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Jan 24, 2017 at 21:09 comment added Danu @CharlesStaats Exactly. Kotschick has another paper (this one, with Terzic) where this is made very explicit (see section 4, page 600 for the description of a setup where this can happen and section 8, page 610 for an explicit example where it does)
Mar 30, 2013 at 17:52 comment added Charles Staats A worthwhile point. I assume what is happening here is that Chern classes are an invariant of the complex vector bundle $T_X$--but different complex structures on the same smooth manifold $X$ can give rise to non-isomorphic descriptions $T_X$ as a complex vector bundle over $X$.
Mar 29, 2013 at 19:09 history answered anonymous CC BY-SA 3.0