Logarithmic vanishing theorems on weakly 1-complete Kähler manifolds

This paper aims to investigate the vanishing theorems of cohomology groups on weakly 1-complete Kähler manifolds. Firstly, we extend a logarithmic vanishing theorem originally proposed by Huang, Liu, Wan and Yang, which says that when \(N\otimes \mathcal{O}_X([\Delta])\) is a \(k\)-positive \(\mathbb{R}\)-line bundle, the \(q\)-th cohomology group of sheaf \(\Omega^{p}(\log D)\otimes L\otimes N\) vanishes for any \(p+q\geq n+k+1\) over a compact Kähler manifold \(X\). The proof employs an approximation theorem. In the local case, the result follows directly as a corollary of the original theorem. In the global case, we construct a solution for any positive real number \(c\), ensuring that the error term with respect to the exact solution converges to zero in the sense of norm. Finally, as applications of our results, we obtain logarithmic generalizations of several classical vanishing theorems and the corresponding corollaries.