I am trying to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for this result, which uses the cellular decomposition of $\mathbb{C}\mathrm{P}^2$ to get $\chi(\mathbb{C}\mathrm{P}^2) = 3$, but I would like to find a proof of this result that relies on concepts like indices of isolated zeros on a vector field. So for the purposes of this question, I would like to utilize the following definition of the Euler characteristic: For a closed orientable manifold $M$ we define $\chi(M) = \sum_i \mathrm{Ind}_{d_i} \mathrm{v}$ where $\mathrm{v}$ is a vector field on $M$ with isolated zeros.
In my first attempt at this problem I thought about finding a vector field on $\tilde{\mathrm{v}}$ on $S^5$, and then using the identification $\mathbb{C}\mathrm{P}^2 \cong S^5/\mathrm{U}(1)$, seeing if $\tilde{\mathrm{v}}$ descended to a vector field on $\mathbb{C}\mathrm{P}^2$ with isolated zeros that lent itself to computing the Euler characteristic. I had difficulty making this work out, so I am unsure if this is a good approach to tackling the problem. Any insights?
Best Answer
There is a canonical way to construct holomorphic vector fields on $\mathbb{C}P^2,$ and that way is described in Zoladek's "Monodromy Group", page 335. If you read the description, it will be pretty clear what the index is (note that if the vector field is holomorphic, it is given by at most quadratic polynomials, otherwise there are poles. There is a large literature [much of it by Iliashenko] studying such fields and the associated foliations.