Symplectic and Kähler structures on $$\mathbb CP^1$$-bundles over $$\mathbb CP^2$$

We show that there exist symplectic structures on a $$\mathbb {CP}^1$$ CP 1 -bundle over $$\mathbb {CP}^2$$ CP 2 that do not admit a compatible Kähler structure. These symplectic structures were originally constructed by Tolman and they have a Hamiltonian $${\mathbb {T}}^2$$ T 2 -symmetry. Tolman’s manifold was shown to be diffeomorphic to a $$\mathbb CP^1$$ C P 1 -bundle over $$\mathbb {CP}^{2}$$ CP 2 by Goertsches, Konstantis, and Zoller. The proof of our result relies on Mori theory, and on classical facts about holomorphic vector bundles over $$\mathbb {CP}^{2}$$ CP 2 .

Bosonic and fermionic Gaussian states from Kähler structures

We show that bosonic and fermionic Gaussian states (also known as ``squeezed coherent states’’) can be uniquely characterized by their linear complex structure JJ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple (G,\Omega,J)(G,Ω,J) of compatible Kähler structures, consisting of a positive definite metric GG, a symplectic form \OmegaΩ and a linear complex structure JJ with J^2=-\mathbb{1}J2=−1. Mixed Gaussian states can also be identified with such a triple, but with J^2\neq -\mathbb{1}J2≠−1. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.

Metallic Kähler and nearly metallic Kähler manifolds

In this paper, we construct metallic Kähler and nearly metallic Kähler structures on Riemannian manifolds. For such manifolds with these structures, we study curvature properties. Also, we describe linear connections on the manifold which preserve the associated fundamental 2-form and satisfy some additional conditions and present some results concerning them.

POWER SERIES PROOFS FOR LOCAL STABILITIES OF KÄHLER AND BALANCED STRUCTURES WITH MILD -LEMMA

By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$ -Bott–Chern numbers.

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.

Correction to: Special Kähler structures, cubic differentials and hyperbolic metrics

A correction to this paper has been published: https://doi.org/10.1007/s00029-021-00643-4

Some Properties of Curvature Tensors and Foliations of Locally Conformal Almost Kähler Manifolds

We investigate a class of locally conformal almost Kähler structures and prove that, under some conditions, this class is a subclass of almost Kähler structures. We show that a locally conformal almost Kähler manifold admits a canonical foliation whose leaves are hypersurfaces with the mean curvature vector field proportional to the Lee vector field. The geodesibility of the leaves is also characterized, and their minimality coincides with the incompressibility of the Lee vector field along the leaves.

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.