Theta-functions on the Kodaira-Thurston manifold

William D. Kirwin; Alejandro Uribe

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arXiv · 0712.4016

Theta-functions on the Kodaira-Thurston manifold

William D. Kirwin·Alejandro Uribe

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Abstract

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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William D. Kirwin, Alejandro Uribe. 2007-12-24. Theta-functions on the Kodaira-Thurston manifold. https://arxiv.org/abs/0712.4016

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