Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications

We investigate a sequence of Yang-Mills connections $A_j$ lying in vector bundles $E_j$ over non-collapsed degenerating closed Einstein 4-manifolds $(M_j, g_ j)$ with uniformly bounded Einstein constants and bounded diameters. We establish a compactness theory modular three types of bubbles. As applications, we get some quantization results for several important topological number associated with the vector bundles, for instance, the first Pontrjagin numbers $p_1(E)$ of vector bundles over Einstein 4-manifolds and the Euler numbers $\chi(M;E)$ of holomorphic vector bundles over K\"{a}hler-Einstein surfaces. Furthermore, we get some quantization results about the volume $v(L_j)$ and certain cohomological numbers (e.g. $dim H^0(M_j;L_j)$) of holomorphic line bundles $L_j$ over non-collapsed degenerating K\"{a}hler-Einstein surfaces $(M_j,J_j,g_j)$ with the aid of the classical vanishing theorems, the classical Hirzebruch-Riemann-Roch type theorems, and the profound convergence theory of K\"{a}hler-Einstein manifolds. In particular, we obtain some interesting identities involving non-collapsed degenerating compact K\"{a}hler-Einstein surfaces with non-zero scalar curvature, which indicate that we can know the Euler number of $M_j$ for large $j$ provided some topological information of the limit orbifold $M_\infty$. For K\"{a}hler-Einstein Del Pezzo surfaces, an interesting implication is that we can provide some preliminary estimates for the number of singularities of various types in $M_\infty$ in an effective way. As an unexpected surprise, we find an identity which connects Milnor numbers for singularities in $M_\infty$ and the correction terms in the Hirzebruch-Riemann-Roch theorem for orbifolds. Some quantization results can be extended to the case of higher dimensional $n$-manifolds.