Complex manifold

Abstract: A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi–Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action both of diffeomorphisms and closed 2-forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology. We introduce in this paper a geometrical structure on a manifold which generalizes both the concept of a Calabi–Yau manifold—a complex manifold with trivial canonical bundle—and that of a symplectic manifold. This is possibly a useful setting for the background geometry of recent developments in string theory; but this was not the original motivation for the author’s first encounter with this structure. It arose instead as part of a programme (following the papers [ 11, 12]) for characterizing special geometry in low dimensions by means of invariant functionals of differential forms. In this respect, the dimension six is particularly important. This paper has two aims, then: first to introduce the general concept, and then to look at the variational and moduli space problem in the special case of six dimensions. We begin with the definition in all dimensions of what we call generalized complex manifolds and generalized Calabi–Yau manifolds .

Abstract: Whether one studies the geometry or analysis in the complex number space C a + l , or more generally, in a complex manifold, one will have to deal with domains. Their boundaries are real hypersurfaces of real codimension one. In 1907, Poincar4 showed by, a heuristic argument tha t a real hypersurface in (38 has local invariants unde r biholomorphie transformations [6]. He also recognized the importance of the special uni tary group which acts on the real hyperquadrics (cf. w Following a remark by B. ~Segre, Elie :Cartan took, up again the problem. In t w o profound papers [1], he gave, among other results, a complete solution of the equivalence problem, tha t is, the problem of finding a complete system of analytic invariants for two real analytic real hypersurfaees in C~ to be locally equivalent under biholomorphic transformations. Let z 1, ..., z n+l be the coordinates in Cn+r We s tudy a real hypersurface M at the origin 0 defined by the equation

Abstract: A manifold is called a complex manifold if it can be covered by coordinate patches with complex coordinates in which the coordinates in overlapping patches are related by complex analytic transformations. On such a manifold scalar multiplication by i in the tangent space has an invariant meaning. An even dimensional 2n real manifold is called almost complex if there is a linear transformation J defined on the tangent space at every point (and varying differentiably with respect to local coordinates) whose square is minus the identity, i.e. if there is a real tensor field h' satisfying