Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold. Suppose $Y$ is a conifold transition on a single nullhomologous Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to H^{2}(X)\rightarrow H^{2}(Y)\rightarrow \mathbb{Z}\to0$ and $c_{1}(X)=c_{1}(Y)$ under the inclusion. Is the square $c_{1}^{2}=0$ of the first Chern class preserved by the conifold transition? Why?
I suppose this is not always true. My reason is as follows: Let $X_{0}=X\backslash S^{3}$. Then there is an isomorphism $H^{4}(X)\rightarrow H^{4}(X_{0})$ and a exact sequence $$0\to\mathbb{Z}\rightarrow H^{4}(Y)\rightarrow H^{4}(X_{0})\to0$$ So there may be some element $z\in\mathbb{Z}\subset H^{4}(Y)$ such that $c_{1}^{2}(Y)=z$ but $c_{1}^{2}(X)=c_{1}^{2}(X_{0})=0$. Is that right? Thank you.