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PreprintarXiv licencearXiv

Non-Kähler Critical Hermitian Metrics of the Dinew--Popovici Functional

  • Hanwen Liu
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Abstract

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 (S1,η1) and (S2,η2). 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 [η1+η2]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.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026nonkähler,
  title = {{Non-Kähler Critical Hermitian Metrics of the Dinew--Popovici Functional}},
  author = {Hanwen Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08113},
  eprint = {2609.08113},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08113},
}
RIS
TY  - UNPB
TI  - Non-Kähler Critical Hermitian Metrics of the Dinew--Popovici Functional
AU  - Hanwen Liu
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08113
UR  - https://arxiv.org/abs/2609.08113
AB  - 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.
ER  -