A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius.

I am working with the complex number, so I will abusively denote $\mathbb{P}_{\mathbb{C}}^{n}$ to $\mathbb{P}^{n}$ here.

I know that the total space $Tot(\mathcal{O}_{\mathbb{P}^{n-1}}(-n))$ of the line bundle over the projective space has such a Kähler metric, thanks to the work of D. Joyce. $Tot(\mathcal{O}_{\mathbb{P}^{n-1}}(-n))$ is actually a crepent resolution of $\mathbb{C}^n/\mu_n$, which has an isolated singularity at $0$. In the paper of D. Joyce “Asymptotically Locally Euclidean metric with holonomy SU(n)”, ALE metric is given explicitly for this case.

I wonder if similar things can be done on the higher dimension case. Let $X$ be the total space of the sheaf of module, $$ \mathcal{O}_{\mathbb{P}^{n-1}}(-d_1)\oplus \cdots \oplus \mathcal{O}_{\mathbb{P}^{n-1}}(-d_r), $$ where $d_1+\cdots+d_r=n$ to make $X$ Calabi-Yau. I wonder it is possible to give a Kähler metric of bounded geometry on $X$. I know that this is a Hermitian vector bundle over $\mathbb{P}^{n-1}$, together with the “radius” function, $$ r(x_1,\ldots,x_n,p_1,\ldots,p_r) = \sum_{j=1}^r \|\mathbf{x}\|^{d_j}|p_j|^2, $$ And it seems possible (although I am not very acquainted with) to write a Kähler metric on $X$ using Calabi ansatz methods to this Hermitian vector bundle, $$ g = u(r)g_{\mathbb{P}^{n-1}}+v(r)g_{F} $$ But I am having a hard time showing that such a metric is of bounded geometry. It is too computationally intensive to compute Riemannian curvature directly.

I wonder if this problem hard to tackle. If so, may I ask how much is known about the bounded-geometry-ness of the Kähler metric on $X$?

Thanks in advance.

$\DeclareMathOperator\Tot{Tot}$A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius.

I am working with the complex number, so I will abusively denote $\mathbb{P}_{\mathbb{C}}^{n}$ to $\mathbb{P}^{n}$ here.

I know that the total space $\Tot(\mathcal{O}_{\mathbb{P}^{n-1}}(-n))$ of the line bundle over the projective space has such a Kähler metric, thanks to the work of D. Joyce. $\Tot(\mathcal{O}_{\mathbb{P}^{n-1}}(-n))$ is actually a crepent resolution of $\mathbb{C}^n/\mu_n$, which has an isolated singularity at $0$. In the paper of D. Joyce “Asymptotically Locally Euclidean metric with holonomy SU(n)”, ALE metric is given explicitly for this case.

I wonder if similar things can be done on the higher dimension case. Let $X$ be the total space of the sheaf of module, $$ \mathcal{O}_{\mathbb{P}^{n-1}}(-d_1)\oplus \cdots \oplus \mathcal{O}_{\mathbb{P}^{n-1}}(-d_r), $$ where $d_1+\cdots+d_r=n$ to make $X$ Calabi-Yau. I wonder it is possible to give a Kähler metric of bounded geometry on $X$. I know that this is a Hermitian vector bundle over $\mathbb{P}^{n-1}$, together with the “radius” function, $$ r(x_1,\ldots,x_n,p_1,\ldots,p_r) = \sum_{j=1}^r \|\mathbf{x}\|^{d_j}|p_j|^2, $$ And it seems possible (although I am not very acquainted with) to write a Kähler metric on $X$ using Calabi ansatz methods to this Hermitian vector bundle, $$ g = u(r)g_{\mathbb{P}^{n-1}}+v(r)g_{F} $$ But I am having a hard time showing that such a metric is of bounded geometry. It is too computationally intensive to compute Riemannian curvature directly.

I wonder if this problem hard to tackle. If so, may I ask how much is known about the bounded-geometry-ness of the Kähler metric on $X$?

Thanks in advance.