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Dec 15, 2010 at 21:49 comment added Martin O Susan Tolman's paper arxiv.org/abs/0903.4918 also considers the 6-dimensional case. From the argument (Remark 2.11) with the $\chi_y$-characteristic one should be able to deduce the other Chern classes of your putative example.
Dec 15, 2010 at 21:02 answer added Dmitri Panov timeline score: 3
Dec 15, 2010 at 16:06 history asked Sil CC BY-SA 2.5