On $X=CP^2\#k{(-CP^2)}$ in $k$ generic points, let $h^i=\dim H^i(T^{1,0}X)$, for $i\ge 0$. First, we know $h^i=0$ for $i\ge 2$. By Riemann–Roch formula, I obtain that $h^0-h^1 = 8-2k$. Would someone be kind to show me how to compute the general formula for $h^0$? Thank you in advance.



Holomorphic vector fields on blow-ups of CP^2
On CP^2, assume the cover is given by $U\_i=\{[z\_0, z\_1, z\_2]| z\_i\ne 0\}$ where $i=0,1,2$, then one can compute that on $U\_0$, $H^0(T^{1,0}CP^2)$ is spanned by $\frac{\partial}{\partial z\_1}$, $\frac{\partial}{\partial z\_2}$, $z\_1\frac{\partial}{\partial z\_1}$, $z\_1\frac{\partial}{\partial z\_2}$, $z\_2\frac{\partial}{\partial z\_1}$, $z\_2\frac{\partial}{\partial z\_2}$, $z\_1z\_2\frac{\partial}{\partial z\_1}+z^2\_2\frac{\partial}{\partial z\_2}$, $z^2\_1\frac{\partial}{\partial z\_1}+z\_1z\_2\frac{\partial}{\partial z\_2}$. If we blow up point at $[1,0,0]$, then only $z\_1\frac{\partial}{\partial z\_1}$, $z\_1\frac{\partial}{\partial z\_2}$, $z\_2\frac{\partial}{\partial z\_1}$, $z\_2\frac{\partial}{\partial z\_2}$, $z\_1z\_2\frac{\partial}{\partial z\_1}+z^2\_2\frac{\partial}{\partial z\_2}$, $z^2\_1\frac{\partial}{\partial z\_1}+z\_1z\_2\frac{\partial}{\partial z\_2}$ can be extended to $CP^2\#(-CP^2)$. Moreover, if we continue blowing up at $[0,1,0]$, one can check that only $z\_1\frac{\partial}{\partial z\_1}$, $z\_1\frac{\partial}{\partial z\_2}$, $z\_2\frac{\partial}{\partial z\_2}$, $z\_1z\_2\frac{\partial}{\partial z\_1}+z^2\_2\frac{\partial}{\partial z\_2}$ are extendable. Continue blowing up at $[0,0,1]$, then on $CP^2\#3(-CP^2)$, the only left vector fields are $z\_1\frac{\partial}{\partial z\_1}$ and $z\_2\frac{\partial}{\partial z\_2}$. And do blow up one more time, e.x. at [1,1,1], then there is no vector fields extendable. This verifies that $h^0=8-2k$ when $k\le 4$ and $h^0=0$ when $k\ge 5$.


