Author
Alfred Gray
Other affiliations: University of Zurich, University of Santiago de Compostela, University of the Basque Country
Bio: Alfred Gray is an academic researcher from University of Maryland, College Park. The author has contributed to research in topics: Symplectic geometry & Complex manifold. The author has an hindex of 15, co-authored 28 publications receiving 1171 citations. Previous affiliations of Alfred Gray include University of Zurich & University of Santiago de Compostela.
Papers
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261 citations
216 citations
TL;DR: In this paper, a special class of compact complex nilmanifolds with nilpotent complex structure is considered, called compact compact complex (CCN) with compact complex structure (CCS) and it is shown that the Dolbeault cohomology H ∗, ∗ ∂̄ (G) is canonically isomorphic to the ∆-cohomology H∗,∗ ∆ ∆ (gC) of the bigraded complex (G, ∆) of complex valued left invariant differential forms.
Abstract: We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if Γ\\G is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology H∗,∗ ∂̄ (Γ\\G) is canonically isomorphic to the ∂̄–cohomology H∗,∗ ∂̄ (gC) of the bigraded complex (Λ∗,∗(gC)∗, ∂̄) of complex valued left invariant differential forms on the nilpotent Lie group G.
150 citations
TL;DR: On decrit plusieurs familles de varietes symplectiques compactes qui generalisent la variete de Kodaira-Thurston as mentioned in this paper.
95 citations
01 Apr 1988
TL;DR: In this article, des exemples de varietes symplectiques compactes sans structure complexe and/ou de structures de Kahler are presented. But they do not mention any structure complex structures.
Abstract: On presente des exemples de varietes symplectiques compactes sans structure complexe et/ou de structures de Kahler
86 citations
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TL;DR: In this paper, it was shown that sixteen classes of almost Hermitian manifolds can be found in the Euclidean space, and that they are Hermitians in a natural way.
Abstract: It is shown that in a natural way there are precisely sixteen classes of almost Hermitian manifolds.
823 citations
714 citations
Book•
17 Jul 2001TL;DR: Symplectic Toric Manifolds as mentioned in this paper are a type of complex manifold that is composed of two or more complex manifold types, and they can be classified into three classes: symplectic, symplectomorphisms, and symmlectic reduction.
Abstract: Symplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kahler Manifolds.- Complex Manifolds.- Kahler Forms.- Compact Kahler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.
538 citations
TL;DR: In this paper, the authors classify real six-dimensional nilpotent Lie algebras for which the corresponding Lie group has a left-invariant complex structure, and estimate the dimensions of moduli spaces of such structures.
348 citations