Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications
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arXiv
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| Format: | Preprint |
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2025
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| author | Chen, Youmin Zhu, Miaomiao |
| author_facet | Chen, Youmin Zhu, Miaomiao |
| contents | 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 $χ(M;E)$ 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. $dim H^0(M_j;L_j)$) 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 $M_\infty$. 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 $M_\infty$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications Chen, Youmin Zhu, Miaomiao Differential Geometry 35B44, 53C07, 53C25 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 $χ(M;E)$ 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. $dim H^0(M_j;L_j)$) 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 $M_\infty$. 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 $M_\infty$ 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. |
| title | Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications |
| topic | Differential Geometry 35B44, 53C07, 53C25 |
| url | https://arxiv.org/abs/2512.19552 |