I'am reaidng complex manifold course : https://www.math.stonybrook.edu/~cschnell/pdf/notes/complex-manifolds.pdf,
p. 100 Kodaira embedding theorem :
I'm now trying to understand the underlined statements.'
Here, the author define the positive holomorphic line bundle as follows (p.92)
Def. A holomorphic line bundle $L\to M$ is positive if its first Chern class $c_1(L)$ can be represented by a closed (1,1)-form $\Omega$ whose associated Hermitian form is positive definite.
For the first underlined statement, why such large integer $l$ exists? I found associated question : https://mathoverflow.net/questions/37827/tensor-product-of-a-line-bundle-with-a-large-multiple-of-another-positive-line-b But any one can explain more in detail?
For the second statement, why $\pi^{*}(L^{l}\otimes K_{M}^{-1}) \otimes (\tilde{L}^{m}\otimes [-E])^n $ is positive? (Here $[-E]$ means the line bundle $\mathcal{O}_{\tilde{M}}(-E)$)
Q.2-1) Pull back of positive line bundle under the blow up is also positive?
Q.2-2) The tensor product of positive line bundles is also positive?
If so, why?
If furthure information is needed, I will upload continuosly
Can anyone helps?
I found a way to bypass our problem. Although $\pi^{*}(L^{l}\otimes K_{M}^{-1})$ may not be positive, the $P_k := \pi^{*}(L^k \otimes K^{-1}_M)\otimes [-E]^n$ in the above image is positive for $k>>0$ by the Heybrechts's Complex Geometry, Lemma 5.3.2 :
So the below argument(underlined statements) is not needed.