The Hodge spectrum of analytic germs on isolated surface singularities. (English. French summary) Zbl 1315.32007
Summary: We use topological methods to prove a semicontinuity property of the Hodge spectra for analytic germs defined on an isolated surface singularity. For this we introduce an analogue of the Seifert matrix (the fractured Seifert matrix), and of the Levine-Tristram signatures associated with it, defined for null-homologous links in arbitrary three dimensional manifolds. Moreover, we establish Murasugi type inequalities in the presence of cobordisms of links.{
}It turns out that the fractured Seifert matrix determines the Hodge spectrum and the Murasugi type inequalities can be read as spectrum semicontinuity inequalities.
MSC:
| 32S25 | Complex surface and hypersurface singularities |
| 32S55 | Milnor fibration; relations with knot theory |
| 14B07 | Deformations of singularities |
| 14D07 | Variation of Hodge structures (algebro-geometric aspects) |
| 14H50 | Plane and space curves |
| 32G20 | Period matrices, variation of Hodge structure; degenerations |
Keywords:
surface singularities; Hodge spectra of analytic germs; semicontinuity of the spectrum; Levine-Tristram signatures; Seifert forms; variation structures; mixed Hodge structureReferences:
| [1] | Arnold, V. I.; Gussein-Zade, S. M.; Varchenko, A. N., Singularities of Differentiable Mappings. II (1984), Nauka: Nauka Moscow · Zbl 0545.58001 |
| [2] | Borodzik, M.; Némethi, A., Hodge-type structures as link invariants, Ann. Inst. Fourier, 63, 1, 269-301 (2013) · Zbl 1275.57018 |
| [3] | Borodzik, M.; Némethi, A., Spectrum of plane curves via knot theory, J. Lond. Math. Soc., 86, 1, 87-110 (2012) · Zbl 1247.32027 |
| [4] | Borodzik, M.; Némethi, A.; Ranicki, A., On the semicontinuity of the mod 2 spectrum of hypersurface singularities, J. Algebr. Geom. (2014), in press |
| [5] | Borodzik, M.; Némethi, A.; Ranicki, A., Codimension 2 embeddings, algebraic surgery and Seifert form |
| [6] | Cisneros-Molina, J. L.; Seade, J.; Snoussi, J., Refinements of Milnor’s fibration theorem for complex singularities, Adv. Math., 222, 937-970 (2009) · Zbl 1180.32009 |
| [7] | Eisenbud, D.; Neumann, W., Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. Math. Stud., vol. 110 (1985), Princeton University Press: Princeton University Press Princeton · Zbl 0628.57002 |
| [8] | Erle, D., Quadratische Formen als Invarianten von Einbettungen der Kodimension 2, Topology, 8, 99-114 (1969) · Zbl 0157.30901 |
| [9] | Farb, B.; Margalit, D., A Primer on Mapping Class Groups, Princeton Math. Ser., vol. 49 (2012), Princeton University Press: Princeton University Press Princeton |
| [10] | Hamm, H., Lokale topologische Eigenschaften komplexer Räume, Math. Ann., 191, 235-252 (1971) · Zbl 0214.22801 |
| [11] | Lê Dũng Tráng, Some remarks on the relative monodromy, (Real and Complex Singularities, Proc. Nordic Summer School, Symp. Math.. Real and Complex Singularities, Proc. Nordic Summer School, Symp. Math., Oslo (1976)), 397-403 · Zbl 0428.32008 |
| [12] | Lê Dũng Tráng, Vanishing cycles on complex analytic sets, Various Problems in Algebraic Analysis, Proc. Sympos.. Various Problems in Algebraic Analysis, Proc. Sympos., Res. Inst. Math. Sci., Kyoto Univ., 1975. Various Problems in Algebraic Analysis, Proc. Sympos.. Various Problems in Algebraic Analysis, Proc. Sympos., Res. Inst. Math. Sci., Kyoto Univ., 1975, Surikaisekikenkyusho Kokyroku, 266, 299-318 (1976) |
| [13] | Dũng Tráng, Lê, Singularités isolées des intersections complêtes, (Introduction à la théorie des singularités, I. Introduction à la théorie des singularités, I, Travaux en Cours, vol. 36 (1988), Hermann: Hermann Paris), 1-48 · Zbl 0687.32010 |
| [14] | Looijenga, E., Isolated Singular Points on Complete Intersections, Lond. Math. Soc. Lect. Note Ser., vol. 77 (1984), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0552.14002 |
| [15] | Milnor, J., On isometries of inner product spaces, Invent. Math., 8, 83-97 (1969) · Zbl 0177.05204 |
| [16] | Milnor, J., Singular Points of Complex Hypersurfaces, Ann. Math. Stud., vol. 61 (1968), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0184.48405 |
| [17] | Némethi, A., The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compos. Math., 98, 23-41 (1995) · Zbl 0851.14015 |
| [18] | Némethi, A.; Steenbrink, J. H.M., On the monodromy of curve singularities, Math. Z., 223, 587-593 (1996) · Zbl 0869.32016 |
| [19] | Neumann, W. D., Invariants of plane curve singularities, (Knots, Braids and Singularities. Knots, Braids and Singularities, Plans-sur-Becs, 1982. Knots, Braids and Singularities. Knots, Braids and Singularities, Plans-sur-Becs, 1982, Monogr. Enseign. Math., vol. 31 (1983), Enseignement Math.: Enseignement Math. Geneva), 223-232 · Zbl 0586.14023 |
| [20] | Neumann, W. D.; Némethi, A.; Pichon, A., Principal analytic link theory in homology sphere links, (Proc. of the Conference in Honor of the 60th Birthday of A. Libgober, Topology of Algebraic Varieties. Proc. of the Conference in Honor of the 60th Birthday of A. Libgober, Topology of Algebraic Varieties, Jaca, Spain, June 2009. Proc. of the Conference in Honor of the 60th Birthday of A. Libgober, Topology of Algebraic Varieties. Proc. of the Conference in Honor of the 60th Birthday of A. Libgober, Topology of Algebraic Varieties, Jaca, Spain, June 2009, Contemp. Math., vol. 538 (2011)), 377-387 · Zbl 1272.32029 |
| [21] | Neumann, W. D.; Pichon, A., Complex analytic realization of links, (Intelligence of Low Dimensional Topology 2006. Intelligence of Low Dimensional Topology 2006, Ser. Knots Everything, vol. 40 (2007), World Sci. Publ.: World Sci. Publ. Hackensack, NJ), 231-238 · Zbl 1146.32013 |
| [22] | Rolfsen, D., Knots and Links, Math. Lect. Ser., vol. 7 (1990), Publish or Perish, Inc.: Publish or Perish, Inc. Houston, TX · Zbl 0854.57002 |
| [23] | Steenbrink, J. H.M., Mixed hodge structures on the vanishing cohomology, (Nordic Summer School/NAVF, Symposium in Math.. Nordic Summer School/NAVF, Symposium in Math., Oslo (1976)), 525-563 · Zbl 0373.14007 |
| [24] | Steenbrink, J. H.M., Mixed Hodge Structures associated with isolated singularities, (Proc. Symp. Pure Math., vol. 40 (2) (1983)), 513-536 · Zbl 0515.14003 |
| [25] | Steenbrink, J. H.M., Semicontinuity of the singularity spectrum, Invent. Math., 79, 557-565 (1985) · Zbl 0568.14021 |
| [26] | Steenbrink, J. H.M., The spectrum of hypersurface singularities, Astérisque, 179-180, 163-184 (1989) · Zbl 0725.14031 |
| [27] | Schrauwen, R.; Steenbrink, J.; Stevens, J., Spectral pairs and the topology of curve singularities, (Complex Geometry and Lie Theory. Complex Geometry and Lie Theory, Sundance, UT, 1989. Complex Geometry and Lie Theory. Complex Geometry and Lie Theory, Sundance, UT, 1989, Proc. Symp. Pure Math., vol. 53 (1991), AMS: AMS Providence), 305-328 · Zbl 0749.14003 |
| [28] | Varchenko, A. N., On the semicontinuity of the spectrum and an upper bound for the number of singular points of projective hypersurfaces, Doklady Akad. Nauk, 270, 6, 1294-1297 (1983) |
| [29] | Varchenko, A. N., On change of discrete invariants of critical points of functions under deformation, Usp. Mat. Nauk, 5, 126-127 (1983) |
| [30] | Zoladek, H., The Monodromy Group, Math Monogr., vol. 67 (2006), Birhäuser Verlag: Birhäuser Verlag Basel · Zbl 1103.32015 |
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.
