The relative symplectic cone and T²-fibrations

In this note we introduce the notion of the relative symplectic cone C_M^V. As an application, we determine the symplectic cone C_M of certain T²-fibrations. In particular, for some elliptic surfaces we verify the conjecture in [17]: If M underlies a minimal Kähler surface with p_g > 0, the symplectic cone C_M is equal to P^c1(M) ∪ P^-c(M), where P^α = {e ∈ H²(M;R)|e·e > 0 and e·α > 0} for nonzero α ∈ H²(M;R) and P⁰ = {e ∈ H²(M;R)|e·e > 0}.