Theta functions on the Kodaira–Thurston manifold

We construct an analog of the classical theta function on an abelian variety for the Kodaira–Thurston nilmanifold, which is defined as a (nonholomorphic) section of a special complex line bundle over the Kodaira–Thurston manifold. The theta functions we introduce are used for a canonical symplectic embedding of the Kodaira–Thurston manifold into a complex projective space (an analog of the Lefschetz theorem).