Complex Symplectic Manifolds 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.

Deformation quantization of moduli spaces of Higgs bundles on a Riemann surface with translation structure

Let X be a compact connected Riemann surface of genus g ≥ 1 equipped with a nonzero holomorphic 1-form. Let MX(r) denote the moduli space of semistable Higgs bundles on X of rank r and degree r(g − 1) + 1; it is a complex symplectic manifold. Using the translation structure on the open subset of X where the 1-form does not vanish, we construct a natural deformation quantization of a certain nonempty Zariski open subset of MX(r).

A finiteness theorem for holonomic DQ-modules on Poisson manifolds

On a complex symplectic manifold we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification (b) in the algebraic case. This extends our previous results in which the symplectic manifold was compact. The main tool is a finiteness theorem for R-constructible sheaves on a real analytic manifold in a non proper situation.

The codimension-three conjecture for holonomic DQ-modules

On a complex symplectic manifold, we prove that any holonomic DQ-module endowed with a lattice and defined outside of an analytic subvariety of codimension 3 of its support extends as an holonomic DQ-module. This is an analogue for DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen.

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 $h^{1,1}(X) > 0$.

Generalized quaternionic manifolds

We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein–Weyl space. In particular, on the product $$Z$$ of any complex symplectic manifold $$M$$ and the sphere, there exists a natural generalized complex structure, with respect to which $$Z$$ is the twistor space of $$M$$ .

On Quantization of Complex Symplectic Manifolds

Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X + 1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X.

Komplexe Analysis

The aim of this workshop was to discuss recent developments in several complex variables and complex geometry. Special emphasis was put on the interaction between model theory and the classification theory of complex manifolds. Other topics included Kähler geometry, foliations, complex symplectic manifolds and moduli theory.

On the Rozansky–Witten weight systems

Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we study these weight systems by using D(X), the derived category of coherent sheaves on X. The main idea (stated here a little imprecisely) is that D(X) is the category of modules over the shifted tangent sheaf, which is a Lie algebra object in D(X); the weight systems then arise from this Lie algebra in a standard way. The other main results are a description of the symmetric algebra, universal enveloping algebra, and Duflo isomorphism in this context, and the fact that a slight modification of D(X) has the structure of a braided ribbon category, which gives another way to look at the associated invariants of links. Our original motivation for this work was to try to gain insight into the Jacobi diagram algebras used in Vassiliev theory by looking at them in a new light, but there are other potential applications, in particular to the rigorous construction of the (1+1+1)-dimensional Rozansky-Witten TQFT, and to hyperkaehler geometry.

Classification of Deformation Quantization Algebroids on Complex Symplectic Manifolds

A (holomorphic) deformation quantization algebroid over a complex symplectic manifold X is a stack locally equivalent to the ring of WKB operators, that is, microdifferential operators with an extra central parameter τ . In this paper, we will show that the (holomorphic) deformation quantization algebroids endowed with an anti-involution are classified by H^2(X; k^*_X) , where k^∗ is a subgroup of the group of invertible series in ℂ\llbracket τ^{−1}\rrbracket . In the formal case, the analogous classification is given by H^2(X; ℂ_X)\llbracket ℏ\rrbracket ^{\mathrm{odd}} , where one sets ℏ = τ^{−1} .

Constructibility and duality for simple holonomic modules on complex symplectic manifolds

Consider a complex symplectic manifold and the algebroid stack of deformation-quantization. For two regular holonomic -modules (i=0,1) supported by smooth Lagrangian submanifolds, we prove that the complex is perverse over the field and dual to the complex .

On period maps that are open embeddings

For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric divisor. We explain that phenomenon and get our hands on the complementary divisor in terms of geometric data.

An Algebra of Deformation Quantization for Star-Exponentials on Complex Symplectic Manifolds

The cotangent bundle T * X to a complex manifold X is classically endowed with the sheaf of k-algebras $${\mathcal{W}_{T*X}}$$ of deformation quantization, where k := $${\mathcal{W}_{\{pt\}}}$$ is a subfield of $${\mathbb{C}[[\hbar, \hbar^{-1}]}$$ . Here, we construct a new sheaf of k-algebras $${\mathcal{W}^t_{T*X}}$$ which contains $${\mathcal{W}_{T*X}}$$ as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If P is any section of order zero of $${\mathcal{W}_{T*X}}$$ , we show that $${{\rm exp}(t\hbar^{-1} P)}$$ is well defined in $${\mathcal{W}^t_{T*X}}$$ .

Quantization of complex Lagrangian submanifolds

Let Λ be a smooth Lagrangian submanifold of a complex symplectic manifold X . We construct twisted simple holonomic modules along Λ in the stack of deformation-quantization modules on X .

On manifolds whose tangent bundle is big and 1-ample

A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample. The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ‘almost’ ample, the present result is yet another extension of the celebrated Mori paper ‘Projective manifolds with ample tangent bundles’ (Ann. of Math. 110 (1979) 593–606). The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results. 2000 Mathematics Subject Classification 14E30, 14J40, 14J45, 14J50.

Stacks of quantization-deformation modules on complex symplectic manifolds

On a complex symplectic manifold X, we construct the stack of quantization-deformation modules, that is, (twisted) modules of microdifferential operators with an extra central parameter τ, a substitute to the lack of homogeneity. We also quantize involutive submanifolds of contact manifolds.

The moduli space of complex Lagrangian submanifolds

Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\ahler metric of Cecotti, Ferrara and Girardello associated to a special K\ahler structure fits into the Legendre transform construction of Lindstr\om and Ro\v cek.

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.

Boothby–Wang fibrations on complex contact manifolds

Since the Boothby–Wang fibrations form an important class of real (strict) contact manifolds, we endeavor to find a similar type of fibration for complex contact manifolds. We use the dual symplectic structures found on complex-symplectic manifolds, in order to generate two related yet distinct Boothby–Wang fibrations. Then we show that the “Whitney Sum” of these S 1 -bundles is the object we are looking for. Examples follow.

On Guan's examples of simply connected non-Kahler compact complex manifolds

The article contains a construction of compact complex symplectic manifolds which are simply connected and don't admit Kahler structure. The manifolds obtained are essentially the same as were constructed by Guan, but the construction itself is geometrically more transparent.