Recently, I encountered the version of the Riemann-Roch theorem for line bundles $\mathscr L$ on a compact Riemann surface $X$: $$\dim H^0(X,\mathscr L) - \dim H^1(X,\mathscr L) = 1 - g + c_1(\mathscr L),$$
with $c_1$ the first Chern-number.
I wonder if we can use this formula to derrive properties of the tangent bundle and the vector fields on $X$?
I first wanted to use that the top Chern class is the Euler-class, i.e. the first Chern number of the tangent bundle is the Euler characteristics. But then I realized, that the tangent bundle is only complex, but not holomorphic, so in fact, we need to set $\mathscr L = T^{(1,0)}X$, the holomorphic part of $TX$ in order to apply Riemann-Roch.
Can we still make some general statements about $c_1(\mathscr L)$ and if yes, what does the resulting formula tell us?
Edit: I made some progress on this questionm however I am not sure about the meaning of the results. Maybe someone can provide some background information.
As the holomorphic tangent bundle $\mathcal{T}_X$ is the dual of the canonical bundle $\Omega_X$, it follows that $c_1(\mathcal T_X)=-c_1(\Omega_X) = 2-2g=\chi(X)$.
Thus we get $c_1(\mathcal T_X)=c_1(TX) = c_1(T^{1,0}X \oplus T^{0,1}X) = c_1(T^{1,0}X)+c_1(T^{0,1}X)$, so $c_1(T^{0,1}X)=0$. What does this mean (maybe in regards to curvature)?
This also implies, that for $g\ge 2$, $\dim H^0(X,\mathcal T_X)=0$, so there is no global holomorphic section of the tangent bundle in this case, i.e. no global holomorphic vector field.
Using Riemann-Roch, we get:
$$\dim H^0(X,\mathcal T_X)- \dim H^1(X,\mathcal T_X) = 1-g+c_1(\mathcal T_X)=3-3g. $$
As for $g=1$, $\mathcal T_X$ is trivial, it follows that $\dim H^0(X,\mathcal T_X)=1.$
For $g=0$, $X=\mathbb P^1$ and $\Omega_{\mathbb P^1}=O(-2)$, so $\mathcal T_{\mathbb P^1} = O(2)$, which is given by the homogeneous polynomials in $z_0,z_1$ of degree $2$. Thus $\dim H^0(X,\mathcal T_X) = 3.$
In conclusion: $$\dim H^1(X,\mathcal T_X) = \begin{cases} 0 &&g=0 ,\\1 && g=1, \\3g-3 && g\ge 2. \end{cases}$$ Is this information any useful or can this be motivated by some geometric insights? Maybe there is a short exact sequence of vector bundles/sheaves starting with $\mathcal T_X$, so atleast in the case of $g=0$ one can conclude surjectivity of map.?