Qfyilyi

Assume we have a closed symplectic manifold $M$ which is the total space of a smooth fibration by half-dimensional tori. Can we infer that $M$ is the total space of a smooth fibration by Lagrangian tori?

This doesn't need to hold. For example, if one takes a $(T^4,omega)$ with a constant symplectic structure $omega$, in order for it to have a fibration by Lagrangian tori one should be able to find a homologically non-trivial $T^2subset T^4$ such that $int_{omega} T^2=0$ which is impossible for general $omega$.

One can also give counter-examples when a half-dimensional torus bundle exists on the symplectic manifold but no Lagrangian torus bundle exists for any symplectic structure on the manifold. To construct such an example consider $M^4$ that fibers over $B=T^2$ with a fiber $F=T^2$ such that the action of $mathbb Z^2=pi_1(B)$ on $H_1(F)=mathbb Z^2$ contains a hyperbolic element. Such a manifold is symplectic by Thurston, but it is not a total space of a Lagrangian torus fibration by the classification of such fibrations in dimension $4$.