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We solve the problem of determining the fundamental degrees of freedomunderlying a generalized K\"ahler structure of symplectic type. For a usualK\"ahler structure, it is well-known that the geometry is determined by acomplex structure, a K\"ahler class, and the choice of a positive $(1,1)$-formin this class, which depends locally on only a single real-valued function: theK\"ahler potential. Such a description for generalized K\"ahler geometry hasbeen sought since it was discovered in 1984. We show that a generalizedK\"ahler structure of symplectic type is determined by a pair of holomorphicPoisson manifolds, a holomorphic symplectic Morita equivalence between them,and the choice of a positive Lagrangian brane bisection, which depends locallyon only a single real-valued function, which we call the generalized K\"ahlerpotential. Our solution draws upon, and specializes to, the many results in thephysics literature which solve the problem under the assumption (which we donot make) that the Poisson structures involved have constant rank. To solve theproblem we make use of, and generalize, two main tools: the first is the notionof symplectic Morita equivalence, developed by Weinstein and Xu to studyPoisson manifolds; the second is Donaldson's interpretation of a K\"ahlermetric as a real Lagrangian submanifold in a deformation of the holomorphiccotangent bundle.

Morita equivalence and the generalized K\"ahler potential