= Solution
On a <complex manifold>, <holomorphic coordinates> give the decomposition $d=\partial+\bar\partial$, with
$$
\partial:\mathcal A^{p,q}\to\mathcal A^{p+1,q},\qquad
\bar\partial:\mathcal A^{p,q}\to\mathcal A^{p,q+1}.
$$
For $\alpha=\sum_{I,J}a_{I\bar J}\,dz^I\wedge d\bar z^J$, the <conjugate Dolbeault operator> $\partial$ and the <Dolbeault operator> $\bar\partial$ are explicitly
$$
\partial\alpha=\sum_{I,J,k}\frac{\partial a_{I\bar J}}{\partial z^k}\,dz^k\wedge dz^I\wedge d\bar z^J,
\qquad
\bar\partial\alpha=\sum_{I,J,k}\frac{\partial a_{I\bar J}}{\partial\bar z^k}\,d\bar z^k\wedge dz^I\wedge d\bar z^J.
$$
These are the type components of the intrinsic <exterior derivative>, so are independent of coordinates. The identity $d^2=0$, separated by type, gives $\partial^2=\bar\partial^2=0$ and $\partial\bar\partial+\bar\partial\partial=0$. A <holomorphic p-form on a complex manifold> is a $(p,0)$-form whose coefficients are <holomorphic functions>; equivalently it is a $(p,0)$-form killed by the <Dolbeault operator>.
A rank-$r$ <holomorphic vector bundle> has local trivializations $E|_{U_i}\cong U_i\times\mathbb C^r$ whose changes are $(x,v)\mapsto(x,g_{ij}(x)v)$, with $g_{ij}:U_i\cap U_j\to\operatorname{GL}_r(\mathbb C)$ holomorphic. These <transition functions of a vector bundle> satisfy $g_{ii}=I$ and $g_{ij}g_{jk}=g_{ik}$. Equivalently, for local frames we use the convention $e_j=e_i g_{ij}$. A <holomorphic section> is a section having holomorphic coordinate functions in every such <holomorphic local frame>; the condition is unchanged by the holomorphic invertible transitions.
The <holomorphic cotangent bundle> has local frames $dz^1,\ldots,dz^n$. Under another <holomorphic coordinate> system $w$, the <chain rule> gives
$$
dw^a=\sum_b\frac{\partial w^a}{\partial z^b}\,dz^b.
$$
The Jacobian matrix is holomorphic and invertible, with holomorphic inverse supplied by the inverse coordinate change. These are precisely holomorphic bundle transitions. A section written as $\sum_b f_b(z)\,dz^b$ is holomorphic exactly when every $f_b$ is holomorphic, so
$$
\boxed{H^0(X,\Omega_X^1)=\{\alpha\in\mathcal A^{1,0}(X):\bar\partial\alpha=0\},}
$$
the space of <holomorphic one-forms>. In particular, being a holomorphic section is not the same as being a $d$-closed one-form on an arbitrary <complex manifold>.
The <complex tautological line bundle> is
$$
\mathcal O(-1)=\{([Z],v)\in\mathbb{CP}^n\times\mathbb C^{n+1}:v\in\mathbb C Z\}.
$$
On $U_i=\{Z_i\ne0\}$ use the nowhere-zero frame $s_i([Z])=Z/Z_i$. Its components are holomorphic affine coordinates and the constant one in position $i$. On overlaps
$$
s_j=\frac{Z_i}{Z_j}s_i,
$$
so the frame changes are nowhere-zero <holomorphic functions>. This proves that $\mathcal O(-1)$ is a <holomorphic line bundle>, and its inclusion into the trivial bundle is a <holomorphic bundle map>.
A global <holomorphic section> composed with that inclusion has $n+1$ component <holomorphic functions> on compact connected <Complex projective space>. Each is constant: its absolute value attains a maximum; in a coordinate ball the one-variable maximum modulus principle along complex lines makes it locally constant, and connectedness propagates that constant globally. Thus the section is a fixed vector $v\in\mathbb C^{n+1}$ lying in every fibre, namely every line $\mathbb C Z$. For $n\geq1$ the intersection of all these lines is zero. Hence
$$
\boxed{H^0(\mathbb{CP}^n,\mathcal O(-1))=0\quad(n\geq1).}
$$
The positive-dimension convention is necessary: $\mathbb{CP}^0$ is one point and this section space is $\mathbb C$.
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