Suppose $X$ is a projective variety over $\mathbb C$ and $T$ an irreducible projective $\mathbb C$-scheme. Let $a,b$ be closed points of $T$.

If we have a flat morphism $X\times T\to T$ such that the fibres $A,B$ over $a,b$ are subvarieties of $X$. Is it necessarily true that $A,B$ are rationally equivalent? If not, can we at least make sure that $A,B$ have the same cohomology class (perhaps with coefficients in $\mathbb Q$ or $\mathbb C$)?

I believe this is true if $T$ is linearly connected (in the sense of [1]). It also makes sense since supposedly $A,B$ ``deforms'' to each other. But is it actually true in general?

[1] Hartshorne, R. Connectedness of the Hilbert scheme. Publications Mathématiques de L’Institut des Hautes Scientifiques 29, 7-48 (1966). https://doi.org/10.1007/BF02684803

Let $X$ be a projective variety over $\mathbb C$ and $T$ an irreducible projective $\mathbb C$-scheme. Let $a,b$ be closed points of $T$.

Suppose we have a flat family $Z\to X\times T\to T$ such that its fibres $A,B$ over $a,b$ are subvarieties of $X$. Is it necessarily true that $A,B$ are rationally equivalent? If not, can we at least make sure that $A,B$ have the same cohomology class (perhaps with coefficients in $\mathbb Q$ or $\mathbb C$)?

I believe this is true if $T$ is linearly connected (in the sense of [1]). It also makes sense since supposedly $A,B$ "deforms" to each other. But is it actually true in general?

[1] Hartshorne, R. Connectedness of the Hilbert scheme. Publications Mathématiques de L’Institut des Hautes Scientifiques 29, 7-48 (1966). https://doi.org/10.1007/BF02684803