PreprintCC BY-NC-ND 4.0arXiv
Higgs-Demailly System and Positivity of Higgs Bundles
Abstract
We prove that the Higgs--Demailly system on a compact Riemann surface admits, for suitable fixed parameters, a smooth admissible solution at its terminal parameter if and only if the Higgs bundle is H-ample. Here H-ampleness is understood in the modified sense of Biswas--Misra--Ray. In particular, the argument gives an analytic proof of the equivalence between this H-ampleness and Griffiths-positive Hitchin--Simpson curvature. The proof extends the Demailly--Pingali--Murakami approach using a priori estimates and Leray--Schauder degree theory. The missing scalar lower bound follows from a Higgs-compatible quotient construction: a blow-up sequence produces a nonzero Higgs quotient of nonpositive degree, contradicting H-ampleness.
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BibTeX
@misc{zhang2026higgsdemailly,
title = {{Higgs-Demailly System and Positivity of Higgs Bundles}},
author = {Liangdi Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00556},
eprint = {2609.00556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00556},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Higgs-Demailly System and Positivity of Higgs Bundles
AU - Liangdi Zhang
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.00556
UR - https://arxiv.org/abs/2609.00556
AB - We prove that the Higgs--Demailly system on a compact Riemann surface admits, for suitable fixed parameters, a smooth admissible solution at its terminal parameter if and only if the Higgs bundle is H-ample. Here H-ampleness is understood in the modified sense of Biswas--Misra--Ray. In particular, the argument gives an analytic proof of the equivalence between this H-ampleness and Griffiths-positive Hitchin--Simpson curvature. The proof extends the Demailly--Pingali--Murakami approach using a priori estimates and Leray--Schauder degree theory. The missing scalar lower bound follows from a Higgs-compatible quotient construction: a blow-up sequence produces a nonzero Higgs quotient of nonpositive degree, contradicting H-ampleness.
ER -