Determinantal capacity and L∞-Estimates
Abstract
We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact Kähler manifolds that provides the link between the L∞-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators. On the analytic side, we introduce 𝒟-subsolutions extending the determinant sublevel condition of Sui--Sun. Using the auxiliary comparison method of Guo--Phong--Tong and pluripotential theory for complex Monge--Ampère equations, we obtain relative L∞-estimates in big cohomology classes under a determinant-entropy bound for viscosity supersolutions and singular reference potentials. Det-ellipticity provides a systematic construction of such subsolutions. On the algebraic side, we study Gårding elliptic polynomial operators of degree d. We characterize the determinant increment bound with exponent d/n by positivity of the determinantal capacity of the top homogeneous part. This is equivalent to balanced-point conditions for the Newton polytopes of rank-one scalarizations and to slope semistability of an associated subspace polymatroid. Together with the analytic hypotheses, these criteria yield relative L∞ estimates for the corresponding fully nonlinear equations.
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BibTeX
@misc{fang2026determinantal,
title = {{Determinantal capacity and $L^\infty$-Estimates}},
author = {Hao Fang and Biao Ma and Jinyang Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24078},
eprint = {2609.24078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24078},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Determinantal capacity and L^∞-Estimates
AU - Hao Fang
AU - Biao Ma
AU - Jinyang Wu
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24078
UR - https://arxiv.org/abs/2609.24078
AB - We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact Kähler manifolds that provides the link between the L^∞-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators. On the analytic side, we introduce 𝒟-subsolutions extending the determinant sublevel condition of Sui--Sun. Using the auxiliary comparison method of Guo--Phong--Tong and pluripotential theory for complex Monge--Ampère equations, we obtain relative L^∞-estimates in big cohomology classes under a determinant-entropy bound for viscosity supersolutions and singular reference potentials. Det-ellipticity provides a systematic construction of such subsolutions. On the algebraic side, we study Gårding elliptic polynomial operators of degree d. We characterize the determinant increment bound with exponent d/n by positivity of the determinantal capacity of the top homogeneous part. This is equivalent to balanced-point conditions for the Newton polytopes of rank-one scalarizations and to slope semistability of an associated subspace polymatroid. Together with the analytic hypotheses, these criteria yield relative L^∞ estimates for the corresponding fully nonlinear equations.
ER -