We first recall some general facts about Poincaré residues. Given a smooth divisor $Z$ in a complex manifold $M$ of dimension $m$, the Poincaré residue map

can be defined as follows. Given a holomorphic $m$ form $\Omega$ on $M$ with a simple pole along $Z$, locally if we choose a defining function $h$ of $Z$, then $h\Omega$ is a holomorphic $m$ form, and we can write

for some locally defined holomorphic $m-1$ form $\tilde{\Omega}$. The Poincaré residue of $\Omega$ along $Z$ is given by

It is straightforward to check that this does not depend on the choice of $h$ and $\tilde{\Omega}$, and gives rise to a well-defined holomorphic volume form $\Omega_{Z}$ globally on $Z$.

If we choose local holomorphic coordinates $z_{1},\cdots,z_{m}$ on $M$, then we may write

At a point on $Z$ where $\frac{\partial h}{\partial z_{1}}\neq 0$, we have then by definition

From the local expression one can see that if $Z$ is an anti-canonical divisor in $M$, and we pick a holomorphic volume form $\Omega_{M}$ on $M\setminus Z$ with a simple pole along $Z$, and then $\Res(\Omega_{M})$ gives a holomorphic volume form $\Omega_{Z}$ on $Z$.

A special case is when we have a globally defined holomorphic function $h:M\rightarrow\mathbb{C}$, and we are given a holomorphic volume form $\Omega$ on $M$, then for each $w\in\mathbb{C}$, we can apply the above construction to the meromorphic form $(h-w)^{-1}\Omega$. In this way we obtain a nowhere vanishing section $\Omega^{\prime}$ of the relative canonical bundle $K_{M}\otimes(h^{*}K_{\mathbb{C}})^{-1}$, on the set where $h$ is a submersion, and it satisfies the equation

We may also view $\Omega^{\prime}$ as a holomorphic varying family of holomorphic volume forms on the fibers of $h$.

Let $\mathcal{S}$ be a two dimensional $A_{k-1}(k\geq 2)$ singularity, which is a hypersurface in $\mathbb{C}^{3}$ with defining equation

Given two positive integers $a_{1}\geq a_{2}$ with $a_{1}+a_{2}=k$, we can define a partial resolution of $\mathcal{S}$ as follows. Let $\overline{\mathcal{S}}$ be the subvariety in the product space $\mathbb{C}^{3}\times\mathbb{C}\mathbb{P}^{2}$ cut out by the following system of equations

where $[u_{1}:u_{2}:u_{3}]$ denotes homogeneous coordinates on $\mathbb{C}\mathbb{P}^{2}$. Alternatively, $\overline{\mathcal{S}}$ can also be described as the closure in $\mathbb{C}^{3}\times\mathbb{C}\mathbb{P}^{2}$ of the graph of the rational map $\mathcal{S}\rightarrow\mathbb{C}\mathbb{P}^{2};(z_{1},z_{2},z_{3})\mapsto[\frac{z_{1}}{z_{3}^{a_{1}}}:\frac{z_{2}}{z_{3}^{a_{2}}}:1]$. On the affine chart $\{u_{i}\neq 0\}$ we shall denote by $v_{j}=u_{j}/u_{i}(j\neq i)$ the affine coordinates.

From another point of view, we can view $\mathcal{S}$ and $\overline{\mathcal{S}}$ as families of algebraic curves by projecting to the $z_{3}$ variable. For $\mathcal{S}$ this is simply the standard nodal degeneration of conics in $\mathbb{C}^{2}$, modified by a base change. The family corresponding to $\overline{\mathcal{S}}$ is isomorphic to $\mathcal{S}$ over any general fiber $\{z_{3}\neq 0\}$, and the special fiber of $\overline{\mathcal{S}}$ is now given by a chain consisting of three components, two of which are given by the proper transforms of the two lines $\{z_{1}=0\}$ and $\{z_{2}=0\}$ in $\mathbb{C}^{2}$, and the middle component is the conic $\{u_{1}u_{2}+u_{3}^{2}=0\}$ in $\mathbb{C}\mathbb{P}^{2}$. In the special case when $a_{1}=a_{2}=1$, $\overline{\mathcal{S}}$ is smooth and the projection map is precisely the minimal resolution of singularity.

It is well-known that $\mathcal{S}$ has a canonical singularity, meaning that the canonical line bundle $K_{\mathcal{S}}$ is trivial. An explicit holomorphic volume form $\Omega_{\mathcal{S}}$ can be written by applying the Poincaré residue to the standard meromorphic $\frac{1}{z_{1}z_{2}+z_{3}^{k}}dz_{1}\wedge dz_{2}\wedge dz_{3}$ on $\mathbb{C}^{3}$. In the chart $\{z_{1}\neq 0\}$, it is given by

Notice $\mathcal{S}$ is isomorphic to the quotient $\mathbb{C}^{2}/\mathbb{Z}_{k}$, and $\Omega_{\mathcal{S}}$ pulls-back to a multiple of the standard holomorphic volume form on $\mathbb{C}^{2}$.

Viewing $\mathcal{S}$ as fibered over $z_{3}\in\mathbb{C}$, we further get a relative holomorphic volume form

One can see $\Omega^{\prime}$ is smooth away from the singularity $\{z_{1}=z_{2}=z_{3}=0\}$, and on each component of the singular fiber it is a meromorphic 1-form with a simple pole along the singularity.

The partial resolution $\overline{\mathcal{S}}$ is a crepant resolution, i.e. the canonical line bundle $K_{\overline{\mathcal{S}}}$ is also trivial. Indeed the pull-back $\Omega_{\overline{\mathcal{S}}}$ of $\Omega_{\mathcal{S}}$ is nowhere vanishing on $\overline{\mathcal{S}}$, and by applying the Poincaré residue to the function $z_{3}$, we then get a meromorphic 1-form on each component of the special fiber. On the conic $\{u_{1}u_{2}+u_{3}^{2}=0\}$ the meromorphic 1-form is given by $v_{1}^{-1}dv_{1}=-v_{2}^{-1}dv_{2}$. The upshot is that we still get a meromorphic section of the relative canonical bundle, which is smooth away from the two singularities $\{u_{1}=u_{3}=z_{1}=z_{2}=z_{3}=0\}$ and $\{u_{2}=u_{3}=z_{1}=z_{2}=z_{3}=0\}$ of $\overline{S}$.

We now recall the set-up in the introduction. Let $n\geq 2$ be an integer. Let $f_{1},f_{2},f$ be homogeneous polynomials of degree $d_{1}\geq d_{2},d_{1}+d_{2}=n+2$ respectively, and let $\mathcal{X}\subset\mathbb{C}\mathbb{P}^{n+1}\times\Delta$ be a family of Calabi-Yau hypersurfaces in $\mathbb{C}\mathbb{P}^{n+1}$ defined by the equation $F_{t}(x)=0$, where

and $t$ is the complex parameter on the unit disc $\Delta\subset\mathbb{C}$. Let $p:\mathcal{X}\rightarrow\Delta$ be the projection map and we denote $\widehat{X}_{t}=p^{-1}(t)$.

We further assume $f_{1},f_{2},f$ are sufficiently general so that the following hold:

1. (i)

   $X_{0}=Y_{1}\cup Y_{2}$, where $Y_{1}=\{f_{1}=0\}$ and $Y_{2}=\{f_{2}=0\}$ are smooth;

2. (ii)

   $\widehat{X}_{t}$ is smooth for $t\neq 0$.;

3. (iii)

   $D=\{f_{1}=f_{2}=0\}$ is a smooth complete intersection;

4. (iv)

   $H=\{f_{1}=f_{2}=f=0\}$ is a smooth complete intersection in $\mathbb{C}\mathbb{P}^{n+1}$.

The total space $\mathcal{X}$ is singular along $H$ and transverse to $H\times\{0\}$ the singularities are locally modeled on a two dimensional ordinary double point. For our purpose we need to perform certain birational transformations to $\mathcal{X}$ keeping the general fibers unchanged.

We first do a base change $t\mapsto t^{n+2}$, and work on the new family, which we still denote by $\mathcal{X}$. Then $\mathcal{X}$ now has singularities along $D\times\{0\}$, transversal to which generically it is a two dimensional $A_{d-1}$ singularity, which becomes worse along $H\times\{0\}$. This is usually referred to as a compounded Du Val (cDV) singularity .

Now we apply the family version of the above model partial resolution to $\mathcal{X}$. Let $\widehat{\mathcal{X}}$ be the subvariety in the projective bundle $\mathbb{P}(\mathcal{O}(d_{2})\oplus\mathcal{O}(d_{1})\oplus\mathbb{C})$ over $\mathbb{C}\mathbb{P}^{n+1}\times\Delta$ cut out by the equations

where naturally we view $f_{i}\in H^{0}(\mathbb{C}\mathbb{P}^{n+1},\mathcal{O}(d_{i}))$, $f\in H^{0}(\mathbb{C}\mathbb{P}^{n+1},\mathcal{O}(n+2))$, and $[s_{1}:s_{2}:s_{3}]$ denotes a point in the fiber of the projective bundle over the point $(x,t)\in\mathbb{C}\mathbb{P}^{n+1}\times\Delta$.

For our discussion in the rest of this section we shall always take $[x_{0}:x_{1}:\cdots:x_{n+1}]$ to be the homogeneous coordinates of a point $x$ on $\mathbb{C}\mathbb{P}^{n+1}$. On the affine chart $\{x_{i}\neq 0\}$ of $\mathbb{C}\mathbb{P}^{n+1}$ we denote by $u=\{u_{j}=x_{j}/x_{i},j\neq i\}$ the affine coordinates, and we view $x_{i}$ as a local trivialization of $\mathcal{O}(1)$. Then on this chart we can view any holomorphic sections of powers of $\mathcal{O}(1)$ as local holomorphic functions. In particular, for a homogeneous function $R(x)$, we denote by $R(u)$ the corresponding inhomogeneous function. On the affine trivialization of the projective bundle $\{s_{i}\neq 0\}$, we denote by $\{\zeta_{j}=s_{j}/s_{i},j\neq i\}$ the affine coordinates on the fibers.

We define

One can see that the new central fiber $\hat{X}_{0}$ consists of a chain of three smooth components intersecting transversally, given by the proper transforms $\hat{Y}_{1},\hat{Y}_{2}$ of $Y_{1},Y_{2}$ respectively and the submanifold $\mathcal{N}$ in the projective bundle $\mathbb{P}(L_{1}\oplus L_{2}\oplus\mathbb{C})$ over $D$ cut out by the equation $s_{1}s_{2}=s_{3}^{2}f(x)$ (so that $\mathcal{N}$ is a quadric bundle over $D$, and singular fibers are over $H$). Notice $\mathcal{N}$ itself is a smooth manifold.

We then have

It is straightforward to see that the normal bundle of $D_{i}$ in $\mathcal{N}$ is $L_{i}^{-1}$.

Next we consider holomorphic volume forms. Viewing $\mathcal{X}$ as an anti-canonical divisor in $\mathbb{C}\mathbb{P}^{n+1}\times\Delta$, then away from $D\times\{0\}$, $\mathcal{X}$ is smooth and we then obtain a holomorphic volume form $\Gamma$. In the affine chart $\{x_{0}\neq 0\}\times\Delta\subset\mathbb{C}\mathbb{P}^{n+1}\times\Delta$, the meromorphic volume form is given by

So the Poincaré residue on $\mathcal{X}$ is

It is easy to check using the equation and the genericity assumptions that $\Gamma$ is indeed holomorphic on $\mathcal{X}\setminus D\times\{0\}$.

Now applying the above discussion to the global function $t$ on $\mathcal{X}$, then we get a holomorphic family of holomorphic volume forms $\Gamma_{t}$ on each $\widehat{X}_{t}$. Differentiating the equation $F_{t}(u)=f_{1}(u)f_{2}(u)+t^{n+2}f(u)=0$, we get

In the above affine chart, on the set where $\frac{\partial F_{t}}{\partial u_{1}}\neq 0$, we have

This is indeed well-defined on $\widehat{X}_{t}$ for $t\neq 0$ and also on $X_{0}\setminus D$. On each component $Y_{i}$ of $X_{0}$, it has a simple pole along $D$. Notice $\Gamma_{t}$ is also the natural holomorphic volume form on $\widehat{X}_{t}$ when we apply the Poincaré residue to the divisor $\widehat{X}_{t}$ in $\mathbb{C}\mathbb{P}^{n+1}$.

Now we pass to the resolution $\widehat{\mathcal{X}}$. Abusing notation we still denote by $\Gamma$ its pull-back.

Now we can apply the previous Poincaré residue to the function $t$ on $\widehat{\mathcal{X}}$. Since the exceptional set of the resolution lies over $D\times\{0\}$, we still get $\Gamma_{t}$ for $t\neq 0$. On the central fiber $\hat{X}_{0}$, we still get $\Gamma_{0}$ on $\hat{Y}_{1}\setminus D_{1}$ and $\hat{Y}_{2}\setminus D_{2}$. Over $\mathcal{N}\setminus(D_{1}\cup D_{2})$, using (7.35) and (7.40) we get the corresponding Poincaré residue

Notice by applying Poincaré residue twice to the complete intersection $D=\{f_{1}=f_{2}=0\}$, we obtain a holomorphic volume form $\Omega_{D}$ on $D$, which in the above local coordinates can be written as

So we get

This means that up to multiplying by $-\sqrt{-1}$, $\Gamma_{\mathcal{N}}$ agrees with the natural holomorphic volume form $\Omega_{0}$ on $\mathcal{N}_{0}$ defined in Section 4.2, under the identification $k_{-}=d_{2},k_{+}=-d_{1}$.