2,347 research outputs found
Convexity package for momentum maps on contact manifolds
Let a torus T act effectively on a compact connected cooriented contact
manifold, and let Psi be the natural momentum map on the symplectization. We
prove that, if dim T > 2, the union of the origin with the image of Psi is a
convex polyhedral cone, the non-zero level sets of Psi are connected (while the
zero level set can be disconnected), and the momentum map is open as a map to
its image. This answers a question posed by Eugene Lerman, who proved similar
results when the zero level set is empty. We also analyze examples with dim T
<= 2.Comment: 39 pages. Contains small corrections and a small simplification of
the argument. To appear in Algebraic and Geometric Topology
The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations
This survey article begins with a discussion of the original form of the
Strominger-Yau-Zaslow conjecture, surveys the state of knowledge concering this
conjecture, and explains how thinking about this conjecture naturally leads to
the program initiated by the author and Bernd Siebert to study mirror symmetry
via degenerations of Calabi-Yau manifolds and log structures.Comment: 44 pages, to appear in the Proceedings of the 2005 AMS Symposium on
Algebraic Geometry, Seattl
On Expansion and Topological Overlap
We give a detailed and easily accessible proof of Gromov's Topological
Overlap Theorem. Let be a finite simplicial complex or, more generally, a
finite polyhedral cell complex of dimension . Informally, the theorem states
that if has sufficiently strong higher-dimensional expansion properties
(which generalize edge expansion of graphs and are defined in terms of cellular
cochains of ) then has the following topological overlap property: for
every continuous map there exists a point that is contained in the images of a positive fraction of
the -cells of . More generally, the conclusion holds if is
replaced by any -dimensional piecewise-linear (PL) manifold , with a
constant that depends only on and on the expansion properties of ,
but not on .Comment: Minor revision, updated reference
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