Let $f:X\longrightarrow B$ be a family of curves, **with $f$ relatively minimal**, over a fixed curve $B$ ($B$ is projective, irreducible and smooth). The fibration $f$ is said locally trivial if all fibers are smooth and isomorphic, and moreover the number $K^2\_f$ is defined as 
$$K^2\_f=K^2\_X-8(g-1)(g(B)-1)$$
where $g$ is the genus of a fiber.



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> **Edit:** $g\ge2$
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I'd like to have a reference for the proof of the following theorem:



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> $K^2\_f=0$ **if and only if** $f$ is locally trivial.
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All authors attribute this theorem to Arakelov and cite the article "S.Ju. Arakelov - family of algberaic curves with fixed degeneracy". In this article I can't find the above theorem! Can you suggest me an alternative reference?



Reference for Arakelov's theorem: $K^2_f=0$ iff $f$ is locally trivial
What you are saying is not entirely correct. 


1. That equation for $K\_f^2$ is not a **definition**. It's a consequence of the definition of $K\_f$ as $K\_X-f^\*K\_B$: If $F$ denotes a fiber of $f$, then since $K\_X\cdot F=\deg K\_F = 2g-2$, 
$$
K\_f^2 = K\_X^2 - 2 K\_X\cdot f^\*K\_B = K\_X^2 - 2 K\_X\cdot (2g(b)-2) F = K\_X^2 - 2(2g-2)(2g(B)-2).
$$
2. The statement you are looking for is not exactly like that. $K\_f^2=0$ is not equivalent to $f$ being locally trivial, but being generically locally trivial. That is, for instance it might have some singular fibers, but the smooth fibers are isomorphic. The right word actually is *isotrivial*, which is defined as two general fibers being isomorphic.


With that said, *Vesselin* already gave you the correct reference, but at the same time there is a whole lot of newer results that deal with these kind of questions.


The modern approach is that first of all regardless of any assumption on isotriviality, the sheaf $f\_\*\mathscr O\_X(mK\_{f})$ is semipositive for large and divisible enough $m$. Then using the fact that the fibers are canonically polarized varieties (that's the real reason you need $g\geq 2$), the natural map 
$$f^\*f\_\*\mathscr O\_X(mK\_{f})\to \mathscr O\_X(mK\_{f})$$ is generically surjective, so with a bit of work you can prove that $K\_f$ is nef and then $K\_f^2\geq 0$ always and then $K\_f^2> 0$ if and only if $K\_f=K\_{X/B}$ is big. That last condition follows from 3.2 in Arakelov's paper as *Vesselin* already mentioned. (By the way, that result is for a *stable* model of $f$ which is not necessarily smooth, but everything works out at the end).


The advantage of this approach is that this works in higher dimensions as well.


