Abstract
We describe some general methods to compute fundamental groups, (co)homology, and irregularity of semi-log-canonical surfaces. As an application, we show that there are exactly two irregular Gorenstein stable surfaces with \(K^2=1\), which have \(\chi (X) = 0\) and \({{\mathrm{Pic}}}^0(X)={\mathbb {C}}^*\) but different homotopy type.
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References
Alexeev, V.: Complete moduli in the presence of semiabelian group action. Ann. Math. 155(3), 611–708 (2002)
Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)
Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge (2nd edn). Springer, Berlin (2004)
Bombieri, E.: Canonical models of surfaces of general type. Inst. Hautes Études Sci. Publ. Math. 42, 171–219 (1973)
Franciosi, M., Pardini, R., Rollenske, S.: Log-canonical pairs and Gorenstein stable surfaces with \(K^2_X=1\). Comp. Math. (2014). doi:10.1112/S0010437X14008045
Kollár, J., Mori, S.: Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998). With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original
Kollár, J.: Moduli of varieties of general type. In: Farkas, G., Morrison, I. editors, Handbook of moduli: Volume II, volume 24 of Advanced Lectures in Mathematics, pp. 131–158. International Press, (2012). arXiv:1008.0621
Kollár, J.: Singularities of the minimal model program, volume 200 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács
Kollár, J.: Moduli of varieties of general type. 2015. Book in preparation
Kollár, J., Shepherd-Barron, N.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)
Lojasiewicz, S.: Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa 3(18), 449–474 (1964)
Liu, W., Rollenske, S.: Geography of Gorenstein stable log surfaces. Trans. Amer. Math. Soc. arXiv:1307.1999 (2013, to appear)
Liu, W., Rollenske, S.: Pluricanonical maps of stable log surfaces. Adv. Math. 258, 69–126 (2014)
May, J.P.: A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1999)
Peters, C.A.M., Steenbrink, J.H.M.: Mixed Hodge structures, volume 52 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin, (2008)
Acknowledgments
The first author is a member of GNSAGA of INDAM. The third author is grateful for support of the DFG through the Emmy Noether program and SFB 701. The collaboration benefited immensely from a visit of the third author in Pisa supported by GNSAGA of INDAM. This project was partially supported by PRIN 2010 “Geometria delle Varietà Algebriche” of italian MIUR. We are indepted to Stefan Bauer, Kai-Uwe Bux, Michael Lönne, Hanno von Bodecker for guidance around the pitfalls of algebraic topology. Kai-Uwe Bux also explained to us how to prove that the fundamental groups computed in Proposition 4.6 are not isomorphic. The third author is grateful to Wenfei Liu for many discussions on stable surfaces and to Filippo Viviani for some helpful email communication.
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Franciosi, M., Pardini, R. & Rollenske, S. Computing invariants of semi-log-canonical surfaces. Math. Z. 280, 1107–1123 (2015). https://doi.org/10.1007/s00209-015-1469-9
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DOI: https://doi.org/10.1007/s00209-015-1469-9