I'm reading the Daniel Huybrechts's Complex Geometry, p.249, Lemma 5.3.2. It is used to prove the Kodaira embedding theorem. Accepting the Lemma as true, I somewhat understand the Kodaira embedding theorem. And I want to understand the Lemma desperately. And through process of trial of understanding, I feel that I am not familier to language of complex geometry (so please understanding) :
I'm mainly trying to understand the underlined statements.
Q.1. How can we use the partition of unity? And why locally near any $E_j$ the curvature $F_{\nabla}$ is of the form $-n_j(2\pi i)p_j^{*}w_{\operatorname{FS}}$ ?
Q.2. Second underlined statement : What " $F_{\nabla}$ semi-positive locally around each $E_j$ and strictly positive for all tangent directions of $E_j$ itself" does exactly means? And why?
Q.3. Finally, why the last underlined statement, "the form $\sigma^{*}(k \alpha + \beta) + (i/2\pi)F_{\nabla}$ is a positive form on $\hat{X}$ for $k \gg 0$" , is true?
Can anyone presents explanation more in detail?
Thanks for reading.
