"Moreover, a holomorphically immersed submanifold $f:M\to P^m(C)$(i.e. $m$ dimensional projective space over $C$) has an induced bundle $f^\star(H)$($H$ is hyperplane section bundle over $P^m$ or dual of universal line bundle of $P^m$) called the hyperplane section bundle of $M$."
$\textbf{Q:}$ Why do I want immersed submanifold here? Shouldn't I want embedding? In other words, I want projectively embedding of varieties? I could not see a good reason that "immersed" is a good notion to study algebraic objects. In particular, this section also involves showing all $H^{1,1}$((1,1) component of Hodge decomposition of $H^2(M)$) comes from holomorphic line bundles. This is basically showing for $M$ algebraic, all degree 2's (1,1) components are all alegbraic cycles. What is the purpose of demanding immersed here? The book did not assume $M$ compact or projective.
Ref. Chern, Complex Manifolds without Potential Theory section 6, pg 55 right before section 7.