Topic

# Hyperkähler manifold

About: Hyperkähler manifold is a research topic. Over the lifetime, 587 publications have been published within this topic receiving 17977 citations. The topic is also known as: hyper-Kähler manifold.

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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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01 Jan 2000

TL;DR: The first known examples of these manifolds were discovered by the author in 1993-5 as mentioned in this paper, and much previously unpublished material which significantly improves the original constructions was presented in this book.

Abstract: The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kahler geometry. Then the Calabi conjecture is proved and used to deduce the existence of compact manifolds with holonomy SU(m) (Calabi-Yau manifolds) and Sp(m) (hyperkahler manifolds). These are constructed and studied using complex algebraic geometry. The second half of the book is devoted to constructions of compact 7- and 8-manifolds with the exceptional holonomy groups 92 and Spin(7). Many new examples are given, and their Betti numbers calculated. The first known examples of these manifolds were discovered by the author in 1993-5. This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.

1,181 citations

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TL;DR: In this paper, a projective algebraic manifold M is a complex manifold in certain projective space CP, N > dim c M = n, and the hyperplane line bundle of CP restricts to an ample line bundle L on M. This bundle L is a polarization on M, and it can be associated with a positive, rf-closed (1, l)-form ωg.

Abstract: A projective algebraic manifold M is a complex manifold in certain projective space CP, N > dim c M = n . The hyperplane line bundle of CP restricts to an ample line bundle L on M. This bundle L is a polarization on M. For the Kahler metric g on M, we can associate a positive, rf-closed (1, l)-form ωg . In any local coordinate system (Zj, , zn) of M , the metric g is expressed by a tensor (^/j)1 0, the hermitian metric h induces a hermitian metric h on L m . Choose an orthonormal basis {STM9 ,S n N}of the space H°{M, L ) of all holomorphic global sec-

652 citations