The Calabi–Yau equation on 4-manifolds over 2-tori

Fino, A.; Li, Y.Y.; Salamon, S.; Vezzoni, L.

doi:10.1090/S0002-9947-2012-05692-3

The Calabi–Yau equation on 4-manifolds over 2-tori
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by A. Fino, Y.Y. Li, S. Salamon and L. Vezzoni PDF

Trans. Amer. Math. Soc. 365 (2013), 1551-1575 Request permission

Abstract:

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 $\mathbb {T}^2$ 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 $\mathbb {T}^2$, 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.

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