Lectures on Lagrangian Torus Fibrations

Comprehensive and visual introduction to the geometry of 4-dimensional symplectic manifolds via 2-dimensional almost-toric diagrams.

Jonny Evans (Author)

9781009372626, Cambridge University Press

Hardback, published 31 July 2023

243 pages
28 x 19 x 2.1 cm, 0.59 kg

'This is a lucid and engaging introduction to the fascinating world of (almost) toric geometry, in which one can understand the properties of Lagrangian and symplectic submanifolds in four dimensions simply by drawing suitable two-dimensional diagrams. The book has many illustrations and intricate examples.' Dusa McDuff, Barnard College, Columbia University

Symington's almost toric fibrations have played a central role in symplectic geometry over the past decade, from Vianna's discovery of exotic Lagrangian tori to recent work on Fibonacci staircases. Four-dimensional spaces are of relevance in Hamiltonian dynamics, algebraic geometry, and mathematical string theory, and these fibrations encode the geometry of a symplectic 4-manifold in a simple 2-dimensional diagram. This text is a guide to interpreting these diagrams, aimed at graduate students and researchers in geometry and topology. First the theory is developed, and then studied in many examples, including fillings of lens spaces, resolutions of cusp singularities, non-toric blow-ups, and Vianna tori. In addition to the many examples, students will appreciate the exercises with full solutions throughout the text. The appendices explore select topics in more depth, including tropical Lagrangians and Markov triples, with a final appendix listing open problems. Prerequisites include familiarity with algebraic topology and differential geometry.

1. The Arnold–Liouville theorem
2. Lagrangian fibrations
3. Global action-angle coordinates and torus actions
4. Symplectic reduction
5. Visible Lagrangian submanifolds
6. Focus-focus singularities
7. Examples of focus-focus systems
8. Almost toric manifolds
9. Surgery
10. Elliptic and cusp singularities
A. Symplectic linear algebra
B. Lie derivatives
C. Complex projective spaces
D. Cotangent bundles
E. Moser's argument
F. Toric varieties revisited
G. Visible contact hypersurfaces and Reeb flows
H. Tropical Lagrangian submanifolds
I. Markov triples
J. Open problems
References
Index.

Subject Areas: Geometry [PBM]