On Cohomology Vanishing with Polynomial Growth on Complex Manifolds with Pseudoconvex Boundary

Abstract

The $\bar{\partial }$ cohomology groups with polynomial growth $H_{p.g.}^{r,s}$ will be studied. It will be shown that, given a complex manifold $M$, a locally pseudoconvex bounded domain $\Omega ⋐M$ satisfying certain geometric boundary conditions and a holomorphic vector bundle $E\to M$, $H_{p.g.}^{r,s}(\Omega ,E)=0$ holds for all $s\ge 1$ if $E$ is Nakano positive and $r=\dim M$. It will also be shown that $H_{p.g.}^{r,s}(\Omega ,E)=0$ for all $r$ and $s$ with $r+s>\dim M$ if, moreover, $\mathrm{rank}E=1$. By the comparison theorem due to Deligne, Maltsiniotis (Astérisque 17 (1974), 141–160) and Sasakura (Inst. Math. Sci. 17 (1981), 371–552), it follows in particular that, for any smooth projective variety $X$, for any ample line bundle $L\to X$ and for any effective divisor $D$ on $X$ such that $[D]{|}_{|D|}\ge 0$, the algebraic cohomology $H_{\mathrm{alg}}^s(X\setminus |D|,{\Omega }_X^r(L))$ vanishes if $r+s>\dim X$.