Let $M$ be an compact Kähler algebraic variety and suppose $K_M$ is semi-ample. Consider the holomorphic map $\pi:X\to \Sigma \subset \mathbb CP^N$ with $Kod(M)=dim_\mathbb C\Sigma$ (here $Kod$ means Kodaira dimension). Does a nonsingular fibre $\pi^{-1}(z)$ has vanishing first Chern class? What about the first Chern class of singular fibers?
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$\begingroup$ This kind of question is better suited to math.stackexhange.com, and you'll get more detailed answers there. $\endgroup$– user5117Commented Aug 20, 2014 at 10:26
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Does a nonsingular fibre $\pi^{−1}(z)$ has vanishing first Chern class?
Yes. Denote a regular fiber $Z$; then $K_M|_ Z= det(N^*Z)\otimes K_Z= K_Z$ by adjunction formula. On the other hand, $K_M$ is trivial on $Z$, because it is a pullback from the base, and $Z$ is a fiber.
This argument proves vanishing of rational $c_1$. For integer $c_1$, the statement is wrong: consider a product of Enriques surface and a curve of genus $g>1$, its canonical bundle is semiample, but the fibers of the corresponding projections are Enriques surfaces, having torsion $c_1$.
What about the first Chern class of singular fibers?
If the map is flat, yes, but not otherwise (take a blowup, for example).
answered Aug 20, 2014 at 1:46
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$\begingroup$ I guess the hypothesis gives only that some multiple $K_M^{\otimes n}$ is a pullback, so $K_Z^{\otimes n}$ is trivial, which gives $c_1=0$ in rational cohomology. $\endgroup$– abxCommented Aug 20, 2014 at 4:36
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$\begingroup$ sure, I meant the rational $c_1$. No idea about integer $c_1$, it's an iteresting question actually. $\endgroup$ Commented Aug 20, 2014 at 6:28
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$\begingroup$ added a counterexample for integer $c_1$, just a product of Enriques and a curve $\endgroup$ Commented Aug 20, 2014 at 6:32