Aspherical Symplectic Manifolds with Positive First Chern Degree

Abstract

For every integer $n\ge5$, we construct a closed aspherical symplectic manifold of real dimension $2n$ with positive first Chern degree. Each example is a flat torus bundle over the same compact hyperbolic surface, with arithmetic monodromy arising from an explicit indefinite quaternion division algebra over $\mathbb{Q}$. We identify its symplectic first Chern class with the pullback of the canonical class of the base and compute its degree explicitly. Asphericity rules out nonconstant pseudoholomorphic spheres for every tame almost complex structure. Thus these examples give a negative answer to the symplectic uniruledness question of Boucksom–Demailly–Păun–Peternell in every even real dimension at least ten.