Dequantization of real convex projective manifolds. (English) Zbl 1225.57011
Litvinov, G. L. (ed.) et al., Tropical and idempotent mathematics. International workshop TROPICAL-07, Moscow, Russia, August 25–30, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4782-4/pbk). Contemporary Mathematics 495, 61-85 (2009).
The paper under review is a survey paper focussing on compactification of parametrization spaces of convex projective structures on closed aspherical \(n\)-manifolds with (Gromov) hyperbolic fundamental group and covered by Euclidean \(n\)-space.
The action of the mapping class group of \(M\) on the parametrization space of convex projective structures on \(M\) extends continuously to an action on the compactification. The construction is reminiscent of Morgan-Shalen’s compactification of Teichmüller space.
The main tools are inspired by tropical geometry, in particular, the tropical semifield and Maslov dequantization. The parameter space turns out to be a real semi-algebraic set and Maslov dequantization is applied to it. The limiting object resulting from dequantization represents the behaviour of the semi-algebraic set near infinity. This is the so-called logarithmic limit set which can be glued to the semi-algebraic set, compactifying it.
Alternately, in tropical geometry, algebraic varieties degenerate to tropical varieties by Maslov dequantization. Thus elements of the logarithmic limit set of the parametrization space of convex projective structures on \(M\) can be interpreted as tropical projective structures.
For the entire collection see [Zbl 1172.00019].
The action of the mapping class group of \(M\) on the parametrization space of convex projective structures on \(M\) extends continuously to an action on the compactification. The construction is reminiscent of Morgan-Shalen’s compactification of Teichmüller space.
The main tools are inspired by tropical geometry, in particular, the tropical semifield and Maslov dequantization. The parameter space turns out to be a real semi-algebraic set and Maslov dequantization is applied to it. The limiting object resulting from dequantization represents the behaviour of the semi-algebraic set near infinity. This is the so-called logarithmic limit set which can be glued to the semi-algebraic set, compactifying it.
Alternately, in tropical geometry, algebraic varieties degenerate to tropical varieties by Maslov dequantization. Thus elements of the logarithmic limit set of the parametrization space of convex projective structures on \(M\) can be interpreted as tropical projective structures.
For the entire collection see [Zbl 1172.00019].
Reviewer: Mahan Mj (Howrah)
MSC:
| 57M50 | General geometric structures on low-dimensional manifolds |
| 14T05 | Tropical geometry (MSC2010) |
