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

2025/12/22 by Youmin Chen, Miaomiao Zhu, Chen, Youmin +1
Mathematics · #35B44 #53C07 #53C25 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2512.19552

openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

Abstract

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 p₁(E) of vector bundles over Einstein 4-manifolds and the Euler numbers χ(M;E) of holomorphic vector bundles over Kähler-Einstein surfaces. Furthermore, we get some quantization results about the volume v(L_j) and certain cohomological numbers (e.g. dim H⁰(M_j;L_j)) of holomorphic line bundles L_j over non-collapsed degenerating Kähler-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 Kähler-Einstein manifolds. In particular, we obtain some interesting identities involving non-collapsed degenerating compact Kähler-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_∞. For Kähler-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_∞ in an effective way. As an unexpected surprise, we find an identity which connects Milnor numbers for singularities in M_∞ 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.

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Related

• Energy Identity for Stationary Yang Mills
• Convergence of Einstein Yang-Mills Systems
• On the blow-up of Yang-Mills fields in dimension four
• Flat connections, Higgs operators, and Einstein metrics on compact Hermitian manifolds
• The Yang-Mills flow and the Atiyah-Bott formula on compact Kahler manifolds