Here by $P^n$ I mean $CP^n$, and what I want to do is to calculate the number of global sections of the holomorphic tangent bundle of $CP^n$. 


If $n=1$, it is well known that $h^0(P^1, TP^1)=h^o(P^1,\mathcal{O}\_{P^1}(2))=3$.


If $n>1$, I did some calculation in local coordinates, and find out that 
$h^0(P^n, TP^n) = n(n+1)$.


I am not sure if this is the correct answer and wonder if anyone else has calculated this before.


Besides, does anybody know the value of $h^1(P^n, TP^n)$? Even the $n=2$ case is enough for me. Many thanks!



Computing the dimension of the module of global holomorphic vector fields for complex projective n-space
The dimension of $H^0(\mathbb P^n,T\mathbb P^n)$ is $(n+1)^2-1$ and 
$h^1(\mathbb P^n, T \mathbb P^n)=0$. Using the Euler sequence (see for instance
Griffiths-Harris, Principles of Algebraic Geometry) you can reduce the computation of these guys to the computation of
the comology of $\mathcal O\_{\mathbb P^n}$ and $\mathcal O\_{\mathbb P^n}(1)$.


For the dimension of the space of holomorphic vector fields on $\mathbb P^n$ it is perhaps easier
to realize that $$Aut(\mathbb P^n) = PSL(n+1, \mathbb C)$$ and its Lie algebra is 
$$\mathfrak{sl}(n+1,\mathbb C)$$. 


