The setting is the following: $f: X \rightarrow T$ is a smooth projective map of complex algebraic varieties, and $L$ is a line bundle on $X$.
My question is the following: is $c_1(L_t)$ (in cohomology) independent on $t \in T$? In the example I am looking at (semiample deformations of ample line bundles, Positivity in Algebraic Geometry, by Lazarsfeld) there is this senstence: "On the other hand, the homomorphism given by cup product with $c_1(L_t)^b$, being defined over $\mathbb{Z}$ and topologically determined, are independent of $t \in T$.".
The goal of the example I am working out is to conclude that the map "capping with $c_1(L_0)^b$" ($X_0$ is a special fibre) is injective (on a certain cohomology group) since it holds true for the fibres $X_t$, $t \neq 0$. So, maybe the question might not make complete sense beyond what is explicitly needed.
What I have guessed so far is that my map gives me a complex analytic family, and hence the fibres are diffeomorphic as real manifolds. So it makes sense to identify their cohomologies. The statement about the line bundle seem very intuitive, but nevertheless I have no clue how to see the claim above.
Both full answers and references are welcome!