I have some questions regarding the proof of the following theorem.
Let $X$ be a projective manifold and $L$ a line bundle on $X$. Then $L$ is ample if and only if for all coherent sheaves $\mathcal{F}$ on $X$ there is an $m_0$ such that for all $m \geq m_0$ and for all $q \geq 1$ one has $H^q(X, \mathcal{F} \otimes L^{\otimes m}) = 0$.
My questions are about the direction $\Leftarrow$, i.e. one wants to show that $L$ is ample under the given conditions.
- The proof starts with
Choose $m$ such that for all $x,y \in X$ the following cohomologies vanish: $H^1(X, \mathfrak{m}_x \otimes L^{\otimes m})$, $H^1(X, \mathfrak{m}_{x,y}\otimes L^{\otimes m})$, $H^1(X, \mathfrak{m}_x^2\otimes L^{\otimes m})$.
- Later in the proof, one gets a holomorphic map $\Phi \colon X \to \mathbb{P}^N$ defined by the sections of $L^{\otimes m}$. The vanishing of $H^1(X, \mathfrak{m}_{x,y}\otimes L^{\otimes m})$ guarantees that $\Phi$ is injective. So far so good. Then the following is stated in the proof without further explanations:
The vanishing of $H^1(X, \mathfrak{m}_{x}^2 \otimes L^{\otimes m})$ gives an embedding.
I think, it is meant that the vanishing gives the closure of the embedding. Why is it like that?
Edit: I found the answer to my second question in Huybrechts, Complex Geometry: an introduction, Springer (2004). Universitext. 5.3. iii), p. 247-248. However, the question 1. is still a question for me.
