*} $$

and the decreasing filtration $F^{\bullet }$ by the simple truncation. To describe the monodromy weight filtration, we consider the resolution of $\Omega ^{\bullet }_{X/\Delta }(\log E)\otimes \mathcal O_E$ as follows. Define a tri-filtered double complex

$$ \begin{align*} (A^{\bullet, \bullet}, d', d", W_{\bullet}, W(M)_{\bullet}, F^{\bullet}) \end{align*} $$

on E by

$$ \begin{align*} \begin{aligned} &A^{p,q}=\frac{\Omega^{p+q+1}_X(\log E)}{W_p\Omega^{p+q+1}_X(\log E)}, \quad d'=(-) \wedge dt/t:A^{p,q} \to A^{p+1, q},\quad d"=d_{dR}:A^{p,q} \to A^{p,q+1} \\ & W_rA^{p,q}=\frac{W_{r+p+1}\Omega^{p+q+1}_X(\log E)}{W_p\Omega^{p+q+1}_X(\log E)}, \quad W(M)_rA^{p,q}=\frac{W_{r+2p+1}\Omega^{p+q+1}_X(\log E)}{W_p\Omega^{p+q+1}_X(\log E)}\\ & F^r A^{p,q}=\frac{F^r\Omega^{p+q+1}(\log E)}{W_p\Omega^{p+q+1}_X(\log E)}. \end{aligned} \end{align*} $$

We have the map

$$ \begin{align*} \begin{aligned} \mu: \Omega^q_{X/\Delta}(\log E)\otimes \mathcal O_E & \to A^{0,q} \\ \omega & \mapsto (-1)^q(dt/t)\wedge \omega\quad \mod{W_0} \end{aligned} \end{align*} $$

which defines a quasi-isomorphism of bifiltered complexes

$$ \begin{align*} \mu: (\Omega^{\bullet}_{X/\Delta}(\log E)\otimes \mathcal O_E, W_\bullet, F^{\bullet}) \to (s(A^{\bullet, \bullet}), W_\bullet, F^{\bullet}), \end{align*} $$

where $s(A^{\bullet , \bullet })$ is the associated single complex. Consider the natural morphism $\nu :A^{p,q} \to A^{p+1, q-1}$ given by $\omega \mapsto \omega \text { (mod}{W_{p+1}})$ . As it commutes with both differentials $d'$ and $d"$ , it induces the endomorphism of the associated simple complex $s(A^{\bullet ,\bullet })$ . Note that this sends $W(M)_r$ to $W(M)_{r-2}$ and $F^p$ to $F^{p-1}$ .

Theorem 2.6 [Reference Peters and Steenbrink25, Theorem 11.21].

The following diagram is commutative:

By taking the residue map, we have

$$ \begin{align*} \begin{aligned} \text{Gr}^{W(M)}_r s(A^{\bullet, \bullet}) & \cong \bigoplus_{k \geq 0,-r}\text{Gr}^W_{r+2k+1}\Omega_X^{\bullet}(\log E)[1] \\ & \cong \bigoplus_{k \geq 0,-r}\Omega^{\bullet}_{E(r+2k+1)}[-r-2k]. \end{aligned} \end{align*} $$

Therefore, the $E_1$ page of the spectral sequence for the monodromy weight filtration $W(M)_\bullet $ is given by

$$ \begin{align*} E_1^{p,q}=\bigoplus_{k \geq 0,p}H^{q+2p-2k}(E(2k-p+1),{\mathbb{C}})(p-k). \end{align*} $$

More explicitly, the $E_1$ -page is given by the following diagram:

where the horizontal arrows are (the alternating sum of) the Gysin morphisms while the antidiagonal arrows are (the alternating sum of) the pullback morphisms. If we write down two morpshisms by G and d, respectively, the differential $d_1:E_1^{p,q} \to E_1^{p+1,q}$ is given by $d_1=G+(-1)^pd$ .

Theorem 2.7 [Reference Peters and Steenbrink25, Theorem 11.22].

The spectral sequence for the filtration $W(M)_\bullet $ degenerates at the $E_2$ -page so that we have

$$ \begin{align*} E_2^{p,q}=E_\infty^{p,q}=\text{Gr}^{W(M)}_{q}H^{p+q}(X). \end{align*} $$

We will also denote the monodromy weight filtration $W(M)_\bullet $ by $W_{\lim {}\bullet }$ .

2.3 The perverse filtration

We briefly review the notion of perverse filtration [Reference de Cataldo and Migliorini6] and its geometric description [Reference de Cataldo and Migliorini7].

Verdier’s dualizing functor on $D^b_c(Y)$ is defined as $\mathbb {D}=\operatorname {\mathrm {Hom}}_{\mathcal O_Y}(-, p^{!}({\mathbb {C}}_{pt}))$ , where $p:Y \to pt$ is a trivial map. We call $p^{!}({\mathbb {C}}_{pt})$ a dualizing complex of Y, and denote it by $\omega _Y$ . In particular, if Y is nonsingular of complex dimension n, $\omega _Y={\mathbb {C}}_Y[2n]$ . Note that the subcategory $\mathcal {P}(Y)$ of perverse sheaves on Y is an abelian category. Also, the support and cosupport condition induces the so-called perverse t-structure $({}^{\mathfrak {p}}D^{b, \geq 0}_c(Y), {}^{\mathfrak {p}}D^{b, \leq 0}_c(Y))$ on $D^b_c(Y)$ whose heart is $\mathcal {P}(Y)$ . Explicitly, it is given by

1. 1. $K^{\bullet } \in {}^{\mathfrak {p}}D^{b, \leq 0}_c(Y)$ if and only if K satisfies the support condition. Also, ${}^{\mathfrak {p}}D^{b, \leq n}_c(Y):={}^{\mathfrak {p}} D^{b, \leq 0}_c(Y)[-n]$

2. 2. $K^{\bullet } \in {}^{\mathfrak {p}} D^{b, \geq 0}_c(Y)$ if and only if K satisfies the cosupport condition. Also, ${}^{\mathfrak {p}}D^{b, \geq n}_c(Y):={}^{\mathfrak {p}} D^{b, \geq 0}_c(Y)[-n]$ .

We denote ${}^{\mathfrak {p}}\tau _{\leq n}:D_c^b(Y) \to {}^{\mathfrak {p}} D^{b, \leq n}_c(Y)$ (resp. ${}^{\mathfrak {p}}\tau _{\geq n}:D_c^b(Y) \to {}^{\mathfrak {p}} D^{b, \geq n}_c(Y)$ ) the natural truncation functor. This induces perverse cohomology functors ${}^{\mathfrak {p}}\mathcal {H}:D^b_c(Y) \to \mathcal {P}(Y)$ defined by ${}^{\mathfrak {p}}\mathcal {H}^k:={}^{\mathfrak {p}} \tau _{\leq 0} \circ {}^{\mathfrak {p}}\tau _{\geq 0} \circ [k]$ . Applying the perverse truncation, one can define the perverse filtration on the hypercohomology of a constructible sheaf $\mathcal K^{\bullet }$ on Y as follows;

Definition 2.9. For $\mathcal K^{\bullet } \in D^b_c(Y)$ , the perverse filtration $P_\bullet $ on $\mathbb H^k(Y, \mathcal K^{\bullet })$ is defined to be

$$ \begin{align*} P_b\mathbb H^k(Y, \mathcal K^{\bullet}):=\operatorname{\mathrm{Im}}\left(\mathbb H^k(Y, {}^{\mathfrak{p}}\tau_{\leq b}\mathcal K^{\bullet}) \to \mathbb H^k(Y, \mathcal K^{\bullet})\right). \end{align*} $$

Let $f:X \to Y$ be a morphism of smooth varieties. Then we can define the perverse (f-)Leray filtration on the cohomology $H^{\bullet }(X, {\mathbb {C}})$ by setting

$$ \begin{align*} P^f_lH^k(X,{\mathbb{C}}):=P^f_l\mathbb H^k(Y, Rf_*{\mathbb{C}}) \end{align*} $$

Theorem 2.10 [Reference Peters and Steenbrink25, Corollary 14.41].

If f is proper, then the spectral sequence for the perverse Leray filtration degenerates at the $E_2$ page. In other words, we have

$$ \begin{align*} \text{Gr}^{P^f}_l\mathbb H^{k}(X,{\mathbb{C}})=E_2^{k-l,l}=\mathbb H^{k-l}(Y, {}^{\mathfrak{p}}\mathcal{H}^l(Rf_*{\mathbb{C}})). \end{align*} $$

We will provide a geometric description of the perverse Leray filtration in case that the base space is either affine or quasi-projective. For this, we introduce some notations used in the next subsection. Let $f:X \to Y$ be a locally closed embedding. Then the restriction functor is given by $(-)|_X=Rf_!f^*$ on $D_c^b(Y)$ , which is exact. If f is closed, then we also have the right derived functor of sections with support in X, denoted by $R\Gamma _X(-)=Rf_*f^!$ .

Next, we provide provide a geometric description of perverse filtrations. We follow the same convention for the indices of filtrations used in [Reference Doran and Thompson12]. Let’s consider the following commutative diagram of varieties

where

• • Y is a smooth complex projective variety of complex dimension n.

• • B is a complex projective variety of complex dimension m. We fix an embedding $B \hookrightarrow \mathbb P^N$ .

• • $\pi :Y \to B$ is a proper morphism.

• • $B_U$ is the affine subvariety of B and $U:=\pi ^{-1}(B_U)$ . We write $\pi _U$ for the restriction of $\pi $ onto U.

Recall that there is a smooth projective variety $F(N,m)$ parametrizing m-flags $\mathfrak {F}=\{F_{-m} \subset \cdots \subset F_{-1}\}$ on $\mathbb P^N$ , where $F_{-p}$ is a codimension p linear subspace. A linear m-flag $\mathfrak {F}$ on $\mathbb P^N$ is general if it belongs to a suitable Zariski open subset of $F(N,m)$ . Similarly, we say a pair of linear m-flags $(\mathfrak {F}_1, \mathfrak {F}_2)$ is general if it belongs to a suitable Zariski open subset of $F(N,m) \times F(N,m)$ .

Fix a general pair of m-flags $({H}_\bullet , {L}_\bullet )$ on $\mathbb P^N$ . Intersecting with B, it gives a pair of flags of subvarieties $({B}_\bullet , {C}_\bullet )$ of B,

$$ \begin{align*} \begin{aligned} \emptyset=B_{-m-1} \subset B_{-m} \subset \dots \subset B_{-1} \subset B_0=B, \\ \emptyset=C_{-m-1} \subset C_{-m} \subset \dots \subset C_{-1} \subset C_0=B, \end{aligned} \end{align*} $$

where ${B}_\bullet :={H}_\bullet \cap B$ and ${C}_\bullet :={L}_\bullet \cap B$ . We set $Y_\bullet =\pi ^{-1}(B_\bullet )$ and $Z_\bullet =\pi ^{-1}(C_\bullet )$ . By following [Reference de Cataldo and Migliorini7, Reference Doran and Thompson12], we define the following flag filtrations.

Note that both two G-filtraions can be defined on other cohomology theories as well. We describe the $E_1$ -page of the spectral sequence for each filtration.

1. 1. The $E_1$ -page of the spectral sequence for the flag filtration $G^{\bullet }$ (of the first kind) on $H^*(U)$ is given by

   $$ \begin{align*} {}^GE_1^{p,q}=H^{p+q}(U \cap Y_p, U \cap Y_{p-1},{\mathbb{C}}) \Longrightarrow H^*(U,{\mathbb{C}}), \end{align*} $$
   and the differential $d_1:{}^GE_1^{p,q} \to {}^GE_1^{p+1,q} $ is the connecting homomorphism of the long exact sequence of cohomology groups of the triple $(Y_{p+1}, Y_p, Y_{p-1})$ . Furthermore, we have
   $$ \begin{align*} {}^GE_\infty^{p,q} =\text{Gr}_G^pH^{p+q}(U,{\mathbb{C}}). \end{align*} $$

2. 2. The $E_1$ -page of the spectral sequence for the flag filtration $G^{\bullet }$ (of the second kind) on $H_c^*(U,{\mathbb {C}})$ is given by

   $$ \begin{align*} {}^GE_{1}^{p,q}=H_{Z_{-p}\cap U-Z_{-p-1}\cap U,c}^{p+q}(U,{\mathbb{C}}) \Longrightarrow H_c^*(U,{\mathbb{C}}), \end{align*} $$
   and the differential $d_1:{}^GE_{1}^{p,q} \to {}^GE_{1}^{p+1,q} $ is the connecting homomorphism of the long exact sequence of cohomology groups with supports $(Z_{-p}, Z_{-p-1}, Z_{-p-2})$ . Furthermore, we have
   $$ \begin{align*} {}^GE_{\infty}^{p,q} =\text{Gr}_G^pH_c^{p+q}(U,{\mathbb{C}}). \end{align*} $$

3. 3. The $E_1$ -page of the spectral sequence for the $\delta $ -flag filtration $\delta ^{\bullet }$ on $H^*(Y)$ is given by

   $$ \begin{align*} {}^\delta E_1^{p,q}=\bigoplus\limits_{i+j=p}H^{p+q}_{Z_{-j}-Z_{-j-1}}(Y, {\mathbb{C}}|_{Y_i-Y_{i-1}}) \Longrightarrow H^*(Y, {\mathbb{C}}). \end{align*} $$

   More explicitly, the $E_1$ -page is given by the following diagram.

   (2.4)

   

   • • For fixed j, the antidiagonal sequence is the same with the $E_1$ -page of the spectral sequence for the G-filtration of the first kind on $H_{Z_{-j}}^*(Y)$ with respects to the induced flag $Z_{-j} \cap Y_\bullet $ . Let’s write $d_I$ for the differential.

   • • For fixed i, the horizontal sequence is the same with the $E_1$ -page of the spectral sequence for the G-filtration of the second kind on $H^*(Y_i)$ with respects to the induced flag $Z_\bullet \cap Y_i$ . Let’s write $d_{II}$ for the differential.

   • • The differential $d_1:{}^\delta E_1^{p,q} \to {}^\delta E_1^{p+1,q}$ is given by $d_1=d_I+(-1)^pd_{II}$ .

   Moreover, we have

   $$ \begin{align*} {}^\delta E_\infty^{p,q}=\text{Gr}_\delta^{p}H^{p+q}(Y,{\mathbb{C}}). \end{align*} $$

3.1 Hybrid LG models

We start to recall the notion of hybrid LG models introduced in [Reference Lee24]. Let’s first introduce some notations. Let $h=(h_1, \dots , h_N):Y \to {\mathbb {C}}^N$ be a N-tuple of (holomorphic) functions and $(z_1,\dots , z_N)$ be the coordinates of the base ${\mathbb {C}}^N$ . For each nonempty subset $I=\{i_1, \dots , i_l\} \subset \{1,\dots , N\}$ , we write $h_I=(h_{i_1}, \dots , h_{i_l}):Y \to {\mathbb {C}}^{|I|}$ and the coordinate $(z_{i_1}, \dots , z_{i_l})$ for the base ${\mathbb {C}}^{|I|}$ , which implicitly determines the natural inclusion ${\mathbb {C}}^{|I|} \subset {\mathbb {C}}^N$ .

When $N=1$ , this definition recovers the usual notion of LG models $(Y, \omega , h:Y \to {\mathbb {C}})$ , where h becomes a locally trivial symplectic fibration with smooth fibers near infinity. One can see that the second condition in Definition 3.1 controls the geometry of the local fibration h near the infinity boundary of the base. For each nonempty subset $I \subset \{1, \dots , N\}$ , let’s write $Y_I$ for a generic fiber of $h_I:Y \to {\mathbb {C}}^{|I|}$ and $h_{Y_I}:Y_I \to {\mathbb {C}}^{N-|I|}$ for the restriction of h into $Y_I$ . Then the induced triple $(Y_I, \omega |_{Y_I}, h_{Y_I})$ can be regraded as a hybrid LG model of rank $N-|I|$ . From this point of view, the condition $(2)-(a)$ in Definition 3.1 is rephrased as the condition that $h_I:Y \to {\mathbb {C}}^{|I|}$ is a local trivialization of the induced hybrid LG models of rank $N-|I|$ .

Associated to the hybrid LG model $(Y, \omega , h:Y \to {\mathbb {C}}^N)$ , we define the ordinary LG model to be a triple $(Y, \omega , \mathsf w:=\Sigma \circ h:Y \to {\mathbb {C}})$ , where $\Sigma :{\mathbb {C}}^N \to {\mathbb {C}}$ is the summation map. The following proposition justifies this terminology.

Proof. We present the proof for the reader’s convenience. Take a hyperplane $H=\{a_1z_1 + \cdots + a_Nz_N=M\}$ , where $a_i \neq 0$ for all i. By changing the coordinate $z_i \mapsto z_i/a_i$ , we reduce to the case where $a_i=1$ for all i. We also further reduce to the case when M is real due to the rotational symmetry. By generality, we take $M>NR$ . First, note that $H \cap (\cap _{i=1}^N\{|z_i|\leq R\})=\emptyset $ . Let $R_i=\{Re(z_i)>R\}$ and the simply connected region

$$ \begin{align*} U_i:=\{Re(z_i)> R\} \cap H=\{Re(z_1 + \cdots + \hat{z_i}+ \cdots + z_n)<M-R\} \cap H \end{align*} $$

for each i. Since $U_i \subset \{|z_i|>R\}$ , one can project $U_i$ to the locus $\{z_i=2R\}$ inside the region $\{|z_i|<R\}$ . The image of the projection is $V_i:=\{z_i=2R, Re(z_1+\cdots +\hat {z_i}+\cdots +z_N)<M-R\}$ which contains $\bigcap _{j \neq i}\{|z_j| \leq R\}$ . Therefore, this projection identifies $h:h^{-1}(U_i) \to U_i$ with $h:h^{-1}(V_i) \to V_i$ due to the local triviality of the hybrid LG model. Moreover, the latter map is completed to $h_{Y_i}:Y_i \to {\mathbb {C}}^{N-1}$ by the inductive argument. In general, for each I, $U_I= \cap _{i \in I}U_i$ is nonempty and simply connected. Since $U_I \subset \{|z_i|>R, i \in I\}$ , one can apply the same argument to get the conclusion.

On the cohomology level, Proposition 3.2 implies that the cohomology group of $\pi ^{-1}(H)$ is (noncanonically) isomorphic to that of the normal crossing union of $Y_i$ ’s. We will use this fact to study the perverse Leray filtration associated to $h:Y \to {\mathbb {C}}^N$ on $H^*(Y)$ .

Consider a general flag of hyperplanes in ${\mathbb {C}}^N$ ,

$$ \begin{align*} \mathfrak{H}:0=H_{-N-1} \subset H_{-N} \subset \cdots \subset H_{-1} \subset H_0={\mathbb{C}}^N \end{align*} $$

which is transversal to the discriminant locus of h in the sense of [Reference de Cataldo and Migliorini7, Definition 5.2.4] and each $H_{-l}$ is not parallel to any coordinate lines. We write $Y_{sm^{(l)}}$ for $h^{-1}(H_{-l})$ so that we have a general flag of subvarieties

$$ \begin{align*} 0 \subset Y_{sm^{(N)}} \subset \cdots \subset Y_{sm^{(1)}}\subset Y \end{align*} $$

which will be used to compute the flag filtration $G^{\bullet }$ (equivalently, the perverse Leray filtration $P^h_\bullet $ ) on $H^*(Y)$ (see Section 2.3). In other words, the $E_1$ -page of the spectral sequence is given by the sequence

(3.1) $$ \begin{align} H^{a-N}(Y_{sm^{(N)}}) \xrightarrow{d_1} H^{a-N+1}(Y_{sm^{(N-1)}}, Y_{sm^{(N)}}) \xrightarrow{d_1} \cdots \xrightarrow{d_1} H^{a-1}(Y_{sm^{(1)}}, Y_{sm^{(2)}}) \xrightarrow{d_1} H^a(Y, Y_{sm^{(1)}}). \end{align} $$

We use the same notation in the proof of Proposition 3.2. Take open (simply connected) regions $\{R_i \subset {\mathbb {C}}^N|i=1, \dots , N\}$ which induce an open covering of $H_{-1}$ , $\{U_i:=R_i \cap H_{-1}|i=1, \dots , N\}$ that yields the gluing property. Let $V_i=\{z_i=const\}$ be the region that $U_i$ projects to. Due to the genericity of the flag, we may assume that $H_{-2} \cap U_i \subset U_i$ projects to a hyperplane that is contained in $\cup _{j \neq i}R_j \cap V_i$ for all i. As both $H_{-1}$ and $H_{-2}$ are not parallel to any coordinate lines, this can be done by scaling M sufficiently large to place $H_{-2}$ far enough from each coordinate line. It ensures that $Y_{sm}\cap h^{-1}(U_i)$ is isotopic to $Y_{i,sm}$ for each i. Inductively, for each k, we could assume that the collection of regions $\{R_i \subset {\mathbb {C}}^N|i=1, \dots , N\}$ yields the gluing property for $H_{-k}$ in $V_I:=\cap _{i \in I}V_i$ for any $|I|=k-1$ . Then the gluing property implies the following:

$$ \begin{align*} Y_{sm^{(k)}} \cap h^{-1}(U_I) \cong \begin{cases} Y_{I,sm^{(k-|I|)}} &|I| < k \\ Y_I & |I| \geq k. \end{cases} \end{align*} $$

Proof. Take the (simply connected) open region $\{R_i \subset {\mathbb {C}}^N|i=1, \dots , N\}$ and the induced cover $\{U_i=R_i \cap H_{-1}\}$ as above. When $k=1$ , the Mayer–Vietoris argument with respect to the open cover $\{h^{-1}(U_i)\}$ and the gluing property implies that $H^a(Y_{sm^{(1)}}, Y_{sm^{(2)}}) \cong \bigoplus _{i=1}^N H^a(Y_i, Y_{i, sm^{(1)}})$ where $H^a(Y_i, Y_{i,sm^{(1)}})) \cong H^a(Y_i,Y_{i,sm})$ . In general, we apply the Mayer–Vietoris sequence to the cohomology group $H^a(Y_{sm^{(k)}}, Y_{sm^{(k+1)}})$ with the induced open cover by $R_i$ ’s. The $E_1$ -page of the spectral sequence is given by

$$ \begin{align*} \bigoplus_{|I|=1}H^a(Y_{I,sm^{(k-1)}},Y_{I,sm^{(k)}}) \xrightarrow{d_1} \dots \xrightarrow{d_1} \bigoplus_{|I|=k-1}H^a(Y_{I,sm^{(1)}},Y_{I,sm^{(2)}}) \xrightarrow{d_1} \bigoplus_{|I|=k}H^a(Y_{I}, Y_{I,sm^{(1)}}) \rightarrow 0. \end{align*} $$

By induction, each direct summand is the direct sum of $H^a(Y_J, Y_{J,sm})$ for some J with $|J|=k$ . Then the differential $d_1$ becomes the alternating sum of the identity morphisms, where the signs are determined by the Mayer–Vietoris sign rule (2.1). Then it follows from a simple combinatorial fact that this sequence is exact except at the first term, and $H^a(Y_{sm^{(k)}}, Y_{sm^{(k+1)}})=\ker (d_1) \cong \bigoplus _{|I|=k}H^a(Y_I, Y_{I,sm})$ .

For any $I \subset J$ with $|J|=|I|+1$ , we write $\rho ^{J}_I$ for the composition of morphisms

$$ \begin{align*} \rho^{J}_I:H^{\bullet}(Y_J, Y_{J, sm}) \hookrightarrow H^{\bullet}(Y_{I,sm}, Y_{I,sm^{(2)}}) \to H^{\bullet+1}(Y_I, Y_{I, sm}), \end{align*} $$

where the first one is given by Lemma 3.5 and the second one is the connecting homomorphism of the long exact sequence of cohomology groups of the triple $(Y_I, Y_{I,sm}, Y_{I, sm^{(2)}})$ . Since we choose an open cover globally, Lemma 3.5 allows one to rewrite the $E_1$ -page of the spectral sequence (3.1) as follows:

$$ \begin{align*} H^{a-N}(Y_{sm^{(N)}}) \xrightarrow{d_1} \bigoplus_{|I|=N-1}H^{a-N+1}(Y_I, Y_{I,sm}) \xrightarrow{d_1} \cdots \xrightarrow{d_1} \bigoplus_{|I|=1}H^{a-1}(Y_I, Y_{I,sm}) \xrightarrow{d_1} H^a(Y, Y_{sm}), \end{align*} $$

where the differential $d_1$ is the signed sum of the induced morphisms $\rho ^{J}_I$ ’s that follows the Mayer–Vietoris sign rule (2.1).

For later use, we introduce the Poincaré dual of $\rho ^J_I$ . For a given hybrid LG model $(Y, h:Y \to {\mathbb {C}}^N)$ of rank N, we will show that there is a canonical isomorphism

(3.2) $$ \begin{align} PD:H^a(Y, Y_{sm},{\mathbb{C}}) \xrightarrow{\cong} H^{2n-a}(Y, Y_{sm}, {\mathbb{C}})^* \end{align} $$

for all $a \geq 0$ (Theorem 7.4). We define the morphism $(\rho ^J_I)^\vee :H^{\bullet }(Y_I, Y_{I,sm}) \to H^{\bullet -1}(Y_J, Y_{J,sm})$ to be the composition $(\rho ^J_I)^\vee =PD_J \circ (\rho ^J_I)^* \circ PD_I^{-1}$ , where $PD_I$ (resp. $PD_J$ ) is the same one in (3.2) for the induced hybrid LG model $(Y_I, h_{Y_I})$ (resp. $(Y_J, h_{Y_J})$ ).

3.2 Extended Fano/LG correspondence

Let X be a smooth (quasi-)Fano manifold and D be an effective simple normal crossing anticanonical divisor with N components $D_1, D_2, \dots , D_N$ . For any index set $I=\{i_1, i_2, \cdots , i_m\} \subset \{1, 2, \cdots , N\}$ , we define

$$ \begin{align*} D_I:=D_{i_1} \cap \cdots \cap D_{i_m}, \qquad D(I):=\Sigma_{j \notin I}D_I \cap D_j. \end{align*} $$

For example, if $I=\{1\}$ , then $D_{\{1\}}=D_1$ and $D(\{1\})=(D_2 \cup \cdots \cup D_k) \cap D_1$ . We also assume that all pairs $(D_I, D(I))$ are (quasi-)Fano. We also write the normal crossing union of l-th intersections by $D\{l\}=\sum _{|I|=l}D_I$ for all $l \geq 0$ .

To elaborate the precise sense of the mirror relations, we introduce some notations. Let $\square $ be a cubical category whose objects are finite subsets of $\mathbb N$ and morphisms $Hom(I,J)$ consists of a single element if $I \subset J$ and else is empty. Given a category $\mathsf {C}$ , we define a cubical object to be a contravariant functor $F:\square \to \mathsf {C}$ , which is also called a cubical diagram of categories. For a cubical object F and $I \subset \mathbb N$ , we write

$$ \begin{align*} \begin{aligned} & F_I:=F(I) \\ & d_{IJ}:=F(I \to J): X_J \to X_I, \quad I \subset J. \end{aligned} \end{align*} $$

We also define a morphism of cubical objects in an obvious way and mainly consider the category of finite-dimensional vector spaces over ${\mathbb {C}}$ , denoted by $\mathsf {Vect_{\mathbb {C}}}$ .

First, on the B-side, consider the natural inclusions $\iota ^J_I:D_J \hookrightarrow D_I$ for $I \subset J$ . We consider a cubical object $\mathfrak {HH}_a(X,D)$ in $\mathsf {Vect}_{\mathbb {C}}$ defined as

$$ \begin{align*} \begin{aligned} &\mathfrak{HH}_a(X,D)_I=\bigoplus_{p-q=a}H^{p,q}(D_I) \\ &\mathfrak{HH}_a(X,D)_{IJ}=\iota^J_{I!}:\bigoplus_{p-q=a}H^{p,q}(D_J) \to \bigoplus_{p-q=a}H^{p,q}(D_I), \end{aligned} \end{align*} $$

where $\iota _!$ ’s are the Gysin morphisms. We also take the Poincaré dual of $\mathfrak {HH}_a(X,D)$ , denoted by $\mathfrak {HH}^c_a(X,D)$ where

$$ \begin{align*} \begin{aligned} &\mathfrak{HH}^c_a(X,D)_I=\bigoplus_{p-q=a}H^{p,q}(D_I) \\ &\mathfrak{HH}^c_a(X,D)_{IJ}=(\iota^J_{I})^*:\bigoplus_{p-q=a}H^{p,q}(D_I) \to \bigoplus_{p-q=a}H^{p,q}(D_J). \end{aligned} \end{align*} $$

On the A-side, let $(Y, \omega , h:Y \to {\mathbb {C}}^N)$ be a hybrid LG model of rank N and $n=\dim _{\mathbb {C}} Y$ . We consider a cubical object $\mathfrak {HH}_a(Y,h) \in \mathsf {Vect}_{\mathbb {C}}$

$$ \begin{align*} \begin{aligned} &\mathfrak{HH}_a(Y,h)_I=H^{n+a-|I|}(Y_I, Y_{I,sm}) \\ &\mathfrak{HH}_a(Y,h)_{IJ}=\rho^J_{I}:H^{n+a-|J|}(Y_J,Y_{J,sm}) \to H^{n+a-|I|}(Y_I,Y_{I,sm}) \end{aligned} \end{align*} $$

for $n \leq a \leq n$ . We also take the Poincaré dual $\mathfrak {HH}^c_a(Y,h)$ , where

$$ \begin{align*} \begin{aligned} &\mathfrak{HH}^c_a(Y,h)_I=H^{n+a-|I|}(Y_I, Y_{I,sm}) \\ &\mathfrak{HH}^c_a(Y,h)_{IJ}=(\rho^J_{I})^\vee:H^{n+a-|I|}(Y_I,Y_{I,sm}) \to H^{n+a-|J|}(Y_I,Y_{J,sm}). \end{aligned} \end{align*} $$

Conjecture 3.7. Let $(X,D)|(Y,\omega , h:Y \to {\mathbb {C}}^N)$ be a (quasi-)Fano mirror pair. For $-n \leq a \leq n$ , there exists isomorphisms of the cubical objects in $\mathsf {Vect}_{\mathbb {C}}$ :

$$ \begin{align*} \mathfrak{HH}_a(X,D) \cong \mathfrak{HH}_a(Y,h), \qquad \mathfrak{HH}^c_a(X,D) \cong \mathfrak{HH}^c_a(Y,h). \end{align*} $$

3.3 Line bundles/Monodromy correspondence

Let $(X,D)$ be a (quasi-)Fano pair, where D is smooth and $(Y,\omega , \mathsf w:Y \to {\mathbb {C}})$ be its mirror LG model. In this case, there is a mirror correspondence between the anticanonical line bundle $-K_X$ and the monodromy T of a generic fiber $\mathsf w^{-1}(t)$ around infinity. Such correspondence can be made precise on the categorical level via the homological mirror symmetry conjecture. On the B-side, tensoring with $-K_X$ provides autoequivalences on the derived category of coherent sheaves on X, $D^b\mathrm {Coh}(X)$ , as well as on $D^b\mathrm {Coh}(D)$ by restriction. On the A-side, the monodromy operator T induces autoequivalences on the relevant Fukaya categories associated with $Y_{sm}$ and $\mathsf w: Y \to {\mathbb {C}}$ .

On the other hand, when D has more than one component, one can ask a more refined version of the above correspondence. On the B-side, we have N line bundles $\mathcal O_X(D_i)$ for $i=1, \dots , N$ , whose sum is the anticanonical line bundle $-K_X$ . Each line bundle induces an autoequivalence on $D^b\mathrm {Coh}(X)$ by taking the tensor product with itself. On the A-side, there are N monodromy operators, each of which is induced by taking a loop $T_i$ near infinity on the base of $h:Y \to {\mathbb {C}}^N$ ,

(3.3) $$ \begin{align} T_i:=(t_1, \dots, t_{i-1}, e^{\sqrt{-1}\theta}t_i, t_{i+1}, \dots, t_N) \quad (0 \leq \theta \leq 2\pi) \end{align} $$

for a generic $(t_1, \dots , t_N) \in {\mathbb {C}}^N$ and $i=1, \dots , N$ . We denote such operators by $\phi _{T_i}$ . Note that the monodromy operator $\phi _{T_i}$ induces not only the automorphism of a generic fiber $Y_i=h_i^{-1}(t)$ but also the automorphism of the induced fibration $h|_{Y_j}:Y_j \to {\mathbb {C}}^{N-1}$ for any $i,j$ . This will play a key role in Section 4. Moreover, note that the composition of $T_i$ ’s is the loop T near infinity on the base of $\mathsf w:Y \to {\mathbb {C}}$ . Each monodromy operator is expected to induce an autoequivalence, denoted by $\phi _{T_i}$ as well, on the relevant Fukaya category of $(Y, \omega , h:Y \to {\mathbb {C}}^N)$ .

The main source of Ansatz 3.9 can be found in [Reference Hanlon15][Reference Hanlon and Hicks16] where the mirror symmetry of smooth toric Fanos has been discussed. See also [Reference Lee24, Section 4.3] for more details.

4.1 Semistable degeneration

Let $\mathfrak X$ be a complex connected analytic space and $\Delta $ be the unit disk. A degeneration is a proper flat surjective map $\pi :\mathfrak X \to \Delta $ such that $\mathfrak X-\pi ^{-1}(0)$ is smooth and the fiber $\mathfrak X_t$ is a compact Kähler manifold for every $t \neq 0$ . The fiber at the zero $\mathfrak X_0:=\pi ^{-1}(0)$ is called the degenerate fiber. Given the degeneration $\pi :\mathfrak X \to \Delta $ and for $t \neq 0$ , we say that $\mathfrak X_t$ degenerates to $\mathfrak X_0$ or equivalently $\mathfrak X_0$ is smoothable to $\mathfrak X_t$ . In particular, if the total space $\mathfrak X$ is smooth and the degenerate fiber $\mathfrak X_0$ is a simple normal crossing divisor of $\mathfrak X$ , then the degeneration $\pi :\mathfrak X \to \Delta $ is called semistable. We define a type of the semistable degeneration to be the dimension of the dual complex of the degenerate fiber.

Due to Friedman [Reference Friedman13], the semistability condition on the degeneration $\pi :\mathfrak X \to \Delta $ controls the behavior of the degenerate fiber in a way that the normal bundle of singular locus of $\mathfrak X_0$ in $\mathfrak X$ is trivial. This property is called d-semistabilty.

Definition 4.1 [Reference Friedman13, Definition 1.13].

Let $X=\bigcup _{i=0}^N X_i$ be a normal crossing variety of pure dimension n whose irreducible component is smooth. We define X to be d-semistable if

(4.1) $$ \begin{align} \bigotimes_{i=0}^N I_{X_i}/I_{X_i}I_D \cong \mathcal O_D, \end{align} $$

where D is the singular locus of X and $I_{D}$ (resp. $I_{X_i}$ ) is the ideal sheaf of $I_{D}$ (resp. $I_{X_i})$ .

From now on, we specialize to the case where the degenerate fiber of the semistable degeneration of type $(N+1)$ consists of $N+1$ irreducible components. In this case, we have an equivalent description of d-semistability, which will be used in the mirror construction. Let’s write $X_c=\bigcup _{i=0}^NX_i$ for the degenerate fiber $\mathfrak X_0$ . For any $i,j \in \{0, \dots , N\}$ , we write $X_{ij}$ for the intersection of $X_i$ and $X_j$ as a divisor of $X_i$ . Since the degeneration $\pi :\mathfrak X \to \Delta $ is semistable, we have $X_c|_{X_i}=\mathfrak X_0|_{X_i} \cong \mathfrak X_t|_{X_i}=0$ for $t \neq 0$ . It implies that in $\operatorname {\mathrm {Pic}}(X_{ij})\cong \operatorname {\mathrm {Pic}}(X_{ji})$ , we have the following relation

$$ \begin{align*} \begin{aligned} 0&=\mathcal O(X_0+\cdots+ X_N)|_{X_i}|_{X_{ij}}\\ & =\mathcal O(X_i)|_{X_{ij}}\otimes \mathcal O(\sum_{j \neq i}\mathcal O(X_j))|_{X_{ji}}. \end{aligned} \end{align*} $$

The right-hand side, denoted by $N(X_{ij})$ , is called the normal class of $X_{ij}$ . A collection of all the normal classes is a $\binom {N}{2}$ -tuple

$$ \begin{align*} N_{X_c}:=(N(X_{ij})) \in \bigoplus_{i<j}\operatorname{\mathrm{Pic}}(X_{ij}). \end{align*} $$

The triviality of the collection of normal classes of $X_c$ implies (4.1). In our case, this is indeed equivalent.

In general, the d-semistability condition is not sufficient to imply the smoothability with a smooth total space. In case that $X_c$ is Calabi–Yau, which is of our main interest, this direction has been studied by Kawamata–Namikawa [Reference Kawamata and Namikawa22].

The following definition is motivated by the type III case [Reference Lee23, Definition 2.1].

Example 4.5. Let $Q_5 \subset \mathbb P^4$ be a smooth quintic 3-fold. In the anticanonical linear system, it degenerates to a normal crossing union of two smooth hyperplanes $H_1$ and $H_2$ and a smooth cubic 3-fold $Q_3$ . For simplicity, we denote it by $Z_c:=Z_1 \cup Z_2 \cup Z_3$ , where $Z_1=H_1$ , $Z_2=H_2$ and $Z_3=Q_3$ . Note that the total space of such degeneration is singular so that one needs to modify $X_c$ to obtain a semistable degeneration. First, consider the intersection between a generic quintic 3-fold and $Z_c$ . It becomes a union of three curves $C_1, C_2 \text { and } C_3$ , where $C_i$ lies in $Z_{jk}$ and $C_i \cap Z_{123}$ are all the same for $\{i,j,k\}=\{1,2,3\}$ . For $\{i,j\}=\{1,2\}$ , we take a blow up of $Z_i$ along $C_3$ , denoted by $\pi _i:\operatorname {\mathrm {Bl}}_{C_j}Z_i \to Z_i$ . Let $E_i$ be an exceptional divisor and write $(-)'$ for the proper transformation of the subvariety $(-)$ . While the proper transform $Z_{i3}'$ is isomorphic to $Z_{i3}$ , $Z_{ij}'$ is the blow up of $Z_{ij}$ along $C_3 \cap Z_{ij}$ . By construction $C_3'$ is disjoint from $Z_{123}'$ . The last step is to blow up $\operatorname {\mathrm {Bl}}_{C_3}Z_1$ along $C_3'$ . If we write the resulting normal crossing variety as $X_c=X_1 \cup X_2 \cup X_3$ , we have

(4.2) $$ \begin{align} \begin{aligned} & (X_1, X_{12} \cup X_{13}) \cong (\operatorname{\mathrm{Bl}}_{C_3'}\operatorname{\mathrm{Bl}}_{C_2}Z_1, Z_{12}' \cup Z_{13}), \\ & (X_2, X_{21} \cup X_{23}) \cong (\operatorname{\mathrm{Bl}}_{C_1}Z_2, Z_{12}' \cup Z_{13}), \\ & (X_3, X_{31} \cup X_{32}) \cong (Z_3, Z_{12} \cup Z_{13}). \\ \end{aligned} \end{align} $$

In [Reference Doran and Thompson12], the authors present a mirror construction for this example by considering this degeneration as an iterative Tyurin degeneration.

4.2 Mirror construction

Let $X_c=\bigcup _{i=0}^NX_i$ be a d-semistable Calabi–Yau n-fold of type $(N+1)$ and X be a smoothing of $X_c$ . For each i, the irreducible component $X_i$ is quasi-Fano with the canonically chosen anticanonical divisor $\bigcup _{j \neq i}X_{ij}$ . Suppose that the mirror hybrid LG model of the pair $(X_i, \cup _{j \neq i} X_{ij})$ is given by $(Y_i, \omega _i, h_i=(h_{i0},\dots , h_{i\hat {i}}, \dots ,h_{iN}):Y_i \to \Delta ^N)$ . Here, we shrink the base of the hybrid LG potential to a sufficiently large polydisks $\Delta ^N$ . We propose a topological construction of a mirror Calabi–Yau manifold of X as a gluing of the hybrid LG models $(Y_i, \omega _i, h_i)$ .

To perform the gluing, we require more topological conditions on the hybrid LG models. In $X_c$ , two divisors $X_{ij} \subset X_i$ and $X_{ji} \subset X_j$ are topologically identified for $i \neq j$ . This should be reflected on the mirror side by requiring that two induced hybrid LG models $(Y_{ij}:=h_{ij}^{-1}(t_j), h_i|_{Y_{ij}}:Y_{ij} \to \Delta ^{N-1})$ and $(Y_{ji}:=h_{ji}^{-1}(t_i), h_j|_{Y_{ji}}:Y_{ji} \to \Delta ^{N-1})$ are topologically the same for $t_i, t_j \in \partial \Delta $ . Furthermore, one can enhance the topological identification by taking into account complex structures and symplectic structures. For instance, once the preferred choice of the topological identification $X_{ij}=X_{ji}$ has been made, the complex isomorphism between $X_{ij}$ and $X_{ji}$ is given by an element $f_{ij} \in \operatorname {\mathrm {Aut}}(X_{ij})$ which is homotopic to the identity. Also, these identifications should be compatible to endow $X_c$ with a well-defined complex structure: For $i\neq j \neq k$ , the composition of the restrictions $ f_{ki}|_{X_{kij}}\circ f_{jk}|_{X_{jki}} \circ f_{ij}|_{X_{ijk}} \in \operatorname {\mathrm {Aut}}(X_{ijk})$ is homotopic to the identity. A mirror counterpart should be the identification given by an element of symplectomorphisms $g_{ij} \in \operatorname {\mathrm {Symp}}(Y_{ij}, \omega _i|_{Y_{ij}}, h_i|_{Y_{ij}})$ which preserves the hybrid LG potentials. For $i\neq j \neq k$ , the composition of the restrictions $g_{ki}|_{(Y_{kij}, h_k)}\circ g_{jk}|_{(Y_{jki}, h_j)} \circ g_{ij}|_{(Y_{ijk}, h_i)} \in \operatorname {\mathrm {Symp}}(Y_{ijk}, \omega _i|_{Y_{ijk}}, h_i|_{Y_{ijk}})$ is required to be homotopic to the identity.

Since there is already a global complex structure on $X_c$ , without loss of generality, we may assume that all such gluing automorphisms are indeed the identity. This follows from perturbing the complex and symplectic structures in the beginning.

In fact, the identification on the bases $\Delta ^N$ along the boundary components is modelled on the normal crossing union. For example, we can consider the base of the i-th hybrid LG model, denoted by $\Delta ^N_{h_i}$ , sits in ${\mathbb {C}}^{N+1}$ as

$$ \begin{align*} \Delta^N_{h_i}\cong\{|z_j|\leq 1, z_i=t_i|j \neq i\} \end{align*} $$

for some $t_i$ with $|t_i|=1$ . Thus, we get a normal crossing union of $(Y_i, \omega _i)$ equipped with the induced map to a normal crossing union of the base $\Delta ^N_{h_i}$ of each potential $h_i$ . Moreover, topologically, we can further glue these bases $\Delta ^N_{h_i}$ along the boundary components until the monodromies associated to $h_i$ come into this procedure. Then the resulting base space becomes topologically the same with ${\mathbb {C}}^N$ , hence we obtain a topological fibration $\tilde {\pi }:\tilde {Y} \to {\mathbb {C}}^N$ . We will give more precise description of $\tilde {\pi }:\tilde {Y} \to {\mathbb {C}}^N$ after Proposition 4.7.

From now on, we assume that the collection of the hybrid LG models $(Y_i, \omega _i, h_i:Y_i \to \Delta ^N)$ satisfies Hypothesis 4.6. Then we can interpret the vanishing of the normal classes of $X_c$ as the relation of the monodromies associated to the hybrid LG models based on Ansatz 3.9. Recall that d-semistability is equivalent to the triviality of normal class in $\operatorname {\mathrm {Pic}}(X_{ij})$

(4.3) $$ \begin{align} \begin{aligned} 0&=\mathcal O(X_0+\cdots+ X_N)|_{X_i}|_{X_{ij}}\\ & =\mathcal O(X_i)|_{X_{ij}}\otimes \mathcal O(\sum_{j \neq i}\mathcal O(X_j))|_{X_{ji}} \end{aligned}\end{align} $$

for any $i \neq j$ . For each hybrid LG model $(Y_i, \omega _i, h_i:Y \to \Delta ^N)$ , we write monodromies induced by the loop along the j-th coordinate and the diagonal by $\phi _{T_{ij}} \text { and } \phi _{T_i}$ , respectively. Then the mirror counterpart of the relation (4.3) corresponds to

(4.4) $$ \begin{align} \phi_{T_{ij}} \circ \phi_{T_{j}} = \mathrm{Id} \in \operatorname{\mathrm{Symp}}(Y_{ij}, \omega_{i}|_{Y_{ij}}, h_{i}|_{Y_{ij}}). \end{align} $$

In other words, we have the following correspondence of monodromies

Proof. Let $[z_0:\cdots :z_N]$ be homogeneous coordinates on $\mathbb P^N$ . Consider the closed subsets $\Delta _i \subset \mathbb P^N$

$$ \begin{align*} \Delta_i=\left\{ |z_j| \leq |z_i| \text{ for } j \neq i\right\} \end{align*} $$

for $i=0, \dots , N$ . First, note that $\cup _{i=0}^N \Delta _i=\mathbb P^N$ . Since $\Delta _i \subset U_i=\{z_i \neq 0\}$ , we have $\Delta _i \cong \Delta ^N$ and any k-th intersection of $\Delta _i$ ’s is homeomorphic to $(S^1)^k \times \Delta ^{N-k}$ . Due to Hypothesis 4.6, we can identify the base of $h_i:Y \to \Delta ^N$ with $\Delta _i$ for all i’s. If we take the coordinates $(t_{i0}, \cdots , t_{i\hat {i}}, \cdots , t_{iN})$ of $\Delta _i$ , where $t_{ik}=\frac {z_k}{z_i}$ , then $\Delta _i$ becomes the closed unit disk. Then the chart map between $\Delta _i$ and $\Delta _j$ is exactly the same as the relation (4.4) because this is given by multiplying $t_{ij}^{-1}$ .

We keep the notation used in the proof of Proposition 4.7. Consider the moment map $\mu :\mathbb P^{N}\to {\mathbb {R}}^{N}$ which is given by

$$ \begin{align*} \begin{aligned} \mu:\mathbb P^{N}&\to {\mathbb{R}}^{N} \\ [z_0:\cdots:z_N] &\mapsto \left(\frac{|z_0|}{\sum_{i=0}^N|z_i|}, \cdots, \frac{|z_{N-1}|}{\sum_{i=0}^N|z_i|}\right). \end{aligned} \end{align*} $$

Note that the image $\operatorname {\mathrm {Im}}(\mu ) \in {\mathbb {R}}^N$ is the standard N-simplex $\Delta $ . Also, a fiber over a k-dimensional face $\sigma $ is $(S^{1})^{k}$ . We consider the dual spine $\Pi ^N$ in $\Delta $ , defined as the subcomplex of the first barycentric subdivision of $\Delta $ spanned by the 0-skeleton of the first barycentric subdivision minus the 0-skeleton of $\Delta $ . Decomposing $\Delta $ along $\Pi ^N$ , we have $N+1$ cubes $\square _0, \dots , \square _N$ , each of which is pulled a product of disks $D^N$ in $\mathbb P^N$ . These are exactly the polydisks $\Delta _0, \dots , \Delta _N$ we have introduced. We illustrate the case of $\mathbb {P}^2$ in Figure 1: the dual spine $\Pi ^2$ is the union of dotted segments that decomposes $\Delta $ into three cubes $\square _0, \square _1$ and $\square _2$ . Also, observe that each k-th intersection of these cubes pulls backs to $(S^1)^{N-k} \times D^k$ , which is the same as the k-th intersection of $\Delta _i$ ’s.

Figure 1

Description of the dual spine $\Pi ^2$ .

Now, we can see how $\tilde {\pi }:\tilde {Y} \to {\mathbb {C}}^N$ sits in the fibration $\pi :Y \to \mathbb P^N$ more rigorously. Take an open cover $\{V^{\prime }_i|i=0, \dots , N\}$ of the N-simplex $\Delta $ such that for each subset $I \subset \{0, \dots , N\}$ , $V^{\prime }_I$ only contains $\square _I$ among $\square _J$ ’s for $|J|=|I|$ . Here, we follow our convention to denote the intersections. Let $V_i$ denote the preimage of $V^{\prime }_i$ under $\mu $ . Suppose we remove the image of a generic $\mathbb P^{N-1} \subset \mathbb P^N$ near the dual spine $\Pi ^N$ . The overlap $V_{ij}$ is now diffeomorphic to $\Delta ^{N-1} \times (S^1 \times [0,1] -\{pt\})$ hence not contracts to $\Delta ^{N-1} \times S^1$ . However, if one instead removes a small closed neighborhood $N_\epsilon (\mathbb P^{N-1})$ of $\mathbb P^{N-1}$ in $\mathbb P^N$ and shrink $V_i$ ’s if necessary, then $V_{ij}$ becomes diffeomorphic to $\Delta ^{N-1}$ (See Figure 2 for $N=2$ ). Since $V_i$ contracts to the base of the hybrid LG model $(Y_i, \omega _i, h_i)$ , the induced symplectic fibration $\pi :{\pi ^{-1}(\mathbb P^N \setminus N_\epsilon (\mathbb P^{N-1}))} \to \mathbb P^N \setminus N_\epsilon (\mathbb P^{N-1})) \cong {\mathbb {C}}^N$ can be seen as $\tilde {\pi }:\tilde {Y} \to {\mathbb {C}}^N$ . In other words, Proposition 4.7 is equivalent to saying that $\tilde {\pi }:\tilde {Y} \to {\mathbb {C}}^N$ is compactifiable (to $\pi :Y \to \mathbb P^N$ ) if the condition (4.4) holds.

Figure 2

Description of the intersection of $V_1$ and $V_2$ in $\mathbb P^2$ .

Proof of Theorem 4.9.

Both items $(1)$ and $(2)$ are proven by the Mayer–Vietoris argument. We prove the item (2) first. By the Mayer–Vietoris sequence, we have

$$ \begin{align*} \begin{aligned} e(X_c)&=\sum_{|I|=1}^{N+1}(-1)^{|I|-1}e(X_I) \\ &=\sum_{|I|=1}^{N+1}(-1)^{|I|-1}(-1)^{n-|I|+1}e(Y_I, Y_{I,sm}) \\ &=(-1)^{n}\sum_{|I|=1}^{N+1}e(Y_I, Y_{I,sm}). \end{aligned} \end{align*} $$

Now, it is enough to show that $e(\tilde {Y})=\sum _{|I|=1}^{N+1}e(Y_I, Y_{I,sm})$ . Take an open cover $\{U_i|i=0, \dots , N\}$ of the base of $\tilde {\pi }:\tilde {Y} \to {\mathbb {C}}^N$ such that $\tilde {\pi }^{-1}(V_I)$ contracts to $Y_{I}$ for all I. By applying the Mayer–Vietoris argument, we have

$$ \begin{align*} e(\tilde{Y})=\sum_{|I|=1}^{N+1} (-1)^{|I|-1}e(Y_I). \end{align*} $$

Since $e(Y_I)=e(Y_I, Y_{I,sm}) + e(Y_{I,sm})$ , Lemma 4.10 implies that

$$ \begin{align*} \begin{aligned} e(\tilde{Y})&=\sum_{|I|=1}^{N+1}(-1)^{|I|-1}(e(Y_I, Y_{I,sm})+e(Y_{I,sm}))\\ &=\sum_{|I|=1}^{N+1}(-1)^{|I|-1}\left(e(Y_I, Y_{I,sm})+\sum_{|J|=1, J \cap I = \emptyset}^{N+1-|I|}(-1)^{|J|-1}e(Y_{I,J})\right). \end{aligned} \end{align*} $$

Here, $Y_{I,J}$ is the same with $Y_{I \cup J}$ by Hypothesis 4.6 (1), but we use different notation to emphasize the Mayer–Vietoris procedure. By rewriting $e(Y_{I,J})=e(Y_{I,J},Y_{I,J,sm})+e(Y_{I,J,sm})$ , we can iteratively apply Lemma 4.10. Then we get $e(\tilde {Y})=\sum _{|I|=1}^{N+1}e(Y_I, Y_{I,sm})$ by taking the resummation.

We apply similar argument to prove an item $(1)$ . It follows from the same method in [Reference Lee23, Proposition 3.2] that the Euler characteristic of the smoothing manifold X is given by

$$ \begin{align*} e(X)=\sum_{|I|=1}^{N+1}(-1)^{|I|}|I|e(X_I). \end{align*} $$

By assumption, we have $e(X)=(-1)^{n}\sum _{|I|=1}^{N+1}|I|e(Y_I, Y_{I,sm})$ . Take an open cover $\{U_i|i=0, \dots , N\}$ of the base of $h:Y \to \mathbb P^N$ as before. Note that for any I, the intersection $U_I$ contracts to $(S^1)^{|I|-1} \times \Delta ^{N+1-|I|}$ . Applying the Mayer–Vietoris argument with respect to the induced open cover $\{h^{-1}(U_i)\}$ , one can see that the Euler characteristic $e(h^{-1}(U_I))$ vanishes for $|I|>1$ because $h^{-1}(U_I)$ contracts to a fiber bundle over $(S^1)^{|I|-1}$ with a fiber $Y_I$ . Therefore,

$$ \begin{align*} e(Y)=\sum_{|I|=1}e(Y_I)=\sum_{|I|=1}e(Y_I, Y_{I,sm})+e(Y_{I,sm}). \end{align*} $$

By iteratively applying Lemma 4.10, we get the conclusion.

6.1 Backgrounds on toric varieties

We recollect some backgrounds about toric varieties. We refer for more details to [Reference Cox, Little and Schenck8]. Let N and M be dual lattices of rank n with the natural bilinear pairing $\langle -,- \rangle :N \times M \to {\mathbb {Z}}$ . We write $N_{\mathbb {R}}:=N \otimes _{\mathbb {Z}} {\mathbb {R}}$ and $M_{\mathbb {R}}=M \otimes _{\mathbb {Z}} {\mathbb {R}}$ . A rational convex polyhedra cone (simply called cone) $\alpha $ in $N_{\mathbb {R}}$ is a convex cone generated by finitely many vectors in N. Associated to a cone $\alpha $ , one can construct an affine toric variety $X_\alpha :=\operatorname {\mathrm {Spec}}({\mathbb {C}}[\alpha ^\vee \cap M])$ where $\alpha ^\vee \subset M_{\mathbb {R}}$ is a dual cone of $\alpha $ defined by

$$ \begin{align*} \alpha^\vee:=\{v \in M_{\mathbb{R}}| \langle v, u\rangle \geq 0 \text{ for all } u \in \alpha\}. \end{align*} $$

Such affine toric varieties can be glued to produce more general toric varieties. This gluing data is combinatorially encoded in a fan $\Sigma \subset N_{\mathbb {R}}$ which is a collection of cones such that

1. 1. each face of a cone in $\Sigma $ is also a cone in $\Sigma $ ,

2. 2. the intersection of two cones in $\Sigma $ is a face of each cone.

Given a fan $\Sigma $ , we define a toric variety $X=X_\Sigma $ by gluing the affine toric varieties $X_\alpha :=\operatorname {\mathrm {Spec}}({\mathbb {C}}[\alpha ^\vee \cap M])$ : Two affine toric varieties $X_\alpha $ and $X_\beta $ are glued over $X_{\alpha \beta }:=\operatorname {\mathrm {Spec}}({\mathbb {C}}[(\alpha \cap \beta )^\vee \cap M])$ . We call $\{X_\alpha \}$ a toric chart of $X_\Sigma $ . If $|\Sigma |=N_{\mathbb {R}}$ , it is called complete, and the corresponding toric variety $X_\Sigma $ is compact.

Let $\Sigma [1]$ be a collection of integral primitive ray generators of $\Sigma $ . Consider the lattice morphism $g:N \to {\mathbb {Z}}^{\Sigma [1]}$ given by $g(v)=(\langle v, \rho \rangle )_{\rho \in \Sigma [1]}$ . This induces a short exact sequence

$$ \begin{align*} 0 \to N \xrightarrow{g} {\mathbb{Z}}^{\Sigma[1]} \to A_{n-1}(X_\Sigma) \to 0, \end{align*} $$

where ${\mathbb {Z}}^{\Sigma [1]}$ is the set of torus invariant Weil divisors and $A_{n-1}(X_\Sigma )$ is the Chow group of $X_\Sigma $ . Applying the functor $\operatorname {\mathrm {Hom}}(-, {\mathbb {C}}^*)$ to the above sequence, we get a short exact sequence

(6.1) $$ \begin{align} 1 \to G \to ({\mathbb{C}}^*)^{\Sigma[1]} \to M \otimes_{\mathbb{Z}} {\mathbb{C}}^* \to 1. \end{align} $$

Let $\{x_\rho \}_{\rho \in \Sigma [1]}$ be a standard basis of rational functions on ${\mathbb {C}}^{\Sigma [1]}$ and V be the vanishing locus of $\{\prod _{\rho \notin \sigma }x_\rho |\sigma \subset \Sigma \}$ . The sequence (6.1) shows that G acts naturally on ${\mathbb {C}}[(x_\rho )_{\rho \in \Sigma [1]}]$ and leaves V invariant. Then the toric variety $X_\Sigma $ is the quotient $({\mathbb {C}}[(x_\rho )_{\rho \in \Sigma [1]}] \setminus V)//G$ and the homogeneous coordinate ring of $X_\Sigma $ is equipped with the grading given by the action of G. The sublocus of $X_\Sigma $ corresponding to $D_\rho =\{x_\rho =0\}$ is exactly the torus invariant divisors associated to the ray generator $\rho $ . A torus invariant divisor $D=\sum _{\rho \in \Sigma [1]} a_\rho D_\rho $ is Cartier if and only if there is some piecewise linear function $\rho $ on $M_{\mathbb {R}}$ which linear on the cones of $\Sigma $ , which takes the integral values on M.

A rational convex polytope $\Delta $ in $M_{\mathbb {R}}$ is the convex hull of finite number of points. We say $\Delta $ is a lattice polytope if every vertex of $\Delta $ is in M. For example, a lattice polytope is given by the intersection of some half spaces cut out by affine hyperplanes

$$ \begin{align*} \Delta=\{v \in M_{\mathbb{R}}|\langle v, n_i \rangle \geq -a_i, n_i \in N, a_i \in \mathbb Z \text{ for }i=1,\dots, s\}. \end{align*} $$

A l-face $\sigma $ is the intersection of $\Delta $ with $n-l$ supporting hyperplanes, and we will denote it by $\sigma \prec \Delta $ . We also write $\Delta [l]$ for the collection of l-faces of $\Delta $ . In particular, a $0$ -face, a $1$ -face and a $(n-1)$ -face are called a vertex, an edge and a facet of $\Delta $ , respectively. For each face $\sigma \prec \Delta $ , the cone $\alpha _\sigma $ dual to $\sigma $ is defined by

$$ \begin{align*} \alpha_\sigma=\{u \in N_{\mathbb{R}}|\langle v, u\rangle \leq \langle v', u \rangle \text{ for all } v \in \sigma \text{ and } v' \in \Delta\}. \end{align*} $$

A collection of dual cones $\alpha _\sigma $ forms a fan $\Sigma ^{\Delta }$ , called a normal fan of $\Delta $ , and we write $X_\Delta $ for the associated toric variety.

To a lattice polytope $\Delta $ , the associated toric variety $X_\Delta $ comes with the divisor $D_\Delta =-\sum _{\rho }a_\rho D_\rho $ (or simply denoted by D) where the sum is taken over all facets $\rho \prec \Delta $ . Equivalently, we get a support function of D, a piecewise linear function $\phi _D$ such that $\phi _D(v_\rho )=-a_\rho $ for the privimite vector $v_\rho $ to the dual cone of the face $\rho \prec \Delta $ . Let $\Delta _D=\{u \in M_{\mathbb {R}}|u \geq \phi _D \text { on } N_{\mathbb {R}}\}$ . Geometrically, $\Delta _D \cap M$ generates the space of sections of the line bundle $\mathcal O_X(D)$ . Note that D is trivial, generated by sections and ample if and only if $\phi _D$ is affine, convex and strictly convex, respectively.

A polytope $\Delta \subset M_{\mathbb {R}}$ is called simplicial, if there are exactly n edges at each vertex and the primary vectors at each vertex span $M_{\mathbb {R}}$ as a vector space. A fan $\Sigma $ in $N_{\mathbb {R}}$ is simplicial if all the maximal cones in $\Sigma $ is simplicial. In particular, if the primary vectors span the lattice, then it is called nonsingular.

6.2 Batyrev mirror pairs

We introduce Batyrev mirror pairs [Reference Batyrev5]. Let $\Delta $ be a simplicial lattice polytope in $M_{\mathbb {R}}$ . A polar dual of the polytope $\Delta $ is defined to be $\Delta ^\circ :=\{u \in N_{\mathbb {R}}|\langle u, v \rangle \geq -1 \text { for }v \in \Delta \}$ . A lattice polytope is called reflexive if its polar dual $\Delta ^\circ $ is also a lattice polytope. This is equivalent to the condition that the zero $0_M$ is the only one interior lattice point of $\Delta $ . Geometrically, the associated toric variety $X_\Delta $ is a Gorenstein Fano variety. From now on, we fix a reflexive polytope $\Delta \subset M_{\mathbb {R}}$ and write $\Sigma _\Delta $ for the fan over the facets of $\Delta $ and $\mathbb P_\Delta :=X_{\Sigma _\Delta }$ for the associated toric variety. Note that $\Sigma _\Delta $ is also the normal fan of the polar dual $\Delta ^\circ $ , so we have $\mathbb P_\Delta =X_{\Sigma _\Delta }=X_{\Sigma ^{\Delta ^\circ }}=X_{\Delta ^\circ }$ .

Consider a general Calabi–Yau hypersurface $V_\Delta $ of $\mathbb P_\Delta $ . Since $\mathbb P_\Delta $ is an orbifold in general, the hypersurface $V_\Delta $ may have singularities. We assume that the hypersurface $V_\Delta $ is $\Delta $ -regular, meaning that the singular locus of $V_\Delta $ is induced from the singular locus of ambient space $\mathbb P_\Delta $ . Then one may desingularize $V_\Delta $ by taking a partial resolution of $\mathbb P_\Delta $ . Such resolution is given by a refinement $\widetilde {\Sigma }_\Delta $ of the fan $\Sigma _\Delta $ whose cone is contained in a cone of $\Sigma _\Delta $ . In this case, to $\Sigma _\Delta $ , one can add all rays pointing to the elements in $\partial \Delta \cap M$ to obtain $\widetilde {\Sigma }_\Delta $ . Batyrev shows that the induced resolution $f: X_{\widetilde {\Sigma }_\Delta } \to \mathbb P_\Delta $ is crepant and this is called a maximal projective crepant partial (MPCP) resolution of $\mathbb P_\Delta $ [Reference Batyrev5, Section 2.2]. In particular, if $X:=f^*(V_\Delta )$ is smooth, then we say $\Delta $ satisfies maximal projective crepant smooth (MPCS) resolution condition. Note that this condition always holds for $n \leq 4$ [Reference Batyrev5, Section 2.2]. Similarly, consider the dual construction for $\Delta ^\circ $ and write $X^\vee $ for a MPCP resolution of $V_{\Delta ^\circ }$ .

6.3 The degeneration-fibration correspondence for Batyrev mirror pairs

We start with reviewing a semistable toric degeneration introduced in [Reference Hu20]. Fix an n-dimensional simplicial polytope $\Delta $ . Let $\Gamma $ be a partition of the polytope $\Delta $ into smaller polytopes $\{\Delta _{(i)}\}$ . We say the partition $\Gamma $ is simplicial if all subpolytopes $\{\Delta _{(i)}\}$ are simplicial polytopes. We define $\sigma $ to be l-face of $\Gamma $ , denoted by $\sigma \prec \Gamma $ , if $\sigma $ is a l-face of $\Delta _{(i)}$ for some i.

From now on, we distinguish vertices in $\Gamma $ from those of $\Delta $ : When we say p is a vertex of $\Gamma $ , it means that p is a vertex of $\Delta _{(i)}$ for some i that is not a vertex of $\Delta $ . The restriction of $\Gamma $ to some face $\sigma \prec \Delta $ is defined to be the partition induced by $\{\Delta _{(i)} \cap \sigma \}$ , and denote it by $\Gamma \cap \sigma $ .

By definition, if $\cap _{k=1}^l\Delta _{(i_k)} \neq \emptyset $ , then it has dimension $n-l+1$ . Also, for any vertex p of $\Gamma $ , there are exactly $n+1$ -edges $\sigma _0, \dots , \sigma _n$ of $\Gamma $ such that $p \prec \sigma _i$ . This allows us to define a dual simplicial complex $K_\Gamma $ whose vertex set is the set of polytopes $\Delta _{(j)}$ ’s in the partition $\Gamma $ . For example, $K_\Gamma $ is l-simplex if and only if there is a l-face of $\Delta $ that contains all vertices of $\Gamma $ .

One can take a minimal integral lifting of $F_\Gamma $ , and we denote it by F.

The semistable degeneration $p:X_{\tilde {\Delta }}\to {\mathbb {C}}$ induces a semistable degeneration of a Calabi–Yau hypersurface X in $X_\Delta $ whose degenerate fiber consists of a generic hypersurface $X_i$ of $X_{\Delta _{(i)}}$ in the linear system $|D_{\Delta _{(i)}}|$ . In other words, the normal crossing variety $X_c=\cup _{i=0}^l X_i$ is d-semistable of type $l+1$ and X is its smoothing.

Next, we assume that the nonsingular polytope $\Delta \subset M_{\mathbb {R}}$ is reflexive and the origin $0_M$ is a unique lattice point.

Let $\Gamma $ be a nonsingular, central, semistable partition of the polytope $\Delta $ . If $K_\Gamma $ is l-dimensional, there is a unique codimension l linear subspace $L \subset M_{\mathbb {R}}$ passing through the origin such that $\Delta \cap L \prec \Delta _{(i)}$ for all i as a $(n-l)$ -face. Consider l primitive vectors given by the intersection of each $\Delta _{(ik)}:=\Delta _{(i)} \cap \Delta _{(k)}$ with the orthogonal complement $L^\perp $ of L. Note that the restriction $\Delta \cap L^\perp $ is also reflexive and simplicial while the induced partition is not necessarily semistable. Thus, there are $l+1$ such primitive vectors $v_0, \dots , v_l$ such that for each i, $\Delta _{(i)}$ contains all $v_j$ except $j=i$ .

Next, consider the dual reflexive polytope $\Delta ^\circ \subset N_{\mathbb {R}}$ whose dual fan is $\Sigma _{\Delta } \subset M_{\mathbb {R}}$ . For simplicity, we assume that $\Delta ^\circ $ satisfies the MPCS resolution condition. We explain what the central semistable partition corresponds to in the dual picture. Let $v_0, \dots , v_l$ be primitive vectors introduced above. We define new fans $\Sigma ^{\prime }_\Gamma \subset \Sigma _\Gamma \subset M_{\mathbb {R}}$ where

1. 1. $\Sigma ^{\prime }_\Gamma $ is generated by primitive vectors in all $\Delta _{(ik)}$ ’s and all $v_i$ ’s;

2. 2. $\Sigma _\Gamma $ is a subfan of $\Sigma ^{\prime }_\Gamma $ that is generated by all primitive vectors lying in $\Delta \cap L$ and all $v_i's$ .

Furthermore, we consider the fan $\Sigma ':=\Sigma _\Delta \cup \Sigma ^{\prime }_\Gamma $ . Geometrically, this refinement amounts to taking a (maximal) projective crepant partial resolution of $X_{\Sigma _{\Delta }}$ , denoted by $\phi _\Delta : X_{\Sigma '} \to X_{\Sigma _\Delta }$ . We also have a blow-down map $\phi _\Gamma :X_{\Sigma '} \to X_{\Sigma _\Gamma }$ which is not necessarily crepant. Consider the projection of the lattices $\Pi _v:M \to M_v$ , where $M_v$ is the sublattice of M generated by $v_i$ ’s. This provides a surjective toric morphism $\pi _v:X_{\Sigma _\Gamma } \to X_{\Sigma _v}$ , where $\Sigma _v \subset M_{v,{\mathbb {R}}}$ is the fan generated by all $v_i$ ’s. In fact, $\pi _v$ is a trivial fibration whose fiber is a toric variety associated to a fan generated by all primitive vectors in $\Delta \cap L$ , denoted by $X_L$ . In summary, we have the following diagrams of toric morphisms

where $\pi _\Gamma :=\phi _\Gamma \circ \pi _v$ . Since $\pi _\Gamma :X_{\Sigma '} \to X_{\Sigma _v}$ is the composition of toric morphisms, a generic fiber is the toric variety $X_L$ . In fact, over the open dense torus $({\mathbb {C}}^*)^l \subset X_{\Sigma _v}$ , the morphism $\pi _\Gamma $ is trivial fibration with fiber $X_L$ . We also present the coordinate description. Consider the homogeneous coordinates of $X_{\Sigma '}$ (resp. $X_{\Sigma _v}$ ), $(z_\sigma |\sigma \in \Sigma ')$ (resp. $(z_{\sigma _0}, \dots , z_{\sigma _l})$ ). In terms of these coordinates, $\pi _\Gamma $ is given by

$$ \begin{align*} \pi_\Gamma=\left(\prod_{\Pi_v(\sigma)=\sigma_0} z_\sigma, \dots, \prod_{\Pi_v(\sigma)=\sigma_l}z_\sigma\right). \end{align*} $$

Let’s take a generic anticanonical divisor Y in $X_{\Sigma '}$ and the induced fibration $\pi :=\pi _\Gamma |_Y$ . Recall that we assume that $\Delta ^\circ $ satisfies the MPCS resolution condition so that Y is nonsingular. Also, it is clear that a generic fiber of $\pi $ is Calabi–Yau as this is a nonsingular general member of the anticanonical divisor of  $X_L$ .

Since $\Gamma $ is nonsingular, the base $X_{\Sigma _v}$ is isomorphic to the projective space $\mathbb P^l$ and we write $\{z_i:=z_{v_i}|i=0, \dots , l\}$ for the homogeneous coordinates. For each i, we take a polydisk $\Delta _i=\{|z_j| \leq |z_i| |j \neq i \} \subset \mathbb P^l$ and set $Y_i:=\pi ^{-1}(\Delta _i)$ and $h_i:=\pi |_{Y_i}:Y_i \to \Delta _i$ . It follows from the genericity condition on Y that the restriction of $\pi :Y \to \mathbb P^l$ over each boundary component of $\Delta _i$ is locally trivial with smooth fibers and intersects transversally to each other. This implies that $(Y_i, h_i:Y \to \Delta _i)$ is a hybrid LG model.

Recall that on the degeneration side, the semistable partition $\Gamma $ provides a semistable degeneration of a nonsingular Calabi–Yau hypersurface of $X_\Delta $ . Each irreducible component $X_i$ of the degenerate fiber $X_c=\cup _{i=0}^l X_i$ is a general hypersurface of the toric variety $X_{\Delta _{(i)}}$ determined by all facets in the facets of $\Delta $ .

One of the major difficulties in proving Conjecture 6.10 and verifying the gluing condition (Ansatz 3.9) is the lack of mirror symmetry results for irreducible components of the degenerate fiber as quasi-Fano varieties. For instance, unlike the Fano case, a toric mirror construction (e.g., Givental’s construction [Reference Givental14]) may not be sufficient and needs further modification, especially in the non-nef case. See [Reference Doran, Harder and Thompson11] for the Tyurin degeneration case. In this regard, this conjecture can be viewed as the reverse construction of the one introduced in Section 4. In other words, this could be one way to obtain a mirror hybrid LG model for a quasi-Fano pair. We further explore this direction in subsequent work. We conclude this section by providing two simple pieces of evidence for Conjecture 6.10.

Example 6.12. Consider the reflexive square $\Delta $ with the semistable partition $\Gamma $ given by the vertical line

where dotted arrows are primitive vectors $v_0$ and $v_1$ . This describes a semistable degeneration of a nonsingular Calabi–Yau hypersurface of $X_\Delta =\mathbb P^1\times \mathbb P^1$ , which is elliptic curve, into the union of two rational curves $X_0$ and $X_1$ intersecting over two points $X_{01}$ . Consider the fans $\Sigma _\Delta $ , $\Sigma '$ and $ \Sigma _v$ described below (Figure 3):

Figure 3

Fans $\Sigma _\Delta $ , $\Sigma '$ and $ \Sigma _v$ from the left.

Geometrically, $X_{\Sigma '}$ is a blow up of $\mathbb P^1 \times \mathbb P^1$ along the four corners and $X_{\Sigma _v} \cong \mathbb P^1$ . The morphism $\pi _\Gamma :X_{\Sigma '} \to X_{\Sigma _v}$ is the one given by the projection to the first factor. In terms of the homogeneous coordinate $\{z_\sigma |\sigma \in \Sigma '[1]\}$ , this is given by

(6.2) $$ \begin{align} \pi_\Gamma=\left[z_{\sigma(-1,1)}z_{\sigma(-1,0)}z_{\sigma(-1,-1)}: z_{\sigma(1,1)}z_{\sigma(1,0)}z_{\sigma(1,1)}\right]. \end{align} $$

Take a generic section Y of $|-K_{X_{\Sigma '}}|$ that is not singular over the locus $\{|z_0|=|z_1|\} \in \mathbb P^1 \cong X_v$ . Then the induced fibration $\pi :Y \to \mathbb P^1$ becomes a double cover with four ramification points. Note that two of them lie near $0 \in \mathbb P^1$ while the other two points lie near $\infty \in \mathbb P^1$ . For $i=0,1$ , we take $Y_i:=\pi ^{-1}(\Delta _i)$ and $\mathsf w_i:=\pi |_{Y_i}:Y_i \to \Delta _i$ . We show that the pair $(Y_i, \mathsf w_i)$ is mirror to the pair $(X_i, X_{01})$ . To describe mirror of $X_0$ (the parallel argument works for $X_1$ ), we first make the rectangle $\Delta {(0)}$ reflexive by shifting the middle vertical line (a facet of $\Delta _{(0)}$ ) to the right by length $1$ . We still denote it by $\Delta _{(0)}$ . Then we apply Givental’s construction to get a mirror of $X_0$ [Reference Givental14, Reference Harder17]. We consider the polar dual of $\Delta _{(0)}$ , denoted by $\Delta ^{(0)}$ , and regard it belongs to M to match the notation we have used. Note that we have a nef partition of $\Delta ^{(0)}=\Delta ^{(0)}_1 + \Delta ^{(0)}_2$ where $\Delta ^{(0)}_1[0]$ are the red vertices and $\Delta ^{(0)}_2[0]$ is blue one in the following picture (left).

The right picture describes the dual partition $\nabla ^{(0)}_1$ and $\nabla ^{(0)}_2$ . Then the LG model for $X_0$ is given by

$$ \begin{align*} Y_0=\left\{\sum_{\rho \in \Delta^{(1)}_1 \cap M}a_\rho z_\rho =0 \right\} \subset ({\mathbb{C}}^*)^2, \qquad \mathsf w_0=\sum_{\rho \in \Delta^{(1)}_2 \cap M}a_\rho z_\rho, \end{align*} $$

where $a_\rho $ ’s are general complex coefficients and $z_\rho =\prod _{i=1}^nz_i^{\langle e_i, \rho \rangle }$ , where $\{e_1, \dots , e_n\}$ is the basis of N. A fiber is of $\mathsf w_0$ is compactified to

$$ \begin{align*} \begin{aligned} &a_{\rho(0,0)}z_{\sigma(0,1)}z_{\sigma(-1,1)}z_{\sigma(-1,0)}z_{\sigma(-1,-1)}z_{\sigma(0,-1)}\\ &+a_{\rho(1,0)}z_{\sigma(0,1)}z_{\sigma(0,-1)}z_{\sigma(1,0)}\\ &+a_{\rho(0,1)}z^2_{\sigma(0,1)}z^2_{\sigma(-1,1)}z_{\sigma(-1,0)}\\ &+a_{\rho(0,-1)}z^2_{\sigma(0,-1)}z^2_{\sigma(-1,-1)}z_{\sigma(-1,0)}=0 \end{aligned} \end{align*} $$

and

$$ \begin{align*} \lambda z_{\sigma(1,0)}-a_{\rho(-1,0)}z_{\sigma(-1,1)}z_{\sigma(-1,0)}z_{\sigma(-1,-1)}=0, \end{align*} $$

where $z_\sigma $ is the homogeneous coordinate of $X_{\nabla ^{(0)}}$ . For generically chosen $a_\rho $ ’s, we see that near $\lambda =0$ , this is a double cover of ${\mathbb {C}}$ with two ramification points. In fact, locally, this morphism is the same with one described in (6.2). As we have the parallel argument for $\Delta _{(1)}$ , we get the conclusion.