Geometric structures on momentangle manifolds
Abstract
A momentangle complex \mathscr{Z}_{\mathscr{K}} is a cell complex with a torus action constructed from a finite simplicial complex {\mathscr{K}}. When this construction is applied to a triangulated sphere {\mathscr{K}} or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Momentangle manifolds and complexes are central objects in toric topology, and currently are gaining much interest in homotopy theory and complex and symplectic geometry. The geometric aspects of the theory of momentangle complexes are the main theme of this survey. Constructions of nonKähler complexanalytic structures on momentangle manifolds corresponding to polytopes and complete simplicial fans are reviewed, and invariants of these structures such as the Hodge numbers and Dolbeault cohomology rings are described. Symplectic and Lagrangian aspects of the theory are also of considerable interest. Momentangle manifolds appear as level sets for quadratic Hamiltonians of torus actions, and can be used to construct new families of Hamiltonianminimal Lagrangian submanifolds in a complex space, complex projective space, or toric varieties.
Bibliography: 59 titles.
 Publication:

Russian Mathematical Surveys
 Pub Date:
 June 2013
 DOI:
 10.1070/RM2013v068n03ABEH004840
 arXiv:
 arXiv:1302.2463
 Bibcode:
 2013RuMaS..68..503P
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Complex Variables;
 Mathematics  Differential Geometry
 EPrint:
 60 pages