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PreprintCC BY-NC-ND 4.0arXiv

Sharp parameter gaps for quantized Mabuchi energies and cscK metrics

  • Kefeng Liu
  • Liangdi Zhang
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Abstract

For polarized manifolds, we show that semistability for the quantized Mabuchi energy at one fixed finite parameter is equivalent to uniform K-stability and to unique existence of constant scalar curvature Kähler (cscK) metrics. We obtain sharp comparisons between quantized Mabuchi slopes at different parameters and with the ordinary Mabuchi slope. In the presence of automorphisms, we give an equivariant criterion using torus twists under the vanishing of the Futaki character. The estimates also hold for finite-energy geodesic rays in compact Kähler classes. On a fixed ruled surface with discrete automorphism group, we construct cscK polarizations with unbounded critical parameters. Their ratios to the α-invariant are also unbounded.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026sharp,
  title = {{Sharp parameter gaps for quantized Mabuchi energies and cscK metrics}},
  author = {Kefeng Liu and Liangdi Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16697},
  eprint = {2610.16697},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16697},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Sharp parameter gaps for quantized Mabuchi energies and cscK metrics
AU  - Kefeng Liu
AU  - Liangdi Zhang
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16697
UR  - https://arxiv.org/abs/2610.16697
AB  - For polarized manifolds, we show that semistability for the quantized Mabuchi energy at one fixed finite parameter is equivalent to uniform K-stability and to unique existence of constant scalar curvature Kähler (cscK) metrics. We obtain sharp comparisons between quantized Mabuchi slopes at different parameters and with the ordinary Mabuchi slope. In the presence of automorphisms, we give an equivariant criterion using torus twists under the vanishing of the Futaki character. The estimates also hold for finite-energy geodesic rays in compact Kähler classes. On a fixed ruled surface with discrete automorphism group, we construct cscK polarizations with unbounded critical parameters. Their ratios to the α-invariant are also unbounded.
ER  -