m-Positivity and Regularisation

2025/10/29 by Dinew, Sławomir, Popovici, Dan
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.25639

Abstract

Starting from the notion of m-plurisubharmonic function introduced recently by Dieu and studied, in particular, by Harvey and Lawson, we consider m-(semi-)positive (1, 1)-currents and Hermitian holomorphic line bundles on complex Hermitian manifolds and prove two kinds of results: vanishing theorems and L²-estimates for the ∂-equation in the context of C^∞ m-positive Hermitian fibre metrics; global and local regularisation theorems for m-semi-positive (1, 1)-currents whose proofs involve the use of viscosity subsolutions for a certain Monge-Ampère-type equation and the associated Dirichlet problem.

Citations

• m-Pseudo-effectivity and a Monge-Ampère-Type Equation for Forms of Positive Degree
• Viscosity solution to complex Hessian equations on compact Hermitian manifolds
• Fully nonlinear elliptic equations on Hermitian manifolds for symmetric functions of partial Laplacians
• m-subharmonic and m-plurisubharmonic functions -- on two problems of Sadullaev
• The Dirichlet Problem for the k-Hessian Equation on a complex manifold
• The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary
• Degenerate complex Hessian equations on compact Kähler manifolds
• From Monge-Ampere equations to envelopes and geodesic rays in the zero\n temperature limit

Related

• Plurisubharmonic functions and positive currents of type (1,1) over almost complex manifolds
• Regularisation of Currents with Mass Control and Singular Morse Inequalities
• Regularity of A Complex Monge-Ampère Equation on Hermitian Manifolds
• m-Positive Stability of Holomorphic Vector Bundles and Moduli Spaces
• Positivity conditions for Hermitian symmetric functions