Morita equivalence and the generalized Kähler potential

Bischoff, Francis; Gualtieri, Marco; Zabzine, Maxim

doi:10.4310/jdg/1659987891

Cited by 2 publications

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“…Formal manifold structure. According to [9,40], the space GK π is parametrized by the closed 2-forms F which tame J and satisfy the algebraic (zero order) identity (2.23)…”

Section: )mentioning

confidence: 99%

“…In the Kähler case, using the dd c J -lemma, the space K α is a convex-linear, Fréchet manifold modeled on C ∞ (M, R) R. Such a description is no longer globally possible in the GK case due to the integrability condition on I (cf. [9,56] for results on local generalized Kähler potentials). Nonetheless, a natural notion of generalized Kähler class has now emerged [9,29,42], which here consists of GK structures defined by deforming an element F 0 ∈ GK π,α by a smooth path of functions φ t ∈ C ∞ (M, R) R in the following nonlinear way [42]:…”

mentioning

confidence: 99%

“…[9,56] for results on local generalized Kähler potentials). Nonetheless, a natural notion of generalized Kähler class has now emerged [9,29,42], which here consists of GK structures defined by deforming an element F 0 ∈ GK π,α by a smooth path of functions φ t ∈ C ∞ (M, R) R in the following nonlinear way [42]:…”

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confidence: 99%

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The Riemannian and symplectic geometry of the space of generalized Kähler structures

Apostolov¹,

Streets²,

Ustinovskiy³

2023

Preprint

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On a compact complex manifold (M, J) endowed with a holomorphic Poisson tensor π J and a deRham class α ∈ H 2 (M, R), we study the space of generalized Kähler (GK) structures defined by a symplectic form F ∈ α and whose holomorphic Poisson tensor is π J . We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger [10] and Goto [36] to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki-Mabuchi extremal vector field [25] and Calabi-Lichnerowicz-Matsushima result [12,55,58] for the Lie algebra of the group of automorphisms of (M, J, π J ). We define a closed 1-form on a GK class, which yields a generalization of the Mabuchi energy and thus a variational characterization of GK structures of constant scalar curvature. Next we introduce a formal Riemannian metric on a given GK class, generalizing the fundamental construction of 60,20]. We show that this metric has nonpositive sectional curvature, and that the Mabuchi energy is convex along geodesics, leading to a conditional uniqueness result for constant scalar curvature GK structures. We finally examine the toric case, proving the uniqueness of extremal generalized Kähler structures and showing that their existence is obstructed by the uniform relative K-stability of the corresponding Delzant polytope. Using the resolution of the Yau-Tian-Donaldson conjecture in the toric case by Chen-Cheng [16] and He [45], we show in some settings that this condition suffices for existence and thus construct new examples.

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“…Formal manifold structure. According to [9,40], the space GK π is parametrized by the closed 2-forms F which tame J and satisfy the algebraic (zero order) identity (2.23)…”

Section: )mentioning

confidence: 99%

mentioning

confidence: 99%

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The Riemannian and symplectic geometry of the space of generalized Kähler structures

Apostolov¹,

Streets²,

Ustinovskiy³

2023

Preprint

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N = ( 2 , 2 ) superfields and geometry revisited

Hull,

Zabzine

2024

J. Phys. A: Math. Theor.

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We take a fresh look at the relation between generalised Kähler geometry and N = (2, 2) supersymmetric sigma models in two dimensions formulated in terms of (2, 2) superﬁelds. Dual formulations in terms of diﬀerent kinds of superﬁeld are combined to give a formulation with a doubled target space and both the original superﬁeld and the dual superﬁeld. For Kähler geometry, we show that this doubled geometry is Donaldson’s deformation of the holomorphic cotangent bundle of the original Kähler manifold. This doubled formulation gives an elegant geometric reformulation of the equations of motion. We interpret the equations of motion as the intersection of two Lagrangian submanifolds (or of a Lagrangian submanifold with an isotropic one) in the inﬁnite dimensional symplectic supermanifold which is the analogue of phase space. We then consider further extensions of this formalism, including one in which the geometry is quadrupled, and discuss their geometry.

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Morita equivalence and the generalized Kähler potential

Cited by 2 publications

References 0 publications

The Riemannian and symplectic geometry of the space of generalized Kähler structures

The Riemannian and symplectic geometry of the space of generalized Kähler structures

N = ( 2 , 2 ) superfields and geometry revisited

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