When the pull-back of positive line bundle by holomorphic map is also positive?

Question

I'm finding condition under which the pullback of positive line bundle by holomorphic map is also positive.

I'm reading the Daniel Huybrechts's Complex Geometry. In his book p.239, he define a holomorphic line bundle over a compact Kahler manifold as positive if its first Chern class $c_1(L) \in H^2(X,\mathbb{R})$ can be represented by a closed positive real (1,1)-form. And in the same page, he argues that a line bundle $L$ is positive if and only if it admits an hermitian structure such that the curvature of the induced Chern connection is positive in the sense of Definition 4.3.15 (his book p.189)

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And as the underliend statement, he argues that in the hermitian holomorphic line bundle case, the curvature $F_{\nabla}$ is positive if and only if the real (1,1)-form $iF_{\nabla}$ is positive in the sense of Definition 4.3.14 :

Now let $f:M \to N$ be a holomorphic map of compact, Kahler manifolds. And let $L \to N$ be a positive (holomorphic) line bundle on $N$. Then as mentioned above, there exists a hermition structure $h$ on $L$ such that the induced curvature $ \alpha := iF_{\nabla_{h}}$ satisfies the definition 4.3.14.

Then we can consider the pull-back bundle with pull-back hermitian structure $(f^{*}L, f^{*}h)$ :

And we can consider its induced Chern connection $\nabla_{f^{*}h}$. So to show that $f^{*}L$ is positive on $M$, it suffices to show that the induced curvature $iF_{\nabla_{f^{*}h}}$ satisfies the Definition 4.3.14.

On the other hand, the author defines the pull-back connection $f^{*}\nabla_{h}$ as follows :

And his book, p.184, Proposition 4.3.7 - (iv), he states that

$$ F_{f^{*} \nabla_{h}} = f^{*}F_{\nabla_{h}}$$

Q. Then my question is, for the $f : M \to N$, when

1. the pushforward $f_{*,x} : T_{x}M \to T_{f(x)}N$ is injective for all $x\in M$
2. $\nabla_{f^{*}h} = f^{*}\nabla_{h}$

?

If $f$ satisfies these conditions, then we can show that the $iF_{\nabla_{f^{*}h}}$ satisfies the Definition 4.3.14 as follows :

Fix $x \in M$ and non-trivial holomorphic tangent vectors $0 \neq v \in T^{1,0}M$. Then

$ -i(iF_{\nabla_{f^{*}h}})(v. \bar{v})(x) = F_{\nabla_{f^{*}h_{,x}}}(v_x, \bar{v}_{x}) = (F_{f^{*}\nabla_{h}})_{,x}(v_x,\bar{v}_x) = (f^{*}F_{\nabla_{h}})_{,x}(v_x,\bar{v}_x) = (F_{\nabla_{h}})_{,f(x)}(f_{*,x}v_x , f_{*,x}\bar{v}_{x}) > 0 $

Q. Under what condition for $f : M \to N$ the 1) and 2) are satisfied?

Q. If $f$ is of the form $\pi : \tilde{M} := Bl_{x}(M) \to M$ (blowing up at one point), then the above 1), 2) are satisfied ?

(This question is originated from the question about Kodaira embedding theorem I uploaded before : In the proof of the Kodaira Embedding theorem)

When the pull-back of positive line bundle is also positive?

Can anyone helps?