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Is there a $2n$ dimensional manifold $M$ and Kahler structures $(J_{1},g_{1}), (J_{2},g_{2})$ on $M$, such that $c_{1}(M,J_{1})$ is positive with respect to $g_{1}$, but $c_{1}(M,J_{2})$ is negative with respect to $g_{2}$?

When $n=1$, this is impossible due to Gauss-Bonnet formula. But I don't know if it is false when $n\ge 2$.

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    $\begingroup$ Chern classes are topological; i.e., the cohomology class does not change when you change, say, the metric or the connection on the vector bundle. $\endgroup$
    – Ted Shifrin
    Commented May 24 at 16:12
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    $\begingroup$ But Chern classes depends on the almost complex structure, aren't they? $\endgroup$
    – Holomodric
    Commented May 24 at 18:04
  • $\begingroup$ The way you stated your question focused on the metrics. See the discussion here and here. I guess if you take the conjugate complex structure, then the sign of the Kähler form does obviously change. $\endgroup$
    – Ted Shifrin
    Commented May 24 at 18:27
  • $\begingroup$ If I take the conjugate complex structure but fix the metric, then the sign of the Kahler form and the first Chern class change at the same time, so I think it does not change the (non-)positivity of the first Chern class. $\endgroup$
    – Holomodric
    Commented May 25 at 4:23
  • $\begingroup$ What precisely do you mean by positivity of a form/class with respect to a metric? $\endgroup$
    – Ted Shifrin
    Commented May 25 at 4:26

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