Čech cohomology / 08

Laurent computations on \(\mathbb{CP}^1\)

On the standard two-chart cover of $$\mathbb{CP}^1$$, Čech cohomology becomes the problem of splitting Laurent series into chart-extendable pieces.

Let

\[U_0=\{Z_0\ne0\}\simeq\mathbb C,\qquad U_\infty=\{Z_1\ne0\}\simeq\mathbb C,\]

with coordinates \(z=Z_1/Z_0\) and \(w=1/z\). The overlap is \(\mathbb C^*\).

Because this is a two-set cover, there are no triple intersections. Every \(1\)-cochain is automatically a cocycle. The only question is whether it is a coboundary.

The structure of \(H^1\)

For a sheaf \(\mathcal F\) on this cover,

\[\check H^1(\mathfrak U,\mathcal F) = \frac{\mathcal F(U_0\cap U_\infty)} {\mathcal F(U_\infty)|_{U_0\cap U_\infty} -\mathcal F(U_0)|_{U_0\cap U_\infty}}.\]

This is the calculus template again: an overlap section is an error term; local sections on the two charts are corrections; the quotient keeps errors that cannot be corrected.

The trivial bundle

For \(\mathcal O\), a section on the overlap has a Laurent expansion

\[f(z)=\sum_{k\in\mathbb Z}a_kz^k.\]

Terms with \(k\ge0\) extend holomorphically to \(U_0\). Terms with \(k<0\) become nonnegative powers of \(w\) and extend to \(U_\infty\). Thus every overlap section splits into two chart sections:

\[H^1(\mathbb{CP}^1,\mathcal O)=0.\]

Twisting by \(\mathcal O(n)\)

For \(\mathcal O(n)\), the transition function changes the splitting rule. The same Laurent monomial may be removable from one chart after twisting but not from the other.

The result is

\[h^0(\mathbb{CP}^1,\mathcal O(n))= \begin{cases} n+1,& n\ge0,\\ 0,& n<0, \end{cases}\]

and

\[h^1(\mathbb{CP}^1,\mathcal O(n))= \begin{cases} 0,& n\ge -1,\\ -n-1,& n\le -2. \end{cases}\]

Čech cohomology explains the formulas by showing which Laurent monomials survive after all allowable local corrections are subtracted.

The first surviving class

For \(\mathcal O(-2)\), one Laurent monomial survives the correction process. This gives

\[H^1(\mathbb{CP}^1,\mathcal O(-2))\cong\mathbb C.\]

The class is not a mystery: it is an overlap term that cannot be extended away by holomorphic sections on the two charts.

How to use this pattern

When a computation reduces to functions on an annulus or punctured coordinate chart, try to write overlap data as a Laurent series. Then divide the terms into:

  1. terms removable from the first chart;
  2. terms removable from the second chart;
  3. terms that survive in the quotient.

Those surviving terms are Čech representatives.