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Topological Euler characteristic is given by the top Chern class of the tangent bundle up to sign.

Comments: Examples don't exist in dimension 2 due to Bogomolov–Miyaoka–Yau inequality. Is that true in higher dimensions? Examples are known if one assumes a weaker condition that the holomorphic Euler characteristic is zero.

Wonder after Jason Starr's answer: Is there a minimal manifold of general type with zero Euler characteristic?

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    $\begingroup$ A product of three hyperbolic curves of genera $g$, $g'$ and $g''$ has topological Euler characteristic equal to $-8(g-1)(g'-1)(g''-1)$. If you blow up that threefold at $4(g-1)(g'-1)(g''-1)$ points, this appears to have topological Euler characteristic equal to $0$. $\endgroup$
    – Jason Starr
    Commented Jun 7, 2021 at 18:30
  • $\begingroup$ cool! so simple. Thank you very much. If you turned it into an answer I will accept it! $\endgroup$
    – guest0803
    Commented Jun 8, 2021 at 1:11

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