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

(7.1)

$$\Res:H^{0}(M,K_{M}\otimes[Z])\rightarrow H^{0}(Z,K_{Z})$$

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

(7.2)

$$h\Omega=dh\wedge\tilde{\Omega}$$

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

(7.3)

$$\Res(\Omega)\equiv\tilde{\Omega}|_{Z}$$

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

(7.4)

$$\Omega=\frac{g}{h}dz_{1}\wedge\cdots dz_{m}.$$

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

(7.5)

$$\Res(\Omega)=\frac{g}{\frac{\partial h}{\partial z_{1}}}dz_{2}\wedge\cdots\wedge dz_{m}.$$

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

(7.6)

$$dh\wedge\Omega^{\prime}=\Omega.$$

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