What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explanation, both talking about curves and surfaces...



what is the cyclic cover trick?
The "cyclic cover trick" can refer to more than one thing. One example is as follows. Let $L$ be an invertible sheaf on a smooth projective scheme such that some power $L^{\otimes d}$ has a global section $s$ whose zero scheme $D$ is a smooth Cartier divisor (all of these smoothness conditions are not strictly necessary). Let $\nu:Y\to X$ be the associated degree $d$ branched cover branched over $D$ and whose restriction over $X\setminus D$ is the $\mathbb{\mu}\_d$-torsor corresponding to $L$. More precisely, $\nu$ is affine and $\nu\_\*\mathcal{O}\_Y$ is the $\mathbb{Z}/d\mathbb{Z}$-graded $\mathcal{O}\_X$-algebra $$\nu\_\*\mathcal{O}\_Y = \mathcal{O}\_X \oplus L^\vee \oplus \dots \oplus (L^\vee)^{\otimes (d-1)}.$$
Of course we have to say what is the multiplication rule on this algebra. But there is a unique multiplication rule that is $\mathbb{Z}/d\mathbb{Z}$-graded and that is compatible with the multiplication rule $s:(L^\vee)^{\otimes d} \to \mathcal{O}\_X$. Now, for every $q$, $\nu\_\*\Omega^p\_Y$ has a natural $\mathbb{Z}/d\mathbb{Z}$-grading, and the graded pieces turn out to be expressible in terms of tensor products of $\Omega^r\_X$ with powers $(L^\vee)^{\otimes s}$. Now assume that $X$ is a $\mathbb{C}$-scheme. Then the Hodge theorem gives surjectivity of the various projections $H^r(Y^{\text{an}};\mathbb{C}) \to H^q(Y,\Omega^p\_Y)$. Both groups have a $\mathbb{\mu}\_d$-action that is equivalent to a $\mathbb{Z}/d\mathbb{Z}$-grading. In particular, vanishing of certain graded pieces of $H^r(Y^{\text{an}};\mathbb{C})$ will imply vanishing of certain cohomology groups $H^q(X,\Omega^p\_X\otimes (L^\vee)^{\otimes s})$. In this way, one can prove the Kodaira-Akizuki-Nakano vanishing theorem. 


So in this case, the "covering trick" is to use a geometric result on the cyclic cover (the Hodge decomposition) combined with computation of the graded pieces to deduce a geometric result on the original scheme. In a similar way, one can prove extensions of the Kodaira vanishing theorem, e.g., the Kawamata-Viehweg vanishing theorem. This is discussed in detail in the book of Kollár and Mori.


