Real symplectic formulation of local special geometry

We consider a formulation of local special geometry in terms of Darboux special coordinates PI=(pi,qi), I=1,…,2n. A general formula for the metric is obtained which is manifestly Sp(2n,R) covariant. Unlike the rigid case the metric is not given by the Hessian of the real function S(P) which is the Legendre transform of the imaginary part of the holomorphic prepotential. Rather it is given by an expression that contains S, its Hessian and the conjugate momenta SI=∂S∂PI. Only in the one-dimensional case (n=1) is the real (two-dimensional) metric proportional to the Hessian with an appropriate conformal factor.