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 $\overline{S}$.