2025/06/01 by Liu, Bingxiao, Marinescu, George, Wang, Huan
#32A22 (Secondary) #32D15 #32F10 #32W10 (Primary) 32W05 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2506.00879
We consider a compact n-dimensional complex manifold endowed with a holomorphic line bundle that is semi-positive everywhere and positive at least at one point. Additionally, we have a smooth domain of this manifold whose Levi form has at least n-q negative eigenvalues (1≤ q≤ n-1) on the boundary. We prove that every ∂b-closed (0,ℓ)-form on the boundary with values in a holomorphic vector bundle admits a meromorphic extension for all q≤ ℓ≤ n-1. This result is an application of holomorphic Morse inequalities on Levi q-concave domains and the Kohn-Rossi extension theorem. We propose a proof of the Morse inequalities by utilizing the spectral spaces of the Laplace operator with ∂-Neumann boundary conditions. To accomplish this objective, we establish a general Nakano-Griffiths inequality with boundary conditions. This leads to a unified approach to holomorphic Morse inequalities and a geometric proof of vanishing theorems for q-concave and q-convex manifolds or domains.