I am a novice in the field of complex geometry. The following question seems to be so simple that no one have explained it in related papers.
In the famous paper---Siu, Yum-Tong Invariance of plurigenera. Invent. Math. 134 (1998) , the main theorem is as follows.
Let $\pi: X \rightarrow \Delta$ be a smooth projective family of compact complex manifolds parametrized by the open unit 1 -disk $\Delta$. Assume that the fibers $X_{t}=\pi^{-1}(t), t \in \Delta$, are of general type. Then for every positive integer $m$ the plurigenus $\operatorname{dim}_{\mathbf{C}} \Gamma\left(X_{t}, m K_{X_{t}}\right)$ is independent of $t \in \Delta$, where $K_{X_{t}}$ is the canonical line bundle of $X_{t}$.
Siu reduced this problem to an extension problem. He wrote: "The Main Theorem is therefore equivalent to the statement that for every $t \in \Delta$ and every positive integer $m$ every element of $\Gamma\left(X_{t}, m K_{X_{t}}\right)$ can be extended to an element of $\Gamma\left(X, m K_{X}\right)$."
Accoding to my understanding, by the Gruert upper-semicontinuity theorem, that for every $t \in \Delta$ and every positive integer $m$ every element of $\Gamma\left(X_{t}, m K_{X_{t}}\right)$ can be extended to an element of $\Gamma\left(X, m K_{X}\right)$ is equivalent to the lower semi-continuity of $\Gamma\left(X_{t}, m K_{X_{t}}\right)$ near $t=0$.
However I am unable to write it down step by step.
According to the definition of the lower semi-continuity and $\Gamma\left(X_{t}, m K_{X_{t}}\right)\in \mathbb{Z}$, we need to find a small positive number $\varepsilon$ such that $\Gamma\left(X_{0}, m K_{X_{0}}\right) \leq \Gamma\left(X_{t}, m K_{X_{t}}\right)$ for any $t<\varepsilon$.
So for a fixed small $t$, we want to establish an injective map $$\Gamma\left(X_{0}, m K_{X_{0}}\right) \to \Gamma\left(X_{t}, m K_{X_{t}}\right).$$
Via the extension theorem, a natural map seens to be the following. $$v: s\mapsto S\mapsto S|_{X_t},$$ where $s\in \Gamma\left(X_{0}, m K_{X_{0}}\right)$, S its extension obtained by the extension theorem.
However, I cannot prove the injectivity of the above map $v$. In fact, for any different sections $s_1$ and $s_2$, we can find some $\varepsilon$ (depends on $s_1$ and $s_2$) such that $v(s_1)\neq v(s_2)$ for $t\leq \varepsilon$. But the difficulty is that the choice of $\varepsilon$ depends on $s_1$ and $s_2$.
So we cannot compare the dimension of $\Gamma\left(X_{0}, m K_{X_{0}}\right)$ with $\Gamma\left(X_{t}, m K_{X_{t}}\right)$ for a fixed $t$.
My question: How to prove the injectivity of the above map $v$. That is to say, how to compare the dimension of $\Gamma\left(X_{0}, m K_{X_{0}}\right)$ with $\Gamma\left(X_{t}, m K_{X_{t}}\right)$ for a fixed $t$.
Any comments will be appreciated.
Siu's extension theorem tells us that the restriction map $\Gamma(X, m K_{X})\to\Gamma(X_{0}, m K_{X_{0}})$ is surjective. This implies that the complex vector space $\Gamma(X_{0}, m K_{X_{0}})$ is isomorphic to $\pi_*\mathcal{O}(mK_X)/\mathfrak{m}_0\cdot\pi_*\mathcal{O}(mK_X)$, where $\mathfrak{m}_0$ is the defining ideal of $0\in\Delta$. Then (by, e.g., Section III.12 of Hartshorne's book) the sheaf $\pi_*\mathcal{O}(mK_X)$ is locally free in a neighborhood $U$ of $0$, and $\Gamma(X_t, m K_{X_t})\cong \pi_*\mathcal{O}(mK_X)/\mathfrak{m}_t\cdot\pi_*\mathcal{O}(mK_X)$ have the same dimension for all $t\in U$.