= Solution
Let $(z^1,\ldots,z^n)$ be a <complex manifold> chart, with $z^j=x^j+iy^j$. The <almost complex structure induced by a complex atlas> is
$$
J\frac{\partial}{\partial x^j}=\frac{\partial}{\partial y^j},
\qquad
J\frac{\partial}{\partial y^j}=-\frac{\partial}{\partial x^j}.
$$
Thus $J^2=-I$. On an overlap, the derivative of a <holomorphic map> is complex linear and hence commutes with multiplication by $i$. The two coordinate definitions of $J$ therefore agree, so $J$ is globally well-defined.
The complexified cotangent bundle splits into the $i$- and $-i$-eigenspaces of $J^*$, locally spanned by $dz^j$ and $d\bar z^j$. A <differential form of type (p, q)> is a sum
$$
\alpha=\sum_{|I|=p,\,|K|=q}\alpha_{I\bar K}\,dz^I\wedge d\bar z^K.
$$
Splitting the <exterior derivative> according to type defines
$$
d=\partial+\bar\partial,
\qquad
\partial:\Omega^{p,q}\to\Omega^{p+1,q},
\qquad
\bar\partial:\Omega^{p,q}\to\Omega^{p,q+1}.
$$
Complex conjugation sends $dz^j$ to $d\bar z^j$ and conjugates the coefficient derivatives. Term by term this gives the <complex conjugation of differential-form type> identity
$$
\bar\partial\bar\alpha=\overline{\partial\alpha}.
$$
Let $J$ act on a $k$-form by
$$
(J\alpha)(v_1,\ldots,v_k)=\alpha(Jv_1,\ldots,Jv_k).
$$
It acts on a $(p,q)$-form by $i^{p-q}$. Therefore $J^{-1}dJ$ multiplies the $\partial$ component by $-i$ and the $\bar\partial$ component by $i$, proving the <d c operator> formula
$$
J^{-1}dJ=i(\bar\partial-\partial)=d^c.
$$
A <holomorphic vector field> is a holomorphic section of $T^{1,0}X$. On the affine chart $U_0\subset\mathbb{CP}^n$, put $w_k=z_k/z_0$. The projection is
$$
\varphi_0(z)=(1,w_1,\ldots,w_n).
$$
Direct differentiation gives
$$
(d\varphi_0)_a\left(\frac{\partial}{\partial z_j}\right)
=
\begin{cases}
\displaystyle \frac1{a_0}\frac{\partial}{\partial w_j},&j\ne0,\\[6pt]
\displaystyle-\frac1{a_0^2}\sum_{k=1}^na_k\frac{\partial}{\partial w_k},&j=0.
\end{cases}
$$
If $\xi(z)$ is linear and homogeneous, the pushed-forward coefficients are consequently $\xi(1,w)$ when $j\ne0$, and $-\xi(1,w)w_k$ when $j=0$. They are holomorphic on $U_0$.
More intrinsically, $\xi(z)\partial_{z_j}$ is a linear vector field $Az$ on $\mathbb C^{n+1}$. Its flow $e^{tA}$ preserves complex lines and induces the projective transformations
$$
[z]\longmapsto[e^{tA}z].
$$
Differentiating gives a globally defined <projectivization of a linear vector field>, whose expression on $U_0$ is the field just computed. This proves extension across the other affine charts.
Finally choose distinct complex numbers $\lambda_0,\ldots,\lambda_n$ and the diagonal field
$$
Z=\sum_{j=0}^n\lambda_jz_j\frac{\partial}{\partial z_j}.
$$
Its projectivization vanishes at $[z]$ precisely when $(\lambda_0z_0,\ldots,\lambda_nz_n)$ is proportional to $z$, so $[z]$ is an eigenline. The distinct eigenvalues leave exactly the $n+1$ coordinate points. Hence $\mathbb{CP}^n$ has a holomorphic vector field with finitely many zeroes.
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