In this article, we study the following bi-nonlocal Kirchhoff-Schr $\ddot{\mathrm{o}}$dinger-Poisson system with critical growth:

$$\{\begin{array}{ll}-{\left({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{2}dx\right)}^r\mathrm{\Delta }u+\varphi u=u^5+\lambda {\left({\int }_{\mathrm{\Omega }}F(x,u)dx\right)}^{s}f(x,u),& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ -\mathrm{\Delta }\varphi =u^2,u>0,& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=\varphi =0,& \mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is a smooth bounded domain, $\lambda >0$, $0\le r<1$, $0<s<\frac{1-r}{3(r+1)}$ and $f(x,u)$ satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.

In this paper, we study the multiplicity results of positive solutions for a class of Kirchhoff type problems with singularity and critical exponents. Combining with the Nehari method and variational method, we prove the existence of positive ground state solutions. Furthermore, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of $Q(x)$.