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PreprintCC BY 4.0arXiv

Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class

  • Frederick Tsz-Ho Fong
  • Junbang Liu
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Abstract

Let X be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class [ω], Jian--Shi--Song proved that there exists a unique cscK metric in the class 2πc1(KX)+ε[ω] when ε ∈ (0,ε0], using the properness criterion of Chen--Cheng. We prove that the cscK metrics in 2πc1(KX)+ε[ω] converge, as ε↓0, in the Gromov--Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of X as conjectured by Jian--Shi--Song. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee--Tosatti--Zhang on the Gromov-Hausdorff convergence of the normalized Kähler--Ricci flow on X.

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Cite

BibTeXRIS
BibTeX
@misc{fong2026gromovhausdorff,
  title = {{Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class}},
  author = {Frederick Tsz-Ho Fong and Junbang Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21934},
  eprint = {2610.21934},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21934},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class
AU  - Frederick Tsz-Ho Fong
AU  - Junbang Liu
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21934
UR  - https://arxiv.org/abs/2610.21934
AB  - Let X be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class [ω], Jian--Shi--Song proved that there exists a unique cscK metric in the class 2πc_1(K_X)+ε[ω] when ε ∈ (0,ε_0], using the properness criterion of Chen--Cheng. We prove that the cscK metrics in 2πc_1(K_X)+ε[ω] converge, as ε↓0, in the Gromov--Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of X as conjectured by Jian--Shi--Song. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee--Tosatti--Zhang on the Gromov-Hausdorff convergence of the normalized Kähler--Ricci flow on X.
ER  -