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I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the following properties:

  • $X$ is a $\mathbb T^n$ bundle over $\mathbb T^n$, but not $\mathbb T^{2n}$.

  • The $\mathbb T^n$ bundle $\pi:X\rightarrow\mathbb T^n$ admits a section.

  • $X$ admits a complex structure.

Note that the work of Cordero-Fernandez-Gray (http://www.sciencedirect.com/science/article/pii/0040938386900509) provides some examples of symplectic manifolds which are $\mathbb T^{p+1}$ bundles over $\mathbb T^{p+1}$, which they call $M(p,0)$, and $M(1,0)$ gives you the Kodaira-Thurston manifold. These symplectic manifolds admit almost complex structures. Another candidate is the iwasawa manifold, but this is topologically a $\mathbb T^4$ bundle over $\mathbb T^2$.

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  • $\begingroup$ It seems if you multiply the Kodaira-Thurston manifold by $T^{2k}$ you get a $T^{k+2}$ bundle over $T^{k+2}$ with desired properties. $\endgroup$ Nov 13, 2014 at 3:54
  • $\begingroup$ @IgorBelegradek Good idea!:) $\endgroup$
    – YHBKJ
    Nov 13, 2014 at 4:03

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