Theta-functions on T^2-bundles over T^2 with the Euler class zero

TL;DR

This paper develops a theta-function analogue for certain 4D symplectic manifolds that are T^2-bundles over T^2 with zero Euler class, enabling their symplectic embedding into complex projective spaces.

Contribution

It introduces a new theta-function construction for these manifolds, extending classical concepts to a symplectic setting and providing a canonical embedding method.

Findings

Constructed theta-functions for T^2-bundles over T^2 with zero Euler class.

Established a symplectic embedding into complex projective spaces.

Extended classical theta-function theory to a new class of symplectic manifolds.

Abstract

We construct an analogue of the classical theta-function on an Abelian variety for closed 4-dimensional symplectic manifolds which are T^2-bundles over T^2 with the zero Euler class. We use our theta-functions for a canonical symplectic embedding of these manifolds into complex projective spaces (an analogue of the Lefschetz theorem).