We solve the problem of determining the fundamental degrees of freedom underlying a generalized Kähler structure of symplectic type. For a usual Kähler structure, it is well-known that the geometry is determined by a complex structure, a Kähler class, and the choice of a positive (1, 1)-form in this class, which depends locally on only a single real-valued function: the Kähler potential. Such a description for generalized Kähler geometry has been sought since it was discovered in 1984. We show that a generalized Kähler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on only a single real-valued function, which we call the generalized Kähler potential. Our solution draws upon, and specializes to, the many results in the physics literature which solve the problem under the assumption (which we do not make) that the Poisson structures involved have constant rank.To solve the problem we make use of, and generalize, two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds; the second is Donaldson's interpretation of a Kähler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle. 1 Generalized Kähler structures of symplectic type 4 2 Introduction to the problem: the Kähler case 8 3 Holomorphic symplectic Morita equivalence 9 4 Generalized Kähler metrics as brane bisections 13 5 The generalized Kähler potential 17 6 The Picard group 22 7 Generalized Kähler metrics via Hamiltonian flows 24 8 Universal local construction via time-dependent flows 27

Given a manifold M with a submanifold N , the deformation space D(M, N ) is a manifold with a submersion to R whose zero fiber is the normal bundle ν(M, N ), and all other fibers are equal to M . This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with N a submanifold transverse to the foliation. New examples include L∞-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around N , in terms of a model structure over ν(M, N ).IMPA, Estrada Dona Castorina 110, Rio de Janeiro,

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