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Excluded homeomorphism types for dual complexes of surfaces. (English) Zbl 1349.14195

Baker, Matthew (ed.) et al., Nonarchimedean and tropical geometry. Based on two Simons symposia, Island of St. John, March 31 – April 6, 2013 and Puerto Rico, February 1–7, 2015. Cham: Springer (ISBN 978-3-319-30944-6/hbk; 978-3-319-30945-3/ebook). Simons Symposia, 133-144 (2016).
Summary: We study an obstruction to prescribing the dual complex of a strict semistable degeneration of an algebraic surface. In particular, we show that if \(\Delta\) is a complex homeomorphic to a 2-dimensional manifold with negative Euler characteristic, then \(\Delta\) is not the dual complex of any semistable degeneration. In fact, our theorem is somewhat more general and applies to a certain class of complexes homotopy equivalent to such a manifold. Our obstruction is provided by the theory of tropical complexes.
For the entire collection see [Zbl 1354.14004].

MSC:

14T05 Tropical geometry (MSC2010)
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[1] Baker, M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2 (6), 613–653 (2008). With an appendix by Brian Conrad · Zbl 1162.14018 · doi:10.2140/ant.2008.2.613
[2] Berkovich, V.G.: Smooth p-adic analytic spaces are locally contractible. Invent. Math. 137, 1–84 (1999) · Zbl 0930.32016 · doi:10.1007/s002220050323
[3] Brown, M., Foster, T.: Rational connectivity and analytic contractibility. Preprint, arXiv:1406.7312
[4] Cartwright, D.: Combinatorial tropical surfaces. Preprint, arXiv:1506.02023
[5] Cartwright, D.: Tropical complexes. Preprint, arXiv:1308.3813
[6] de Fernex, T., Kollár, J., Xu, C.: The dual complex of singularities. Adv. Stud. Pure Math. [in honor of Kawamata’s 60th birthday] (to appear). Preprint, arXiv:1212.1675
[7] Gallier, J., Xu, D.: A Guide to the Classification Theorem for Compact Surfaces. Springer, Berlin (2013) · Zbl 1270.57001 · doi:10.1007/978-3-642-34364-3
[8] Gubler, W.: Local and canonical heights of subvarieties. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (4), 711–760 (2003) · Zbl 1170.14303
[9] Gubler, W., Rabinoff, J., Werner, A.: Skeletons and tropicalization. Preprint, arXiv:1404.7044 · Zbl 1370.14024
[10] Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001
[11] Helm, D., Katz, E.: Monodromy filtrations and the topology of tropical varieties. Can. J. Math. 64 (4), 845–868 (2012) · Zbl 1312.14145 · doi:10.4153/CJM-2011-067-9
[12] Kapovich, M., Kollár, J.: Fundamental groups of links of isolated singularities. J. Am. Math. Soc. 27 (4), 929–952 (2014) · Zbl 1307.14005 · doi:10.1090/S0894-0347-2014-00807-9
[13] Kollár, J.: Simple normal crossing varieties with prescribed dual complex. Algebra Geom. 1 (1), 57–68 (2014) · Zbl 1307.14059 · doi:10.14231/AG-2014-004
[14] Kulikov, V.S.: Degenerations of K3 surfaces and Enriques surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 41 (5), 1008–1042 (1977) · Zbl 0367.14014
[15] Morrison, D.R.: Semistable degenerations of Enriques’ and hyperelliptic surfaces. Duke Math. J. 48 (1), 197–249 (1981) · Zbl 0476.14015 · doi:10.1215/S0012-7094-81-04813-4
[16] Perrson, U.: On degenerations of algebraic surfaces. Mem. Am. Math. Soc. 11(189), pp. xv+144 (1977)
[17] Persson, U., Pinkham, H.: Degeneration of surfaces with trivial canonical bundle. Ann. Math. (2) 113 (1), 45–66 (1981) · Zbl 0426.14015 · doi:10.2307/1971133
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