In a <holomorphic coordinate> chart, $T^{1,0}X$ has frame $\partial/\partial z^1,\ldots,\partial/\partial z^n$. Under a holomorphic coordinate change $w=w(z)$, the <chain rule> gives
$$
\frac{\partial}{\partial z^j}=\sum_k\frac{\partial w^k}{\partial z^j}\frac{\partial}{\partial w^k}.
$$
The Jacobian is an invertible matrix of <holomorphic functions>. These frames therefore make $T^{1,0}X$ the <holomorphic tangent bundle>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 1 / b / Solution

{title2=$T^{1,0}X$}

The holomorphic tangent bundle of a <complex manifold> $X$ is locally spanned in <holomorphic coordinates> by $\partial/\partial z^1,\ldots,\partial/\partial z^n$. Under a holomorphic coordinate change $w=w(z)$, the <chain rule> gives
$$
\frac{\partial}{\partial z^j}
=\sum_k\frac{\partial w^k}{\partial z^j}\frac{\partial}{\partial w^k},
$$
whose coefficient matrix is holomorphic; these frames therefore define a <holomorphic vector bundle>.

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