1. Can somebody give an example of a complex manifold whose first Chern class is a torsion class? In general it seems that Chern classes may have torsion part as well as free part. However when using Chern-Weil theory computing in terms of de Rham cohomology we loose the torsion part.
  2. Is there an example which has both free and torsion part?
  3. Is there a Kähler manifold whose Chern classes involving torsion? Using the Kähler form to compute the first Chern class may result in losing the torsion.

On a complex manifold $X$ the exponential sequence of sheaves of abelian groups $$0 \to \mathbb Z \to\mathcal O \stackrel {exp (2i\pi \cdot)}{\to}\mathcal O^*\to 0$$ yields a long exact sequence in cohomology containing the fragment $$\cdots \to H^1(X,\mathcal O) \to H^1(X,\mathcal O^*) \stackrel {c_1}{\to } H^2(X,\mathbb Z) \to H^2(X,\mathcal O) \to \cdots $$ Now suppose $X$ is an Enriques surface: such a surface is projective algebraic and thus certainly Kähler. By definition, it satisfies $$H^1(X,\mathcal O) =H^2(X,\mathcal O)=0 $$ [in algebraic geometry slang: irregularity=geometric genus =0] so that our fragment above reduces to the isomorphism $$ 0\to H^1(X,\mathcal O^*) \stackrel {c_1}{\to } H^2(X,\mathbb Z)\to 0 $$
Finally, also by definition, the canonical bundle $\omega_X$ of an Enriques surface is non trivial but its square is trivial.
The isomorphism above then yields that $c_1(\omega_X)\neq 0$ but $2c_1(\omega_X)=0$ , and this gives the required example since $c_1(T_X)=-c_1(\omega _X)$ ( I assume that by "first Chern class of a manifold" you mean that of its tangent bundle).

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    $\begingroup$ For the second question, the blowup of an Enriques surface is an example where $c_1$ has non-trivial free and torsion parts. $\endgroup$ – Michael Albanese Dec 17 '18 at 2:08

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