Theta-functions on the Kodaira-Thurston manifold

Details Theta-functions on the Kodaira-Thurston manifold Theta-functions on the Kodaira-Thurston manifold

2007-12-24

arxiv.org/abs/0712.4016v1

Mathematics

Differential Geometry

29 pages, 1 figure

Scientific paper

The Kodaira--Thurston M manifold is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kaehler. We describe a construction of theta-functions associated to M which parallels the classical theory of theta-functions associated to the torus (from the point of view of representation theory and geometry), and yields pseudoperiodic complex-valued functions on R^4. There exists a three-step nilpotent Lie group G which acts transitively on the Kodaira--Thurston manifold M in a Hamiltonian fashion. The theta-functions discussed in this paper are intimately related to the representation theory of G in much the same way the classical theta-functions are related to the Heisenberg group. One aspect of our results which has not appeared in the classical theory is a connection between the representation theory of G and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in M.

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