= Holomorphic Laurent-series cohomology of twists on the projective line
{title2=$H^1(\mathbb{CP}^1,\mathcal O(n))=\mathcal O(\mathbb C^*)/[\mathcal O(\mathbb C)+z^n\mathcal O(\mathbb C)(1/z)]$}
For analytic <holomorphic sections>, the two-chart <Čech cohomology> uses convergent <Laurent series>, rather than only Laurent polynomials. A frame on the infinity chart equals $z^n$ times a finite-chart frame. Entire finite-chart functions remove Laurent powers $k\ge0$, and infinity-chart functions remove powers $k\le n$. The positive and negative tails converge to entire chart functions, so only the finite gap $n<k<0$ remains. Thus the first cohomology has basis $z^{n+1},\ldots,z^{-1}$ when $n\le-2$ and is zero when $n\ge-1$. The same transition relation gives global sections of dimension $\max(n+1,0)$.
Back to article page
