On numerically pluricanonical cyclic coverings

Vik. S. Kulikov ; V. M. Kharlamov

Izvestiya. Mathematics , Tome 78 (2014) no. 5, pp. 986-1005.

Résumé

We investigate some properties of cyclic coverings $f\colon Y\to X$ (where $X$ is a complex surface of general type) branched along smooth curves $B\subset X$ that are numerically equivalent to a multiple of the canonical class of $X$. Our main results concern coverings of surfaces of general type with $p_g=0$ and Miyaoka–Yau surfaces. In particular, such coverings provide 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.

Détail

Keywords: numerically pluricanonical cyclic coverings of surfaces, irreducible components of moduli spaces of surfaces.

Bibliographie

[1] Vik. S. Kulikov, V. M. Kharlamov, “On real structures on rigid surfaces”, Izv. Math., 66:1 (2002), 133–150 | DOI | DOI | MR | Zbl

[2] M. Manetti, “On the moduli space of diffeomorphic algebraic surfaces”, Invent. Math., 143:1 (2001), 29–76 | DOI | MR | Zbl

[3] V. Kharlamov, Vik. S. Kulikov, “Deformation inequivalent complex conjugated complex structures and applications”, Turkish J. Math., 26:1 (2002), 1–25 | MR | Zbl

[4] F. Catanese, “Moduli spaces of surfaces and real structures”, Ann. of Math. (2), 158:2 (2003), 577–592 | DOI | MR | Zbl

[5] Vik. S. Kulikov, V. M. Kharlamov, “Surfaces with $\operatorname{DIF}\ne\operatorname{DEF}$ real structures”, Izv. Math., 70:4 (2006), 769–807 | DOI | DOI | MR | Zbl

[6] M. V. Nori, “Zariski's conjecture and related problems”, Ann. Sci. École Norm. Sup. (4), 16:2 (1983), 305–344 | MR | Zbl

[7] E. Bombieri, “Canonical models of surfaces of general type”, Inst. Hautes Études Sci. Publ. Math., 42:1 (1973), 171–219 | DOI | MR | Zbl

[8] W. P. Barth, K. Hulek, C. A. M. Peters, A. Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), 4, 2nd ed., Springer-Verlag, Berlin, 2004, xii+436 pp. | MR | Zbl

[9] I. Reider, “Vector bundles of rank 2 and linear systems on algebraic surfaces”, Ann. of Math. (2), 127:2 (1988), 309–316 | DOI | MR | Zbl

[10] Vik. S. Kulikov, “Old and new examples of surfaces of general type with $p_g=0$”, Izv. Math., 68:5 (2004), 965–1008 | DOI | DOI | MR | Zbl

[11] D. Mumford, Lectures on curves on an algebraic surface, Ann. of Math. Stud., 59, Princeton Univ. Press, Princeton, N.J., 1966, xi+200 pp. | MR | Zbl | Zbl

[12] D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Oxford Univ. Press, London, 1970, viii+242 pp. | MR | Zbl | Zbl

[13] I. Bauer, F. Catanese, R. Pignatelli, “Surfaces of general type with geometric genus zero: a survey”, Complex and differntial geometry, Springer Proc. Math., 8, Springer, Heidelberg, 2011, 1–48 | DOI | MR | Zbl

[14] P. Burniat, “Sur les surfaces de genre $P_{12}>1$”, Ann. Mat. Pura Appl. (4), 71:1 (1966), 1–24 | DOI | MR | Zbl

[15] M. Mendes Lopes, R. Pardini, “A new family of surfaces with $p_{g}=0$ and $K^2=3$”, Ann. Sci. École Norm. Sup. (4), 37:4 (2004), 507–531 | DOI | MR | Zbl

[16] G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Stud., 78, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo, 1973, v+195 pp. | MR | Zbl

[17] M. Gromov, P. Pansu, “Rigidity of lattices: an introduction”, Geometric topology: recent developments (Montecatini Terme, 1990), Lect. Notes in Math., 1504, Springer, Berlin, 1991, 39–137 | DOI | MR | Zbl

[18] E. Witten, “Monopoles and four-manifolds”, Math. Res. Lett., 1:6 (1994), 769–796 | DOI | MR | Zbl

[19] R. Brussee, “The canonical class and the $C^\infty$-properties of Kähler surfaces”, New York J. Math., 2 (1996), 103–146 (electronic) | MR | Zbl

[20] Yum-Tong Siu, “The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds”, Ann. of Math. (2), 112:1 (1980), 73–111 | DOI | MR | Zbl