Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T01:23:49.411Z Has data issue: false hasContentIssue false
  • English
  • Français

NOTES ON VANISHING CYCLES AND APPLICATIONS

Part of: Singularities Theory of singularities and catastrophe theory Differential equations in the complex domain Local theory

Published online by Cambridge University Press:  29 October 2020

LAURENŢIU G. MAXIM Open the ORCID record for LAURENŢIU G. MAXIM [Opens in a new window]
LAURENŢIU G. MAXIM*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, MadisonWI53706, USA
*
e-mail: maxim@math.wisc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

Keywords

hypersurface singularitiesMilnor fibrationvanishing cyclesnearby cyclesperverse sheavescharacteristic classes

MSC classification

Primary: 14B05: Singularities
Secondary: 14B07: Deformations of singularities 32S05: Local singularities 32S20: Global theory of singularities; cohomological properties 32S25: Surface and hypersurface singularities 32S30: Deformations of singularities; vanishing cycles 32S50: Topological aspects 34M35: Singularities, monodromy, local behavior of solutions, normal forms 58K30: Global theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 3 , December 2020 , pp. 371 - 415
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Finnur Larusson

The author was supported by the Simons Foundation Collaboration Grant #567077 and by the Sydney Mathematical Research Institute.

References

A’Campo, N., ‘Le nombre de Lefschetz d’une monodromie’, Indag. Math. 35 (1973), 113–118. CrossRefGoogle Scholar
Behrend, K., ‘Donaldson-Thomas type invariants via microlocal geometry’, Ann. Math. (2) 170 (2009), 1307–1338. CrossRefGoogle Scholar
Beilinson, A. A., ‘How to glue perverse sheaves’, in: K-Theory, Arithmetic and Geometry (Moscow, 1984–1986), Lecture Notes in Mathematics, 1289 (Springer, Berlin, 1987), 42–51.Google Scholar
Beilinson, A. A., Bernstein, J. N. and Deligne, P., ‘Faisceaux pervers’, in: Analysis and Topology on Singular Spaces, I (Luminy, 1981). Astérisque, Vol. 100 (Société Mathématique de France, Paris, 1982), 5–171. Google Scholar
Blanc, A., Robalo, M., Toën, B. and Vezzosi, G., ‘Motivic realizations of singularity categories and vanishing cycles’, J. Éc. Polytech. Math. 5 (2018), 651–747.CrossRefGoogle Scholar
Brasselet, J.-P., Schürmann, J. and Yokura, S., ‘Hirzebruch classes and motivic Chern classes for singular spaces’, J. Topol. Anal. 2(1) (2010), 1–55.CrossRefGoogle Scholar
Brav, C., Bussi, V., Dupont, D., Joyce, D. and Szendröi, B., ‘Symmetries and stabilization for sheaves of vanishing cycles’, J. Singul. 11 (2015), 85–151 (with an Appendix by Jörg Schürmann).Google Scholar
Brieskorn, E., ‘Beispiele zur Differentialtopologie von Singularitäten’, Invent. Math. 2 (1966), 1–14.CrossRefGoogle Scholar
Budur, N. and Saito, M., ‘Multiplier ideals, V-filtration, and spectrum’, J. Algebraic Geom. 14 (2005), 269–282.Google Scholar
Bussi, V., Joyce, D. and Meinhardt, S., ‘On motivic vanishing cycles of critical loci’, J. Algebraic Geom. 28(3) (2019), 405–438.Google Scholar
Cappell, S., Maxim, L., Schürmann, J. and Shaneson, J., ‘Characteristic classes of complex hypersurfaces’, Adv. Math. 225(5) (2010), 2616–2647.CrossRefGoogle Scholar
Deligne, P., ‘La conjecture de Weil. I’, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.CrossRefGoogle Scholar
Deligne, P., Katz, N. and Katz, N. M., Groupes de Monodromie en Géométrie Algébrique. I, Lecture Notes in Mathematics, 288 (Springer, Berlin–New York, 1972).CrossRefGoogle Scholar
Deligne, P., Katz, N. and Katz, N. M., Groupes de Monodromie en Géométrie Algébrique. II, Lecture Notes in Mathematics, 340 (Springer, Berlin–New York, 1973).CrossRefGoogle Scholar
Denef, J. and Loeser, F., ‘Motivic Igusa zeta functions’, J. Algebraic Geom. 7(3) (1998), 505–537. Google Scholar
Denef, J. and Loeser, F., ‘Motivic exponential integrals and a motivic Thom-Sebastiani theorem’, Duke Math. J. 99(2) (1999), 285–309.CrossRefGoogle Scholar
Denef, J. and Loeser, F., ‘Geometry on arc spaces of algebraic varieties’, in: European Congress of Mathematics, Vol. I (eds. C. Casacuberta, R. M. Miró-Roig, J. Verdera and S. Xambó-Descamps) Progress in Mathematics, 201 (Birkhäuser, Basel, 2001), 327–348.CrossRefGoogle Scholar
Dimca, A., Topics on Real and Complex Singularities, Advanced Lectures in Mathematics (Friedr. Vieweg & Sohn, Braunschweig, 1987). CrossRefGoogle Scholar
Dimca, A., Singularities and Topology of Hypersurfaces (Springer, New York, 1992). CrossRefGoogle Scholar
Dimca, A., Sheaves in Topology (Springer-Verlag, Berlin, 2004). CrossRefGoogle Scholar
Dimca, A. and Saito, M., ‘Some consequences of perversity of vanishing cycles’, Ann. Inst. Fourier (Grenoble) 54(6) (2004), 1769–1792. CrossRefGoogle Scholar
Dimca, A. and Saito, M., ‘Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations’, J. Algebraic Geom. 21(2) (2012), 247–271.CrossRefGoogle Scholar
Dimca, A. and Saito, M., ‘Number of Jordan blocks of the maximal size for local monodromies’, Compos. Math. 150(3) (2014), 344–368. CrossRefGoogle Scholar
Draisma, J., Horobeţ, E., Ottaviani, G., Sturmfels, B. and Thomas, R., ‘The Euclidean distance degree of an algebraic variety’, Found. Comput. Math. 16(1) (2016), 99–149. Google Scholar
Durfee, A. H., ‘Fifteen characterizations of rational double points and simple critical points’, Enseign. Math. (2) 25(1–2) (1979), 131–163.Google Scholar
Fulton, W. and Johnson, K., ‘Canonical classes on singular varieties’, Manuscripta Math. 32 (1980), 381–389. CrossRefGoogle Scholar
Gaitsgory, D., ‘Construction of central elements in the affine Hecke algebra via nearby cycles’, Invent. Math. 144(2) (2001), 253–280. Google Scholar
Goresky, M. and MacPherson, R., Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 14 ( Springer, Berlin, 1988).Google Scholar
Guibert, G., ‘Motivic vanishing cycles and applications’, in: Singularity Theory (World Scientific, Hackensack, NJ, 2007), 613–623. CrossRefGoogle Scholar
Hotta, R., Takeuchi, K. and Tanisaki, T., D-modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, 236 (Birkhäuser, Boston, 2008).Google Scholar
Ishii, S., ‘On isolated Gorenstein singularities’, Math. Ann. 270 (1985), 541–554.Google Scholar
Kashiwara, M., ‘Vanishing cycle sheaves and holonomic systems of differential equations’, in: Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, 1016 (Springer, Berlin, 1983), 134–142.CrossRefGoogle Scholar
Kashiwara, M., ‘The Riemann–Hilbert problem for holonomic systems’, Publ. Res. Inst. Math. Sci. 20(2) (1984), 319–365.CrossRefGoogle Scholar
Kashiwara, M. and Schapira, P., Sheaves on Manifolds, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 292 (Springer, Berlin, 1994).Google Scholar
Kato, M. and Matsumoto, Y., ‘On the connectivity of the Milnor fiber of a holomorphic function at a critical point’, Manifolds Proc. Int. Conf., Tokyo, 1973 (University of Tokyo Press, Tokyo, 1975), 131–136.Google Scholar
Kervaire, M. A. and Milnor, J. W., ‘Groups of homotopy spheres’, Ann. Math. (2) 77 (1963), 504–537. CrossRefGoogle Scholar
Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations. Preprint, arXiv:0811.2435, 2008. Google Scholar
Landman, A., ‘On the Picard–Lefschetz transformation for algebraic manifolds acquiring general singularities’, Trans. Am. Math. Soc. 181 (1973), 89–126. CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in Algebraic Geometry. II (Springer, Berlin, 2004).Google Scholar
Lê, D. T., Topologie des singularités des hypersurfaces complexes, Singularités à Cargèse (Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972), Astérisque, 7–8 (Société Mathematique de France, Paris, 1973), 171–182.Google Scholar
Lê, D. T., ‘Some remarks on relative monodromy’, in: Real and Complex Singularities, Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, August 5–25, 1976 (ed. P. Holm) (Sijthoff and Noordhoff, Alphen aan den Rijn, 1977), 397–403. Google Scholar
Lê, D. T., ‘Sur les cycles évanouissants des espaces analytiques’, C. R. Acad. Sci. Paris Sér. A-B 288(4) (1979), A283–A285.Google Scholar
MacPherson, R., ‘Chern classes for singular algebraic varieties’, Ann. of Math. (2) 100 (1974), 423–432.CrossRefGoogle Scholar
MacPherson, R., Intersection Homology and Perverse Sheaves, Unpublished Colloquium Lectures, 1990. Google Scholar
Massey, D. B., ‘Critical points of functions on singular spaces’, Topol. Appl. 103 (2000), 55–93. CrossRefGoogle Scholar
Massey, D. B., ‘The Sebastiani–Thom isomorphism in the derived category’, Compos. Math. 125 (2001), 353–362.Google Scholar
Massey, D. B., ‘Natural commuting of vanishing cycles and the Verdier dual’, Pac. J. Math. 284 (2016), 431–437. Google Scholar
Maulik, D. and Toda, Y., ‘Gopakumar–Vafa invariants via vanishing cycles’, Invent. Math. 213(3) (2018), 1017–1097.CrossRefGoogle Scholar
Maxim, L., Intersection Homology and Perverse Sheaves, with Applications to Singularities, Graduate Texts in Mathematics, 281 (Springer, New York, 2019). CrossRefGoogle Scholar
Maxim, L., Topological methods in algebraic geometry and algebraic statistics, Rev. Roumaine Math. Pures Appl. 65(3) (2020), 311–325. Google Scholar
Maxim, L., Păunescu, L. and Tibăr, M., ‘Vanishing cohomology and Betti bounds for complex projective hypersurfaces’, Preprint, arXiv:2004.07686, 2020. Google Scholar
Maxim, L., Păunescu, L. and Tibăr, M., ‘The vanishing cohomology of non-isolated hypersurface singularities’, Preprint, arXiv:2007.07064, 2020. Google Scholar
Maxim, L., Rodriguez, J. and Wang, B., ‘Euclidean distance degree of the multiview variety’, SIAM J. Appl. Algebra Geom. 4(1) (2020), 28–48. CrossRefGoogle Scholar
Maxim, L., Rodriguez, J. and Wang, B., ‘Defect of Euclidean distance degree’, Adv. in Appl. Math. 121 (2020), 102101. CrossRefGoogle Scholar
Maxim, L., Rodriguez, J. and Wang, B., ‘A Morse theoretic approach to non-isolated singularities and applications to optimization’, Preprint, arXiv:2002.00406, 2020. Google Scholar
Maxim, L., Saito, M. and Schürmann, J., ‘Hirzebruch–Milnor classes of complete intersections’, Adv. Math. 241 (2013), 220–245.CrossRefGoogle Scholar
Maxim, L., Saito, M. and Schürmann, J., ‘Spectral Hirzebruch–Milnor classes of singular hypersurfaces’, Math. Ann. 377(1–2) (2020), 281–315.Google Scholar
Maxim, L., Saito, M. and Schürmann, J., ‘Thom–Sebastiani theorems for filtered D-modules and for multiplier ideals’, Int. Math. Res. Not. IMRN 2020(1), 91–111.CrossRefGoogle Scholar
Milnor, J. W., ‘Construction of universal bundles II’, Ann. of Math. (2) 63 (1956), 430–436.CrossRefGoogle Scholar
Milnor, J. W., ‘Singular points of complex hypersurfaces’, Ann. Math. Stud. 61 (1968), 1.Google Scholar
Mirković, I. and Vilonen, K., ‘Geometric Langlands duality and representations of algebraic groups over commutative rings’, Ann. of Math. (2) 166(1) (2007), 95–143.CrossRefGoogle Scholar
Oka, M., ‘On the homotopy types of hypersurfaces defined by weighted homogeneous polynomials’, Topology 12 (1973), 19–32. CrossRefGoogle Scholar
Parusiński, A. and Pragacz, P., ‘Characteristic classes of hypersurfaces and characteristic cycles’, J. Algebraic Geom. 10(1) (2001), 63–79.Google Scholar
Reich, R., ‘Notes on Beilinson’s “How to glue perverse sheaves”’, J. Singul. 1 (2010), 94–115.CrossRefGoogle Scholar
Saito, M., ‘Modules de Hodge polarisables’, Publ. Res. Inst. Math. Sci. 24(6) (1988), 849–995.CrossRefGoogle Scholar
Saito, M., ‘Mixed Hodge modules’, Publ. Res. Inst. Math. Sci. 26(2) (1990), 221–333.Google Scholar
Saito, M., ‘Decomposition theorem for proper Kähler morphisms’, Tohoku Math. J. (2) 42(2) (1990), 127–147. CrossRefGoogle Scholar
Saito, M., ‘On Steenbrink’s conjecture’, Math. Ann. 289(4) (1991), 703–716. Google Scholar
Sakamoto, K., ‘Milnor fiberings and their characteristic maps’, Manifolds Proc. Int. Conf., Tokyo, 1973 (University of Tokyo Press, Tokyo, 1975), 145–150.Google Scholar
Scherk, J. and Steenbrink, J., ‘On the mixed Hodge structure on the cohomology of the Milnor fibre’, Math. Ann. 271 (1985), 641–665.CrossRefGoogle Scholar
Schürmann, J., Topology of Singular Spaces and Constructible Sheaves, Monografie Matematyczne, 63 (Birkhauser, 2003). Google Scholar
Schürmann, J., ‘Specialization of motivic Hodge–Chern classes’ Preprint, arXiv:0909.3478, 2009.Google Scholar
Sebastiani, M. and Thom, R., ‘Un résultat sur la monodromie’, Invent. Math. 13 (1971), 90–96.CrossRefGoogle Scholar
Siersma, D. and Tibăr, M., ‘Vanishing homology of projective hypersurfaces with 1-dimensional singularities’, Eur. J. Math. 3(3) (2017), 565–586.CrossRefGoogle Scholar
Steenbrink, J., ‘The spectrum of hypersurface singularities’, Astérisque 179–180 (1989), 163–184.Google Scholar
Szendröi, B., ‘Cohomological Donaldson–Thomas theory’, in: String–Math 2014, Proceedings of Symposia in Pure Mathematics, 93 (American Mathematical Society, Providence, RI, 2016), 363–396.Google Scholar
Varchenko, A., ‘Asymptotic Hodge structure in the vanishing cohomology’, Math. USSR, Izv. 18 (1982), 465–512.Google Scholar
Verdier, J.-L., ‘Spécialisation des classes de Chern’, Astérisque 82–83 (1981), 149–159.Google Scholar
Verdier, J.-L., ‘Extension of a perverse sheaf over a closed subspace, Differential systems and singularities (Luminy, 1983)’, Astérisque 130 (1985), 210–217.Google Scholar
Zhu, X., ‘An introduction to affine Grassmannians and the geometric Satake equivalence’, in: Geometry of Moduli Spaces and Representation Theory, IAS/Park City Mathematics Series, 24 (American Mathematical Society, Providence, RI, 2017), 59–154.CrossRefGoogle Scholar