The nondegenerate generalized Kähler Calabi–Yau problem

We formulate a Calabi–Yau-type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri’s Calabi–Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kähler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeroes of a moment map associated to a Hamiltonian action and finding a Kempf–Ness functional. Lastly we indicate the naturality of generalized Kähler–Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf–Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kähler metrics. On a hyper-Kähler background, we establish global existence and weak convergence of the flow.

Deformation classes in generalized Kähler geometry

We describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.

Pluriclosed flow on generalized Kähler manifolds with split tangent bundle

We show that the pluriclosed flow preserves generalized Kähler structures with the extra condition [J_{+},J_{-}]=0 , a condition referred to as “split tangent bundle.” Moreover, we show that in this case the flow reduces to a nonconvex fully nonlinear parabolic flow of a scalar potential function. We prove a number of a priori estimates for this equation, including a general estimate in dimension n=2 of Evans–Krylov type requiring a new argument due to the nonconvexity of the equation. The main result is a long-time existence theorem for the flow in dimension n=2 , covering most cases. We also show that the pluriclosed flow represents the parabolic analogue to an elliptic problem which is a very natural generalization of the Calabi conjecture to the setting of generalized Kähler geometry with split tangent bundle.

Generalized Kähler geometry and current algebras in classical N=2 superconformal WZW model

I examine the Generalized Kähler (GK) geometry of classical [Formula: see text] superconformal WZW model on a compact group and relate the right-moving and left-moving Kac–Moody superalgebra currents to the GK geometry data using biholomorphic gerbe formulation and Hamiltonian formalism. It is shown that the canonical Poisson homogeneous space structure induced by the GK geometry of the group manifold is crucial to provide [Formula: see text] superconformal [Formula: see text]-model with the Kac–Moody superalgebra symmetries. Then, the biholomorphic gerbe geometry is used to prove that Kac–Moody superalgebra currents are globally defined.

Generalized Kähler Geometry and current algebras in SU(2) × U(1) N = 2 superconformal WZW model

We examine the Generalized Kähler Geometry (GKG) of quantum N = 2 superconformal WZW model on SU(2) × U(1) and relate the right-moving and left-moving Kac–Moody superalgebra currents to the GKG data of the group manifold using Hamiltonian formalism.