Hermitian-Einstein metrics from noncommutative $U\left(1 \right)$ instantons

We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean ${\mathbb R}^4$ to symmetric two-tensors introduced in "Gravitational instantons from gauge theory," H. S. Yang and M. Salizzoni, Phys. Rev. Lett. (2006) 201602, [hep-th/0512215]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose $U(1)$ instantons on a noncommutative ${\mathbb C}^2$ as the self-dual two-form, from which we derive a family of Hermitian-Einstein metrics. We also discuss the condition when the metric becomes K\"ahler.