Topic

# Complex manifold

About: Complex manifold is a research topic. Over the lifetime, 2842 publications have been published within this topic receiving 55245 citations.

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TL;DR: In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).

Abstract: Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.

2,336 citations

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TL;DR: A geometrical structure on even-dimensional manifolds is defined in this paper, which generalizes the notion of a Calabi-Yau manifold and also a symplectic 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 .

1,275 citations

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TL;DR: In this paper, Cartan gave a complete solution of the equivalence problem, which is, among other results, the problem of finding a complete system of analytic invariants for two real analytic real hypersurfaees in Cn+l to be locally equivalent under biholomorphic transformations.

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

1,004 citations

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01 Jan 1980

TL;DR: In this paper, the authors present an introduction to the basics of analysis and geometry on compact complex manifolds and provide tools which are the building blocks of many mathematical developments over the past 30 years.

Abstract: * Presents a concise introduction to the basics of analysis and geometry on compact complex manifolds
* Provides tools which are the building blocks of many mathematical developments over the past 30 years
* The new edition contains a 40 page appendix which updates the text for the modern reader
* Includes exercises and examples which are ideal for use in a classroom setting
In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations Subsequent chapters then develop such topics as Hermitian exterior algebra and the Hodge *-operator, harmonic theory on compact manifolds, differential operators on a Kahler manifold, the Hodge decomposition theorem on compact Kahler manifolds, the Hodge-Riemann bilinear relations on Kahler manifolds, Griffiths's period mapping, quadratic transformations, and Kodaira's vanishing and embedding theorems
The third edition of this standard reference contains a new appendix by Oscar Garcia-Prada which gives an overview of certain developments in the field during the decades since the book first appeared
From reviews of the 2nd Edition:
"the new edition of Professor Wells' book is timely and welcomean excellent introduction for any mathematician who suspects that complex manifold techniques may be relevant to his work"
- Nigel Hitchin, Bulletin of the London Mathematical Society
"Its purpose is to present the basics of analysis and geometry on compact complex manifolds, and is already one of the standard sources for this material"
- Daniel M Burns, Jr, Mathematical Reviews

997 citations