Let $\pi:\mathcal X\to B$ be a deformation of a compact complex manifold $X=\pi^{-1}(0)$, then for any $t\in B$, the first Chern class $c_1(X)=c_1(X_t)$?

I know the Chern class of a manifold depends on the complex structure, for example, the same diffeomorphism type of $\mathbb C^1$ with a different complex structure may have a different Chern class, but also the same differential manifold with different complex sturctures may have the same Chern class, for example, the deformations of a Calabi-Yau manifold have the same Chern class $c_1=0$ while the complex structures are different. So my question is: is the Chern class a deformation invariant?

topologicalinvariants? $\endgroup$