In the talk, it will be discussed some properties of cyclic coverings
$f: Y\to X$ of complex surfaces of general type $X$ branched along
smooth curves $B\subset X$ that are numerically equivalent to a
multiple of the canonical class of $X$. The main results concern
coverings of surfaces of general type with $p_g=0$ and Miyaoka--Yau
surfaces; in particular, they provide new examples of
multicomponent moduli spaces of surfaces with given Chern numbers
as well as new examples of surfaces that are not deformation
equivalent to their complex conjugates.
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