This question is a follow-up to this recent question of mine:
Comparing notions of degree of vector bundle
In that question, the definition of the degree of a vector bundle is discussed — in particular, the complications of doing so when the base complex manifold has dimension larger than $1$.
In that question, the "degree" of a vector bundle $V$ on a $d$-dimensional complex projective variety $X\stackrel{\kappa}{\to}\mathbb P^n$ (where $d\leq n$) was defined to be the number
$\deg_\kappa(V)=c_1(\kappa^*\mathcal O(1))^{d-1}.c_1(V)$,
where $\mathcal O(1)$ is the hyperplane bundle on $\mathbb P^n$. The subscript in $\deg_\kappa$ is not usual notation, but for the purpose of this question, I want to emphasize that the degree here depends on the embedding $\kappa$.
If we have two embeddings $\kappa:X\to\mathbb P^n$ and $\iota:X\to\mathbb P^m$ (where necessarily $d\leq m,n$), is it true that $\deg_\kappa(V)>0$ if and only if $\deg_\iota(V)>0$? If so, why?
Also, since smooth projective varieties are Kähler, if we were to define the degree of $V$ using the Kähler class on $X$ instead of $\kappa$, would the degree defined using the Kähler class have the same sign as $\deg_\kappa(V)$?
Let $V = \mathcal{O}(1,-1)$ on $X = \mathbb{P}^1 \times \mathbb{P}^1$. Let the two embeddings be from the line bundles $\mathcal{O}(n,1)$ and $\mathcal{O}(1,n)$.
The degree of $V$ is $1-n$ for the first embedding and $n-1$ for the second. So, no.
I don't know the answer to your second question.