×

Boundary complexes and weight filtrations. (English) Zbl 1312.14049

Let \(X\) be a complex algebraic variety of dimension \(n\) and \(\overline{X}\) be its compactification such that boundary \(\partial X\) has a nonsingular neighborhood in which it is a divisor \(D\) with simple normal crossings. One defines the CW-complex \(\Delta(\partial X)\) whose \(k\)-dimensional simplices correspond to irreducible components of intersections of \(k+1\)-dimensional components of \(D\) and where the inclusions of faces correspond to the inclusions of subvarieties. The complex \(\Delta(\partial X)\) was first introduced and studied by V. I. Danilov [Mat. Sb., Nov. Ser. 97(139), 146–158 (1975; Zbl 0321.14010)] that has apparently escaped the attention of the author of the paper under review (as well as of many others working in this area). The English translation of the original paper also has many illuminating notes by its translator F. Zak [Math. USSR, Sb. 26, 137–149 (1976; Zbl 0334.14008)]. One of the main results of Danilov is the homotopy invariance of \(\Delta(\partial X)\).
The paper under review contains several improvements and generalizations of Danilov’s results. Thus the author treats the case when the singular locus of \(X\) is not proper – he considers the simple homotopy type instead of the ordinary homotopy type, and uses a weak factorization theorem to deduce its invariance. Special attention is paid to the case, also considered by Danilov, of the resolution complex associated to a resolution of singularities. For example, the author gives an example of an isolated rational singularity whose resolution is not of the homotopy type of a point (although it is known to be of the rational homotopy type of a point). The author also discusses the relationship (well known under some special restrictive assumptions (see [V. S. Kulikov and P. F. Kurchanov, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 36, 1–217 (1989; Zbl 0881.14003)]) between the cohomology of the boundary and of the resolution complexes to some parts in the mixed Hodge structure of \(X\).

MSC:

14E30 Minimal model program (Mori theory, extremal rays)
14B05 Singularities in algebraic geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14T05 Tropical geometry (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] D. Abramovich, K. Karu, K. Matsuki, and J. Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), 531-572. · Zbl 1032.14003 · doi:10.1090/S0894-0347-02-00396-X
[2] A. Andreotti and T. Frankel, The Lefschetz theorem on hyperplane sections, Ann. of Math. (2) 69 (1959), 713-717. · Zbl 0115.38405 · doi:10.2307/1970034
[3] D. Arapura, P. Bakhtary, and J. Włodarczyk, The combinatorial part of the cohomology of a singular variety, preprint, 2009, arXiv:
[4] —, Weights on cohomology, invariants of singularities, and dual complexes, preprint, 2011, arXiv:
[5] E. Babson and P. Hersh, Discrete Morse functions from lexicographic orders, Trans. Amer. Math. Soc. 357 (2005), 509-534. · Zbl 1050.05117 · doi:10.1090/S0002-9947-04-03495-6
[6] V. Berkovich, An analog of Tate’s conjecture over local and finitely generated fields, Internat. Math. Res. Notices 13 (2000), 665-680. · Zbl 1068.14502 · doi:10.1155/S1073792800000362
[7] A. Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159-183. · Zbl 0441.06002 · doi:10.2307/1999881
[8] S. Boucksom, C. Favre, and M. Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008), 449-494. · Zbl 1146.32017 · doi:10.2977/prims/1210167334
[9] H. Bruggesser and P. Mani, Shellable decompositions of cells and spheres, Math. Scand. 29 (1971), 197-205. · Zbl 0251.52013
[10] M. Cohen, A course in simple-homotopy theory, Grad. Texts in Math., 10, Springer-Verlag, New York, 1973. · Zbl 0261.57009
[11] P. Deligne, Poids dans la cohomologie des variétés algébriques, Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Canad. Math. Congress, 1, pp. 79-85, Montreal, 1975. · Zbl 0334.14011
[12] A. Durfee, A naive guide to mixed Hodge theory, Singularities, part 1 (Arcata, 1981), Proc. Sympos. Pure Math., 40, pp. 313-320, Amer. Math. Soc., Providence, RI, 1983. · Zbl 0521.14003
[13] F. El Zein, Mixed Hodge structures, Trans. Amer. Math. Soc. 275 (1983), 71-106. · doi:10.1090/S0002-9947-1983-0678337-5
[14] C. Favre and M. Jonsson, The valuative tree, Lecture Notes in Math., 1853, Springer-Verlag, Berlin, 2004. · Zbl 1064.14024 · doi:10.1007/b100262
[15] —, Eigenvaluations, Ann. Sci. École Norm. Sup. (4) 40 (2007), 309-349. · Zbl 1135.37018 · doi:10.1016/j.ansens.2007.01.002
[16] —, Dynamical compactifications of \({\mathbf C}^{2},\) Ann. of Math. (2) 173 (2011), 211-248. · Zbl 1244.32012 · doi:10.4007/annals.2011.173.1.6
[17] R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), 90-145. · Zbl 0896.57023 · doi:10.1006/aima.1997.1650
[18] H. Gillet and C. Soulé, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996), 127-176. · Zbl 0863.19002 · doi:10.1515/crll.1996.478.127
[19] F. Guillén and V. Navarro Aznar, Un critère d’extension des foncteurs définis sur les schémas lisses, Publ. Math. Inst. Hautes Études Sci. 95 (2002), 1-91. · Zbl 1075.14012 · doi:10.1007/s102400200003
[20] P. Hacking, The homology of tropical varieties, Collect. Math. 59 (2008), 263-273. · Zbl 1198.14059 · doi:10.1007/BF03191187
[21] M. Hanamura and M. Saito, Weight filtration on the cohomology of algebraic varieties, preprint, 2006..
[22] A. Hatcher, Algebraic topology, Cambridge Univ. Press, Cambridge, 2002. · Zbl 1044.55001
[23] D. Helm and E. Katz, Monodromy filtrations and the topology of tropical varieties, Canad. J. Math. 64 (2012), 845-868. · Zbl 1312.14145 · doi:10.4153/CJM-2011-067-9
[24] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-203, 205-326. · Zbl 0122.38603 · doi:10.2307/1970486
[25] A. Hovanskiĭ, Newton polyhedra and torical varieties, Funkcional. Anal. i Priložen. 11 (1977), 56-64, 96 (Russian); English translation in Funct. Anal. Appl. 11 (1977), 289-296.
[26] E. Hrushovski and F. Loeser, Nonarchimedean topology and stably dominated types, preprint, 2010.
[27] S. Ishii, On isolated Gorenstein singularities, Math. Ann. 270 (1985), 541-554. · Zbl 0541.14002 · doi:10.1007/BF01455303
[28] M. Kapovich and J. Kollár, Fundamental groups of links of isolated singularities, preprint, 2011, arXiv:
[29] K. Karčjauskas, A generalized Lefschetz theorem, Funkcional. Anal. i Priložen. 11 (1977), 80-81 (Russian); English translation in Funct. Anal. Appl. 11 (1977), 312-313.
[30] —, Homotopy properties of algebraic sets, Studies in topology, III, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 83 (1979), 67-72, 103.
[31] E. Katz and A. Stapledon, The tropical motivic nearby fiber, Compositio Math. (to appear). · Zbl 1249.14021
[32] M. Kerz and M. Saito, Cohomological Hasse principle and McKay principle for weight homology, preprint, 2011, arXiv:
[33] J. Kollár, Lectures on resolution of singularities, Ann. of Math. Stud., 166, Princeton Univ. Press, Princeton, NJ, 2007.
[34] S. Kovács, Rational, log canonical, Du Bois singularities: On the conjectures of Kollár and Steenbrink, Compositio Math. 118 (1999), 123-133. · Zbl 0962.14011 · doi:10.1023/A:1001120909269
[35] D. Kozlov, Combinatorial algebraic topology, Algorithms Comput. Math., 21, Springer-Verlag, Berlin, 2008. · Zbl 1130.55001
[36] V. Kulikov and P. Kurchanov, Complex algebraic varieties: Periods of integrals and Hodge structures, Algebraic geometry, III, pp. 1-217, 263-270, Encyclopaedia Math. Sci., 36, Springer-Verlag, Berlin, 1998. · Zbl 0881.14003 · doi:10.1007/978-3-662-03662-4_1
[37] K. Matsuki, Lectures on factorizations of birational maps, preprint, 2000.v1. · doi:10.2748/tmj/1178207758
[38] C. McCrory and A. Parusiński, The weight filtration for real algebraic varieties, Topology of stratified spaces, pp. 121-160, Math. Sci. Res. Inst. Publ., 58, Cambridge Univ. Press, Cambridge, 2011. · Zbl 1240.14012
[39] D. Morrison, Semistable degenerations of Enriques’ and hyperelliptic surfaces, Duke Math. J. 48 (1981), 197-249. · Zbl 0476.14015 · doi:10.1215/S0012-7094-81-04813-4
[40] M. Nagata, Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2 (1962), 1-10. · Zbl 0109.39503
[41] S. Payne, Boundary complexes of varieties, March 25, 2009, lecture at MSRI, available in streaming video through \(\langle\)http://www.msri.org/web/msri/online-videos/\(\rangle.\)
[42] —, Toric vector bundles, branched covers of fans, and the resolution property, J. Algebraic Geom. 18 (2009), 1-36. · Zbl 1161.14039 · doi:10.1090/S1056-3911-08-00485-2
[43] C. Peters and J. Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3), 52, Springer-Verlag, Berlin, 2008. · Zbl 1138.14002 · doi:10.1007/978-3-540-77017-6
[44] V. Shokurov, Complements on surfaces, J. Math. Sci. (New York) 102 (2000), 3876-3932. · Zbl 1177.14078 · doi:10.1007/BF02984106
[45] J. Steenbrink and J. Stevens, Topological invariance of the weight filtration, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), 63-76. · Zbl 0539.14016
[46] D. Stepanov, A note on the dual complex associated to a resolution of singularities, preprint, 2005.v1.
[47] —, Combinatorial structure of exceptional sets in resolutions of singularities, preprint, 2006.v1.
[48] —, A remark on the dual complex of a resolution of singularities, Uspekhi Mat. Nauk 61 (2006), 185-186. · doi:10.4213/rm1697
[49] —, A note on resolution of rational and hypersurface singularities, Proc. Amer. Math. Soc. 136 (2008), 2647-2654. · Zbl 1144.14002 · doi:10.1090/S0002-9939-08-09289-7
[50] S. Takayama, Local simple connectedness of resolutions of log-terminal singularities, Internat. J. Math. 14 (2003), 825-836. · Zbl 1058.14024 · doi:10.1142/S0129167X0300196X
[51] J. Tevelev, Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), 1087-1104. · Zbl 1154.14039 · doi:10.1353/ajm.2007.0029
[52] A. Thuillier, Géométrie toroï dale et géométrie analytique non archimédienne, Manuscripta Math. 123 (2007), 381-451. · Zbl 1134.14018 · doi:10.1007/s00229-007-0094-2
[53] B. Totaro, Topology of singular algebraic varieties, Proceedings of the International Congress of Mathematicians, vol. II (Beijing, 2002), pp. 533-541, Higher Education Press, Beijing, 2002. · Zbl 1057.14030
[54] A. Varchenko, Zeta-function of monodromy and Newton’s diagram, Invent. Math. 37 (1976), 253-262. · Zbl 0333.14007 · doi:10.1007/BF01390323
[55] A. Weber, Pure homology of algebraic varieties, Topology 43 (2004), 635-644. · Zbl 1072.14023 · doi:10.1016/j.top.2003.09.001
[56] J. Włodarczyk, Toroidal varieties and the weak factorization theorem, Invent. Math. 154 (2003), 223-331. · Zbl 1130.14014 · doi:10.1007/s00222-003-0305-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.