Theta functions on the Kodaira–Thurston manifold
نویسندگان
چکیده
The Kodaira–Thurston manifold M is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kähler. We describe a construction of θ-functions associated to M which parallels the classical theory of θ-functions associated to the torus (from the point of view of representation theory and geometry), and yields pseudoperiodic complex-valued functions on R. There exists a three-step nilpotent Lie group G̃ which acts transitively on the Kodaira– Thurston manifold, and on the universal cover of M in a Hamiltonian fashion. The θ-functions discussed in this paper are intimately related to the representation theory of G̃ in much the same way that the classical θ-functions are related to the Heisenberg group. One aspect of our results is a connection between the representation theory of G̃ and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in M ; in particular, we show that G̃invariant special Lagrangian foliations can be detected by a simple algebraic condition on certain subalgebras of the Lie algebra of G̃. Crucial to our generalization of θ-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 lowlying spectrum of ∆.
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تاریخ انتشار 2008