Kulikov surfaces form a connected component of the moduli space (Q2840042)

In the paper under review, the authors study the moduli space of Kulikov surfaces. A Kulikov surface \(S\) is a surface of general type with invariants: \(p_g(S)=0\) and \(K^2_S=6\) which is obtained as a \((\mathbb{Z}/3\mathbb{Z})^2\)-cover of \(\mathbb{P}^2\) branched in a particular configuration of lines. Such surfaces were introduced by \textit{V. S. Kulikov} in [Izv. Math. 68, No. 5, 965--1008 (2004); translation from Izv. Ross. Akad. Nauk Ser. Mat. 68, No. 5, 123--170 (2004; Zbl 1073.14055)], and they present similar features as Burniat surfaces (see [\textit{I. C. Bauer} and \textit{F. Catanese}, EMS Series of Congress Reports, 49--76 (2011; Zbl 1264.14052); Invent. Math. 180, No. 3, 559--588 (2010; Zbl 1219.14051)]).NEWLINENEWLINEFirst of all, the authors calculate the fundamental group of a Kulikov surface and its first homology group \(H_1(S, \mathbb{Z})=(\mathbb{Z}/3\mathbb{Z})^3\). They use this calculation to prove the main result of the paper, namely that Kulikov surfaces are closed under homotopy equivalence. This means that any compact complex surface which is homotopy equivalent to a Kulikov surface is given by the construction above. This implies that Kulikov surfaces form a connected component of the Gieseker moduli space of surfaces of general type with invariants \(\chi=1\) and \(K^2=6\). Moreover, this component has dimension 1. These results are very similar of the results obtained for Burniat surfaces in [\textit{I. C. Bauer} and \textit{F. Catanese}, ``Burniat surfaces, III: deformations of automorphisms and extended Burniat surfaces'', \url{arXiv:1012.3770}; \textit{M. Mendes Lopes} and \textit{R. Pardini}, Topology 40, No. 5, 977--991 (2001; Zbl 1072.14522)].NEWLINENEWLINESecondly, the fact that the geometric genus \(p_g\) of Kulikov surfaces is zero force the authors to a deeper analysis of these surfaces. In particular, one wants to study the birationality of the bicanonical map and the Chow group of \(0\)-cycles of degree zero 1\(A^0_0(S)\). Indeed, the authors are able to prove that the bicanonical map is birational. And finally, they prove that these surfaces verify Bloch's conjecture, i.e., \(A^0_0(S)=0\). To prove this last proposition they use the computer algebra program \texttt{MAGMA} (the script of the program can be found in the article).