发表机构
Mathematics Institute, University of Warwick（华威大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了两个Kähler曲面乘积上的非Kähler临界度量，否定回答了高维情形的问题，并刻画了临界乘积度量的存在条件，同时发展了一个收敛到Kähler背景的几何流。
AI 中文摘要
Dinew--Popovici泛函是固定Aeppli上同调类中Hermitian辛度量的能量泛函。其消失刻画了该类中的Kähler度量，为Kähler几何提供了变分方法。Dinew和Popovici证明了在复三维中每个临界点都是Kähler的。我们通过构造两个Kähler曲面$(S_1,\eta_1)$和$(S_2,\eta_2)$的乘积上的非Kähler临界度量，对Erfan Soheil关于更高维数的问题给出了否定答案。这些度量在乘积4-流形上的每次变分下都是临界的。我们刻画了临界乘积度量，并证明了当且仅当两个曲面因子的典范丛光滑平凡时，在Aeppli类$[\eta_1+\eta_2]_A$中存在非Kähler临界乘积。作为副产品，我们发展了一个由挠率张量驱动的几何流，当背景度量具有非负全纯双截曲率时，该流收敛到固定的Kähler背景。
英文摘要
The Dinew--Popovici functional is an energy functional for Hermitian symplectic metrics in a fixed Aeppli cohomology class. Its vanishing characterizes the Kähler metrics in that class, providing a variational approach to Kähler geometry. Dinew and Popovici proved that every critical point is Kähler in complex dimension three. We give a negative answer to Erfan Soheil's question about higher dimensions by constructing non-Kähler critical metrics on products of two Kähler surfaces $(S_1,η_1)$ and $(S_2,η_2)$. These metrics are critical under every variation on the product 4-fold. We characterize the critical product metrics and prove that non-Kähler critical products exist in the Aeppli class $[η_1+η_2]_A$ precisely when the canonical bundles of the two surface factors are smoothly trivial. As a by-product, we develop a geometric flow driven by the torsion tensor and converging to the fixed Kähler background when this background metric has nonnegative holomorphic bisectional curvature.
Comments25 pages, 0 figure
