$\overline{\partial}$ cohomology of the complement of a semi-positive anticanonical divisor of a compact surface

TL;DR

This paper studies the $ar{ ext{d}}$ cohomology of the complement of a semi-positive anticanonical divisor in a compact complex surface, revealing new insights into the geometric and cohomological properties of such complements.

Contribution

It provides a detailed analysis of the $ar{ ext{d}}$ cohomology group of the complement of a semi-positive anticanonical divisor on a compact surface, a topic not extensively explored before.

Findings

Computed the $ar{ ext{d}}$ cohomology group $H^1(M, \\mathcal{O}_M)$ for the complement of the divisor.

Established conditions under which the cohomology group vanishes or is non-trivial.

Connected the curvature properties of the anticanonical bundle to the cohomological behavior of the complement.

Abstract

Let $X$ be a non-singular compact complex surface such that the anticanonical line bundle admits a smooth Hermitian metric with semi-positive curvature. For a non-singular hypersurface $Y$ which defines an anticanonical divisor, we investigate the $\overline{\partial}$ cohomology group $H^1(M, \mathcal{O}_M)$ of the complement $M=X\setminus Y$.