Lemma 19.9.15744C8C462232

Proof. If $n=1$, take $\pi $ to be the identity and any point as the last member of the flag; the remaining assertions are then immediate. If $n\geq 2$, blow up a point $p\in X$. The local description of the blowup of $\mathbb C^ n$ at the origin shows that $\widetilde X$ is smooth and that its exceptional divisor is $E\simeq \mathbb P^{n-1}$. A nested sequence of linear subspaces of $E$, together with $\widetilde X$, gives a flag

Each member is smooth, compact and connected, and is a prime divisor in the preceding member. The modification is an isomorphism off $E$.

Let $s$ be a section of $\pi ^*L^ m$. Off $E$, it gives a section of $L^ m$ on $X\setminus \{ p\} $. In a trivialization of $L^ m$ on a coordinate ball centered at $p$, its coefficient is holomorphic on the punctured ball, and by Hartogs' extension theorem, valid since $n\geq 2$, it extends holomorphically across $p$. The extension glues with the section away from $p$ to a section of $L^ m$ on $X$. Its pullback agrees with $s$ off $E$, hence everywhere by the identity theorem. This proves surjectivity; injectivity and compatibility with products follow because pullback is an isomorphism on the dense open set $\widetilde X\setminus E$. $\square$