University lecture series· 2021· Book chapter
Generalized Complex Geometry
- 656citations
- 2021year
Short summary
Generalized complex geometry, encompassing complex and symplectic geometries, exhibits novel phenomena like B-field actions and applies to manifolds lacking standard complex/symplectic structures.
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Key points
- Generalized complex geometry naturally incorporates B-field actions and applies to manifolds lacking standard complex or symplectic structures.
- A generalized Darboux theorem provides a local normal form for this geometry.
- Generalized Kähler geometry is shown to be equivalent to a bi-Hermitian geometry with torsion.
- Generalized complex submanifolds correspond to D-branes in string theory.
AI-generated from the title and abstract; the full text is not read.
Abstract
Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.
The authors' abstract, as published at the source. University lecture series, 2021 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics