Hi, these are three questions regarding extendability of holomorphic vector fields on complex projective space to its blow up along a subvariety.
Let $\mathbb{P}^n$ be the complex projective space, with homogeneous coordinates of a point $p\in\mathbb{P}^n$ $$p=[z\_0:\ldots:z\_n]$$


1) suppose we blow up the point $p\_0=[1:0:\ldots:0]$, which are the global holomorphic vector fields of $\mathbb{P}^n$ that extend to the blow up?


I made some computations in the affine chart $\lbrace z\_0\neq0 \rbrace$ and modulo (fatal) errors i obtain that my vector fields are of the form $$l\_0\frac{\partial}{\partial z\_0}+\sum\_{i=1}^n c\_iz\_i\frac{\partial}{\partial z\_i}$$
with $l\_0\in\mathbb{C}[z\_0,\ldots,z\_n]$ homogeneous of degree 1 and $c\_i\in\mathbb{C}$.
So vector fields that extend to the blow up span a vector subspace of dimension $2n$ of $H^0(T\mathbb{P}^n)$. Am i right? 


2)In a similar fashion i find that if i blow up a k-codimensional variety of type $$\lbrace z\_k=\ldots=z\_n=0 \rbrace$$ with $k>1$ my vector fields span a vector subspace of dimension 
 $n(k+2)$ of $H^0(T\mathbb{P}^n)$. Again, am i right?


3) If i blow up a smooth curve of sufficiently high degree/genus do i still have holomorphic vector fields that extend to the blow up? 


Thank you in advance.



Holomorphic vector fields on $\mathbb{P}^n$ that extend to the blow up
**1)** No. There are many more vector fields. The vector fields you are looking for are precisely those which vanish at $p\_0$. Since $h^0( \mathbb P^n, T \mathbb P^n) = (n+1)^2 -1$ and you are imposing $n$ linearly independent conditions you should get $n^2 + n - 1$ vector fields. In homegeneous coordinates they can be written as
$$
l\_0 \frac{\partial}{\partial z\_0} + \sum\_{i=1}^n \sum\_{j=1}^n c\_{ij} z\_i \frac{\partial}{\partial z\_j} .
$$
Notice that the homogeneous vector field $\sum\_{i=0}^n z\_i \frac{\partial}{\partial z\_i}$ does not contribute to the counting since it corresponds to the zero vector field on $\mathbb P^n$.


**2)** You should get $k^2 + (n+1-k) n -1$ vector fields. As pointed out in the comments the
result remains unchanged if we swap $k$ and $n-k+1$. The point is that $PGL(n+1)$ acts on $\mathbb P^n$ as well as on the dual projective space $\check{\mathbb P}^n$. If we look at the subgroup preserving a linear subspace of codimension $k$ on $\mathbb P^n$, then the natural action on $\check{\mathbb P}^n$ will preserve its dual: a linear subspace of codimension $n-k+1$. 


**3)** You will get a holomorphic vector field on the blow-up if and only if the curve is left invariant by the original vector field. 


If your vector field vanishes on a linear subspace of dimension at least two then every curve contained in it will be invariant by the vector field. Thus on the blow-up along these curves you will still get holomorphic vector fields.


If instead you restrict your attention to vector fields with zero set of dimension at most one then while you can find curves of arbitrarily high degree invariant by this class of vector fields (think on orbits of $\mathbb C^\*$-actions on $\mathbb P^n$), they all have (geometric) genus zero. 


