Theta functions on the Kodaira-Thurston manifold
William Kirwin and Alejandro Uribe
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Submission date: 26. Sep. 2008
published in: Transactions of the American Mathematical Society, 362 (2010) 2, p. 897-932
DOI number (of the published article): 10.1090/S0002-9947-09-04852-1
MSC-Numbers: 53D, 11F27, 43A
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The Kodaira-Thurston manifold M 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 .
There exists a three-step nilpotent Lie group G which acts transitively on the Kodaira-Thurston manifold, and on the universal cover of 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 that the classical theta-functions are related to the Heisenberg group. One aspect of our results is a connection between the representation theory of G and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in M; in particular, we show that G-invariant special Lagrangian foliations can be detected by a simple algebraic condition on certain subalgebras of the Lie algebra of G.
Crucial to our generalization of theta-functions is the spectrum of the Laplacian 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 this Laplacian.