Let $V$ be a smooth projective algebraic surface and let $\pi\colon V'\to V$ be a blowup of a point $p\in V$. I would like to ask if the following is true:
$h^{0}(V,\Theta_{V})=h^{0}(V', \Theta_{V'})$,
where $\Theta_{X}$ denotes the tangent sheaf to $X$. I am particularly interested in the case of rational surfaces.
As far as I know, we have $\pi_{*}\Theta_{V'}=\mathfrak{m}_{p}\cdot \Theta_{V}$ and $\pi^{*}\Theta_{V}=\Theta_{V'}\langle E \rangle (E)$, where $E$ is the exceptional curve of $\pi$ and $\Theta_{X}\langle D \rangle$ denotes the subsheaf of $\Theta_{X}$ consisting of vector fields tangent to a divisor $D$. However, I have no intuition on what the functors $\pi^{*}$ and $\pi_{*}$ do with the sheaf cohomology (and how to apply it to my original question). If someone knows a comprehensive guide I can learn such things, I would be very grateful for a reference.
Thank you for any help.