Logarithmic vanishing theorems on Weakly 1-complete Kahler manifolds

Abstract In this paper, we prove a logarithmic vanishing theorem on weakly $1$-complete k\"ahler manifold, which is a generalization of Huang-Liu-Wan-Yang's result on compact k\"ahler manifold. We first briefly introduce the local case of the theorem, which can be obtained as a corollary of the compact k\"ahler case. Next, we prove the global case of the theorem. The difficulty here is how to find a continuous solution ψ from a sequence ψᵥ of discrete solutions of equation φᵥ=∂ψᵥ, such that for each v∈ R, φᵥ=∂ψ. In this paper, the continuous solution is given by using the approximation theorem. 2020 Mathematics Subject Classification. 32Q15,14F17.