Singularization Problems for Projective Clusters in High-Dimensional Algebraic Geometry and Their Applications to Complex Domains

The singularization solution of high-dimensional algebraic geometric projective clusters is based on minimal models. In this paper, we obtain a singularization theorem of projective clusters from the Kähler metric, combined with the singularization theorem of projective clusters of Siu-Yau and Mok under the different conditions that the Ricci curvature of a 2-dimensional complete noncompact Kähler projective cluster manifold is positive and bounded when M is an n-dimensional complete noncompact Kähler projective cluster manifold. M is biholomorphic pure with a proposed projective cluster if and only if $k_r(x_0) \ge -\frac{c}{1+r^2}$ is satisfied and the general Sobolev inequalities $\|f\|_t \le C_0 \|\nabla f\|_q^\theta \|f\|_s^{1-\theta}, \forall f \in C_0^\infty(M)$, $\frac{1}{t} = \frac{\theta}{p} + \frac{1-\theta}{s}$, and $\int_M \text{Ric}^n < \infty$ are satisfied. Then we introduce $L^2$ all-pure multiple inverse canonical cross section estimates, Bézout estimates and Gauss-Bonnet integral estimates to prove the validity of the singularization theorem. And combining the singularization theorem with the related theorem of high-dimensional algebraic clusters, we design the singular branching solution problem of rooted functions, in order to illustrate the feasibility of the application of the singularization of projective clusters to the complex domain.