MiS Preprint Repository

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 $R^4$.

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.