具有常全纯截面曲率的平衡Bismut挠率平行流形
Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature
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中文总结 AI 辅助
本文证明了所有维数下平衡Bismut挠率平行流形的常全纯截面曲率猜想，并构造反例否证满秩B猜想。
中文摘要 AI 辅助
非Kähler几何中一个长期存在的猜想认为，任何具有常全纯截面曲率的紧致Hermitian流形要么是Kähler流形，要么是Chern平坦的。该猜想在二维情形已被证明成立，但在三维或更高维情形下，除若干特殊情况外，仍属开放问题。本文研究Bismut挠率平行（简称BTP）的紧致Hermitian流形，即Bismut联络具有平行挠率的流形。对于非平衡的BTP流形，Chen和Zheng证明了该猜想。他们还利用Zhao和Zheng对这类三维流形的分类结果，确认了三维平衡BTP流形的猜想。在最近的工作中，我们通过详细的逐例分析，确认了四维平衡BTP流形的猜想。在本文中，我们建立了所有维数下平衡BTP流形的猜想。我们还通过构造一个紧致平衡BTP五维流形，其满秩$B$既非Kähler也非Chern平坦，从而否证了满秩$B$猜想；该流形的Chern全纯截面曲率是非恒定的。
英文摘要
A long-lasting conjecture in non-Kähler geometry says that any compact Hermitian manifold with constant holomorphic sectional curvature must be either Kähler or Chern flat. The conjecture is known to be true in dimension 2 but remains open in dimensions 3 or higher, except in several special cases. In this article we consider compact Hermitian manifolds that are Bismut torsion-parallel (BTP for brevity), which means that the Bismut connection has parallel torsion. For BTP manifolds that are not balanced, the conjecture was proved by Chen and Zheng. They also confirmed the conjecture for balanced BTP manifolds in dimension 3, utilizing a classification result for such threefolds by Zhao and Zheng. In a recent work, we confirmed the conjecture for balanced BTP manifolds in dimension 4 through a detailed case-by-case analysis. In this article, we establish the conjecture for balanced BTP manifolds in all dimensions. We also disprove the full-rank $B$ conjecture by constructing a compact balanced BTP fivefold with full-rank $B$ that is neither Kähler nor Chern flat; its Chern holomorphic sectional curvature is nonconstant.
发表机构
- Halıcıoğlu Data Science Institute, University of California San Diego（加州大学圣地亚哥分校 Halıcıoğlu 数据科学研究所）
- School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）
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