I have a proper map of complex manifolds $$f: X \to D,$$ where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ is a complex torus. This induces a monodromy representation on homology $$\mu: H_1(F, \mathbb C) \xrightarrow{\cong} H_1(F, \mathbb C).$$ Using a Mayer-Vietoris sequence, one obtains a short exact sequence $$0 \to H_2(F, \mathbb C) \xrightarrow{i_*} H_2(X \setminus F_0, \mathbb C) \xrightarrow{\delta} H_1(F, \mathbb C)^\mu \to 0.$$ Here $F_0 = f^{-1}(0)$ is the central fiber, and $H_1(F, \mathbb C)^\mu$ is the subgroup of $\mu$-invariant classes. Given such a $[\sigma] \in H_1(F)^\mu$, which is represented by a cycle $\sigma$, I can find a preimage in $H_2(X \setminus F_0)$ in the following way:

Choose an $S^1 \subset D \setminus 0$ which has winding number $1$. Cover it by open subsets over which $f$ becomes locally trivial (in the $C^\infty$-sense, using Ehresmann's theorem). Then extend $\sigma$ over the products, and obtain a 2-chain $\alpha$ which lives over $S^1$. When coming back to $F$, this produces a 1-cycle $\mu(\sigma)$, which is homologous to $\sigma$, as $[\sigma]$ is $\mu$-invariant. So $\sigma - \mu(\sigma) = \partial \beta$ for some 2-cycle $\beta$ on $F$. Then $\alpha + \beta$ is closed and $\delta([\alpha + \beta]) = [\sigma]$ by construction.

My question is: Is the cycle $\alpha + \beta$ a boundary in $X$? If not in general, is it possible to create a preimage of $[\sigma]$ which vanishes in $H_2(X)$?

I think this works for $H_1$ in case of $H_2$, because one can take a disk $D' \subset X$ which intersects the reduction of $F_0$ transversally. The boundary of $D'$ then pushes-down to a multiple of the generator of $H_1(D \setminus 0)$. But I'm not sure how to make a similar argument precise for $H_2$.

Dually, one might ask: Given a closed differential 2-form $\omega$ on $X$ such that $\omega|_F$ is exact, is $\omega|_{X \setminus F_0}$ also exact?