Special complex manifolds

In this memoire we present the theory of symplectic quotients and the interactions with the theory of kahler quotients. The starting point is the following natural question: in the presence of a symplectic action of a Lie group G on a symplectic manifold pM, ωq can one define in a coherent way the quotient of M by G (so that the result will have a natural symplectic structure)? Even if we assume that G is compact and the action is free, we see that in general M {G cannot be symplectic (for dimension reasons). In order to obtain a symplectic quotient, one needs a new ingredient: a moment map. The fundamental theorem of the theory states that, assuming that G acts freely and properly around the zero locus of the moment map, then the G-quotient of this zero locus has a natural structure of a symplectic manifold. We will solve in detail the existence and unicity problems for moment maps. The main tool needed here is Lie algebra cohomology. We will continue with explicit computations of moment maps and explicit descriptions of symplectic quotients. We will prove that many interesting manifolds (projective spaces, Grassman manifolds, flag manifolds) can be obtained as symplectic quotients. We will conclude with a general principle which emphasizes an interesting relation between symplectic geometry and complex geometry: Let X be a Kähler manifold, K be a compact Lie group and G " K CˆX Ñ X be a holomorphic action on X which restricts to a Hamiltonian symplectic action of K which is free around µ´1p0q. Then the corresponding symplectic quotient has a natural structure of complex manifold. We will illustrate this principle in the examples we study.

Let M be a 2m-dimensional topological manifold. A coordinate atlas {(U, φ U : U → C m )} is called holomorphic if the transition functions φ U • φ −1 V are holomorphic functions between subsets of C m ; in this case the coordinate charts φ U are called local holomorphic coordinates. The manifold M is called complex if it admits a holomorphic atlas. Two holomorphic atlases are called equivalent if their union is a holomorphic atlas. An equivalence class of holomorphic atlases on M is called a complex structure.

European Journal of Mathematics

Let M be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration π with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure (I, g, ω) and a symplectic flat connection ∇ such that the metric g is locally the Hessian of a function. We prove that any Lagrangian subvariety Z ⊂ M which intersects smooth fibers of π and smoothly projects to π(Z) is a toric fibration over its image π(Z) in B, and this image is also special Kähler. This answers a question of N. Hitchin related to Kapustin-Witten BBB/BAA duality.

Annals of Mathematics, 2011

Generalized complex geometry encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define and study generalized complex branes, which interpolate between flat bundles on Lagrangian submanifolds and holomorphic bundles on complex submanifolds.

Communications in Mathematical Physics, 2007

We study generalized Kähler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kähler four-manifolds for which the induced complex structures yield opposite orientations.

Geometriae Dedicata, 2008

In this paper we discuss relations between symplectic forms and smooth structures on closed manifolds. Our main motivation is the problem if there exist symplectic structures on exotic tori. This is a symplectic generalization of a problem posed by Benson and Gordon. We give a short proof of the (known) positive answer to the original question of Benson and Gordon that there are no Kähler structures on exotic tori. We survey also other related results which give an evidence for the conjecture that there are no symplectic structures on exotic tori.

Journal of Geometry and Physics, 2010

We study a class of complex structures on the generalized tangent bundle of a smooth manifold M endowed with a torsion free linear connection, ∇. We introduce the concept of ∇-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M induced by complex structures on the generalized tangent bundle of M.

2004

American Journal of Computational Mathematics, 2016

A charming feature of symplectic geometry is that it is at the crossroad of many other mathematical disciplines. In this article we review the basic notions with examples of symplectic structures and show the connections of symplectic geometry with the various branches of differential geometry using important theorems.

2009

We consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P. We assume that the general fiber is isomorphic to a product of two elliptic curves. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface. 1. Preliminaries First we define our main objects of study, irreducible symplectic manifolds or hyperKähler manifolds. Definition 1.1. A compact complex Kähler manifold X is called irreducible symplectic if it is simply connected and if H(X, ΩX) is spanned by an everywhere non-degenerate 2-form ω. Any holomorphic two-form σ induces a homomorphism TX → ΩX . The two-form is everywhere non-degenerate if and only if TX → ΩX is bijective. The last condition in the definition implies that h(X) = h(X) = 1 and KX ∼= OX , i.e., c1(X) = 0. Definition 1.2. A compact connected 4n-dimensional Riemannian manifold (M, g) is called irreducible hyper-Kä...