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On numerically pluricanonical cyclic coverings. (English. Russian original) Zbl 1386.14071

Izv. Math. 78, No. 5, 986-1005 (2014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 5, 143-166 (2014).
The paper studies some properties of totally \(d\) cyclic coverings \(Y\) of a smooth projective surface \(X\) of general type branched along smooth curves \(B\subset X\) that are numerically equivalent to a multiple of the canonical class of \(X\). In particular, the authors prove that the moduli space of \(Y\) consists of at least two connected components under some condition of \(\text{Tor}_d\text{Pic}(X)\).
The paper under review concerns mainly the cases when \(X\) are surfaces of general type with \(p_g=0\) or Miyaoka-Yau surfaces (surfaces of general type with \(c_1^2=3c_2\)). In particular, the authors show that if \(X\) is a Miyaoka-Yau surface with a suitable condition then the number of connected components of \(Y\) is equal to the number of orbits of the action of \(\text{Aut}(X)\) on \(\text{Tor}(X)\). Finally, the authors find new examples of multi-component moduli spaces of surfaces with given Chern numbers and new examples of surfaces that are not deformation equivalent to their complex conjugates.
We mention that there are many interesting examples of surfaces that are not deformation equivalent to their complex conjugates in the paper [Ann. Math. (2) 158, No. 2, 577–592 (2003; Zbl 1042.14011)] by F. Catanese.

MSC:

14E20 Coverings in algebraic geometry
14J29 Surfaces of general type
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
32Q55 Topological aspects of complex manifolds
14J10 Families, moduli, classification: algebraic theory
14J17 Singularities of surfaces or higher-dimensional varieties

Citations:

Zbl 1042.14011
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