2,347 research outputs found

    Convexity package for momentum maps on contact manifolds

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    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

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    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

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    We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let XX be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension dd. Informally, the theorem states that if XX has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of XX) then XX has the following topological overlap property: for every continuous map X→RdX\rightarrow \mathbf{R}^d there exists a point p∈Rdp\in \mathbf{R}^d that is contained in the images of a positive fraction μ>0\mu>0 of the dd-cells of XX. More generally, the conclusion holds if Rd\mathbf{R}^d is replaced by any dd-dimensional piecewise-linear (PL) manifold MM, with a constant μ\mu that depends only on dd and on the expansion properties of XX, but not on MM.Comment: Minor revision, updated reference
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