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Theta functions on the Kodaira-Thurston manifold

Authors: William D. Kirwin and Alejandro Uribe
Journal: Trans. Amer. Math. Soc. 362 (2010), 897-932
MSC (2000): Primary 53Dxx; Secondary 11F27, 43A30, 22E70
Published electronically: August 17, 2009
MathSciNet review: 2551510
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Abstract: The Kodaira-Thurston manifold $ M$ is a compact, $ 4$-dimensional nilmanifold which is symplectic and complex but not Kähler. We describe a construction of $ \vartheta$-functions associated to $ M$, which parallels the classical theory of $ \vartheta$-functions associated to the torus (from the point of view of representation theory and geometry), and which yields pseudoperiodic complex-valued functions on $ \mathbb{R}^4.$

There exists a three-step nilpotent Lie group $ \widetilde{G}$ which acts transitively on the Kodaira-Thurston manifold and on the universal cover of $ M$ in a Hamiltonian fashion. The $ \vartheta$-functions discussed in this paper are intimately related to the representation theory of $ \widetilde{G}$ in much the same way that the classical $ \vartheta$-functions are related to the Heisenberg group. One aspect of our results is a connection between the representation theory of $ \widetilde{G}$ and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in $ M$; in particular, we show that $ \widetilde{G}$-invariant special Lagrangian foliations can be detected by a simple algebraic condition on certain subalgebras of the Lie algebra of $ \widetilde{G}.$

Crucial to our generalization of $ \vartheta$-functions is the spectrum of the Laplacian $ \Delta$ acting on sections of certain line bundles over $ M$. One corollary of our work is a verification of a theorem of Guillemin-Uribe describing the structure (in the semiclassical limit) of the low-lying spectrum of $ \Delta$.

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Additional Information

William D. Kirwin
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig, Germany

Alejandro Uribe
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043

Received by editor(s): March 10, 2008
Published electronically: August 17, 2009
Additional Notes: The first author was supported in part by the Max Planck Institute for Mathematics in the Sciences (Leipzig).
The second author was supported in part by NSF Grant DMS-0401064.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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