The Calabi–yau Equation on 4-manifolds over 2-tori

This paper pursues the study of the Calabi–Yau equation on certain symplectic non-Kähler 4-manifolds, building on a key example of Tosatti and Weinkove in which more general theory had proved less effective. Symplectic 4-manifolds admitting a 2-torus fibration over a 2-torus base T2 are modelled on one of three solvable Lie groups. Having assigned an invariant almost-Kähler structure and a volume form that effectively varies only on T2, one seeks a symplectic form with this volume. Our approach simplifies the previous analysis of the problem and establishes the existence of solutions in various other cases.