仿射二次曲面上的具有消失Bismut Ricci形式的完全pluriclosed度量

中文摘要 AI 辅助

我们研究了仿射二次曲面$Q_3\cong TS^3$上的一族SO(4)-不变的pluriclosed Hermitian度量$g_c$，其Bismut Ricci形式消失，这族度量先前在物理学文献中有所描述。对于每个实数参数$c$，我们证明了定义奇异初值问题的实解析解的局部存在性和唯一性，以及相关度量在奇异轨道$S^3$附近的正性。然后，我们建立了$|c|\leq1$时的全局存在性和完备性，以及$|c|>1$时的有限时间退化。这族完备度量穷尽了仿射二次曲面$Q_3$上所有SO(4)-不变的Bismut Hermitian-Einstein度量，并将Stenzel的Kähler Ricci平坦度量与Chamseddine-Volkov/Maldacena-Nuñez首次考虑的度量连接起来。我们还确定了主导渐近行为，并推导出显式的数量曲率公式，证明了非Kähler成员的严格正性，并确定了在端点$c=1$处渐近数量曲率的变化。最后，我们证明所有这些度量都不是Bismut平坦的，且具有完全的Bismut和乐群SU(3)，为此类流形提供了新的例子。

英文摘要

We study a one-parameter family of SO(4)-invariant pluriclosed Hermitian metrics $g_c$ with vanishing Bismut Ricci form on the affine quadric $Q_3\cong TS^3$, previously described in the physics literature. For every real parameter $c$, we prove local existence and uniqueness of a real-analytic solution to the defining singular initial-value problem, together with positivity of the associated metric near the singular orbit $S^3$. We then establish global existence and completeness for $|c|\leq1$ and finite-time degeneration for $|c|>1$. The complete family exhausts all SO(4)-invariant Bismut Hermitian-Einstein metrics on the affine quadric $Q_3$, and joins Stenzel's Kähler Ricci-flat to a metric that was first considered by Chamseddine-Volkov/Maldacena-Nuñez. We also determine the leading asymptotics and derive an explicit scalar curvature formula, proving strict positivity for the non-Kähler members and identifying the change in asymptotic scalar curvature at the endpoint $c=1$. Finally, we show that all these metrics are not Bismut flat and have full Bismut holonomy SU(3), providing new examples of such manifolds.

发表机构

• Università degli Studi di Firenze（佛罗伦萨大学）
• Università degli Studi di Torino（都灵大学）

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