Complex Symplectic Structure Research Articles

Complex symplectic Lie algebras with large Abelian subalgebras

We present two constructions of complex symplectic structures on Lie algebras with large Abelian ideals. In particular, we completely classify complex symplectic structures on almost Abelian Lie algebras. By considering compact quotients of their corresponding connected, simply connected Lie groups we obtain many examples of complex symplectic manifolds which do not carry (hyper)kähler metrics. We also produce examples of compact complex symplectic manifolds endowed with a fibration whose fibers are Lagrangian tori.

Symmetric and skew-symmetric complex structures

On a complex manifold (M,J), we interpret complex symplectic and pseudo-Kähler structures as symplectic forms with respect to which J is, respectively, symmetric and skew-symmetric. We classify complex symplectic structures on 4-dimensional Lie algebras. We develop a method for constructing hypersymplectic structures from the above data. This allows us to obtain two families of hypersymplectic structures on a 4-step nilmanifold.

Complex symplectic structures on Lie algebras

We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called complex symplectic oxidation , to construct certain complex symplectic Lie algebras of dimension 4 n + 4 from those of dimension 4 n . We specialize this construction to the nilpotent case and apply complex symplectic oxidation to obtain all eight-dimensional nilpotent complex symplectic Lie algebras.

Complex symplectic structures and the $$\partial {\bar{\partial }}$$ ∂ ∂ ¯ -lemma

In this paper we study complex symplectic manifolds, i.e., compact complex manifolds $X$ which admit a holomorphic $(2, 0)$-form $\sigma$ which is $d$-closed and non-degenerate, and in particular the Beauville-Bogomolov-Fujiki quadric $Q_\sigma$ associated to them. We will show that if X satisfies the $\partial \bar{\partial}$-lemma, then $Q_\sigma$ is smooth if and only if $h^{2,0}(X) = 1$ and is irreducible if and only if $h^{1,1}(X) > 0$.

A Thouless formula and Aubry duality for long-range Schrödinger skew-products

In this paper, we study the dynamical properties of a class of ergodic linear skew-products which includes the linear skew-products defined by quasi-periodic Schrödinger operators and their duals, in Aubry sense, when the potential is a trigonometric polynomial. Notably, these linear skew-products preserve an adapted complex-symplectic structure. We prove a Thouless formula relating the sum of the positive Lyapunov exponents and the logarithmic potential associated with the density of states of the corresponding operator. In particular, for quasi-periodic Schrödinger operators and their duals, we prove an identity for the upper Lyapunov exponent of the skew-product and the sum of the positive Lyapunov exponents of their dual, which generalizes the well-known formula for the Almost Mathieu. We illustrate these identities with some numerical illustrations.

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h is an integrable complex structure J with a closed non-degenerate ( 2 , 0 ) -form. It is determined by J and the real part Ω of the ( 2 , 0 ) -form. Suppose that h is a semi-direct product g ⋉ V , and both g and V are Lagrangian with respect to Ω and totally real with respect to J . This note shows that g ⋉ V is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω and J are isomorphic. The geometry of ( Ω , J ) on the semi-direct product g ⋉ V is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra g . By further exploring a relation between ( J , Ω ) with hypersymplectic Lie algebras, we find an inductive process to build families of complex symplectic algebras of dimension 8 n from the data of the 4 n -dimensional ones.

On the symplectic structure of instanton moduli spaces

We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU ( 2 ) instantons on both commutative and non-commutative R 4 . We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero–Moser spaces to the instanton moduli spaces.

K�hler Reduction of Metrics with Holonomy G 2

A torsion-free G_2 structure admitting an infinitesimal isometry is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to G_2.

A Fourier-Mukai approach to spectral data for instantons

We study SU(r) instantons on elliptic surfaces with a section and show that they are in one-one correspondence with spectral data consisting of a curve in the dual elliptic surface and a line bundle on that curve. We use relative Fourier-Mukai transforms to analyse their properties and, in the case of the K3 and abelian surface, we show that the moduli space of instantons has a natural Lagrangian fibration structure with respect to the canonical complex symplectic structures.

Complex Symplectic Geometry of Quasi-Fuchsian Space

We study the complex symplectic geometry of the space QF(S) of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1. We prove that QF(S) is a complex symplectic manifold. The complex symplectic structure is the complexification of the Weil–Petersson symplectic structure of Teichmuller space and is described in terms which look natural from the point of view of hyperbolic geometry.

Almost Complex Poisson Manifolds

In this paper we consider complex Poisson manifolds and extendthe concept of complex Poisson structure, due to Lichnerowicz to themore general concept of almost complex Poisson structures. Examples ofsuch structures and the associated generalized foliation are given.Moreover, some properties of the complex symplectic structures as wellas of the holomorphic complex Poisson structures are studied.

Real sector of the nonminimally coupled scalar field to self-dual gravity

A scalar field nonminimally coupled to gravity is studied in the canonical framework, using self-dual variables. The corresponding constraints are first class and polynomial. To identify the real sector of the theory, reality conditions are implemented as second class constraints, leading to three real configurational degrees of freedom per space point. Nevertheless, this realization makes nonpolynomial some of the constraints. The original complex symplectic structure reduces to the expected real one, by using the appropriate Dirac brackets. For the sake of preserving the simplicity of the constraints, an alternative method preventing the use of Dirac brackets, is discussed. It consists of converting all second class constraints into first class by adding extra variables. This strategy is implemented for the pure gravity case.

Real and complex symplectic structures. Application to concentration in degree for microfunctions at the boundary

Real and complex symplectic structures. Application to concentration in degree for microfunctions at the boundary