Invariant Hodge forms and equivariant splittings of algebraic manifolds

Let T be a complex torus acting holomorphically on a compact algebraic manifold M and let $ev_∗ :π₁(T,1) → π₁(M,x₀)$ be the homomorphism induced by $ev: T ∋ t ↦ tx₀ ∈ M. We show that for each T-invariant Hodge form Ω on M there is a holomorphic fibration p:M → T whose fibers are Ω-perpendicular to the orbits. Using this we prove that M is T-equivariantly biholomorphic to T × M/T if and only if there is a subgroup Δ of π₁(M) and a Hodge form Ω on M such that $π₁(M) = im ev_∗ × Δ$ and $∫_{β×δ} Ω = 0$ for all $β ∈ im ev_∗$ and δ ∈ Δ.