Holomorphic Euler characteristic of complex line bundle

Question

Let $X = \mathbb{CP^1}\times\mathbb{CP^1}$. How to show that the holomorphic Euler characteristic $\chi _h(\underline{ \mathbb{C}})$ of the trivial complex line bundle $\underline{C} = C \times X$ equals 1?

Answer 1

The holomorphic Euler characteristic of a vector bundle $E$ over a compact complex manifold $X$ of dimension $n$ is given by

$$\chi(X, E) = \sum_{i=0}^n(-1)^i\dim H^i(X, E).$$

If $E$ is the trivial line bundle $\mathcal{O}$, then $H^i(X, \mathcal{O}) \cong H^{0,i}_{\bar{\partial}}(X)$ by Dolbeault's Theorem. Therefore, we have

$$\chi(X, \mathcal{O}) = \sum_{i=0}^n(-1)^i\dim H^{0,i}_{\bar{\partial}}(X) = \sum_{i=0}^n(-1)^ih^{0,i}(X).$$

Now note that $X = \mathbb{CP}^1\times\mathbb{CP}^1$ has dimension $n = 2$, so

$$\chi(\mathbb{CP}^1\times\mathbb{CP}^1, \mathcal{O}) = h^{0,0}(\mathbb{CP}^1\times\mathbb{CP}^1) - h^{0,1}(\mathbb{CP}^1\times\mathbb{CP}^1) + h^{0,2}(\mathbb{CP}^1\times\mathbb{CP}^1) = 1 - 0 + 0 = 1.$$