M ar 2 01 1 THE CALABI – YAU EQUATION ON 4 - MANIFOLDS OVER 2 - TORI

This paper pursues the study of the Calabi–Yau equation on certain sym-plectic non-Kähler 4-manifolds, building on a key example of Tosatti–Weinkove [24] in which more general theory had proved less effective. Symplectic 4-manifolds admitting a 2-torus fibration over a 2-torus base T 2 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 T 2 , 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.