Morita equivalence and the generalized Kähler potential

Abstract

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

Non-Kähler Calabi-Yau geometry and pluriclosed flow

“…In particular, if we fix G a compact Lie group with bi-invariant metric g, and let I and J denote leftand right-invariant complex structures compatible with g, then the triple (g, I, J) is generalized Kähler [36]. Furthermore, there always exist infinite dimensional families of local deformations of a given generalized Kähler structure using natural classes of Hamiltonian flows [10,33]. These families are analogous to Kähler classes, and it is natural to seek classification of generalized Kähler structures up to equivalence via these deformations.…”

Generalized Ricci Flow

“…Lemma 7.77 shows that a generalized Kähler metric is determined by a local potential function in the special case [I, J] = 0. These local potentials conjecturally exist in full generality, with suggestive results appearing in [17,129] dealing with most cases. 7.4.5.…”

“…We next construct a natural class of deformations using Ω-Hamiltonian diffeomorphisms first appearing in [6] (cf. [16,17,85,91,102] for further developments), indicating a fundamental difference between generalized Kähler and classical Kähler geometry, namely that the basic deformations occur in a nonlinear space. The starting point is to reduce the construction of nondegenerate generalized Kähler structures purely in terms of the holomorphic symplectic structures.…”

Generalized Kähler-Ricci flow on toric Fano varieties

“…In [46,11], the so-called flow deformation of a symplectic type generalized Kähler structure is introduced, generalizing the Joyce deformations in the non-degenerate case [5]. Let (g, I, J) be a generalized Kähler structure of symplectic type on M , with corresponding real Poisson tensor σ = [I, J]g −1 , and f ∈ C ∞ (M ) a smooth function on M .…”

“…In §4 we discuss toric generalized Kähler structures, unify the discussions of [15] and [85], [86], and explicitly describe the relevant associated Poisson tensors. This leads to a derivation of the underlying Kähler metric and the associated scalar potentials, and we derive the relationship of these scalar potentials to the general Hamiltonian flow constructions of [5,46,11]. Also we are able to derive a potential function in this setting for the Bismut-Ricci curvature.…”