On the links of simple singularities, simple elliptic singularities and cusp singularities

• Autorzy: Kasuya N.
• Języki publikacji: EN

Abstrakty

EN

This is a survey article about the study of the links of some complex hypersurface singularities in C3. We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group SU (2), Nil3 or Sol3, respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of S5. Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in S5 as codimension two contact submanifolds.

Słowa kluczowe

EN

contact structures; complex singularities

PL

struktury kontaktowe; osobliwości złożone

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2015
• Tom: Vol. 48, nr 2
• Strony: 289--312
• Opis fizyczny: Bibliogr. 32 poz., rys.

Twórcy

autor

Kasuya N.

• Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-Ku, Tokyo 153-8914, Japan

Bibliografia

• [1] V. I. Arnol’d, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps I, Monograph in Math., vol. 82, Birkhäuser, 1985.
• [2] M. Bhupal, B. Ozbagci, Canonical contact structures on fibred singularity links, Bull. London Math. Soc. 46(3) (2014), 576–586.
• [3] E. Brieskorn, Singular elements of semisimple algebraic groups, Actes Congres Intern. Math., vol. 2, 1970, 279–284.
• [4] C. Caubel, A. Nemethi, P. Popescu-Pampu, Milnor open books and Milnor fillable contact 3-manifolds, Topology 45(3) (2006), 673–689.
• [5] I. Dolgachev, On the link space of a Gorenstein quasihomogeneous surface singularity, Math. Ann. 265 (1983), 529–540.
• [6] A. Durfee, Fifteen characterizations of rational double points and simple critical points, Enseign. Math. 25 (1979), 131–163.
• [7] F. Ehlers, W. D. Neumann, J. Scherk, Links of surface singularities and CR space forms, Comment. Math. Helv. 62 (1987), 240–264.
• [8] A. M. Gabriélov, Dynkin diagrams for unimodal singularities, Funktsional. Anal. I Prilozhen. 8(3) (1974), 1–6.
• [9] F. Hirzebruch, Hilbert modular surfaces, Enseign. Math. 19 (1973), 183–281.
• [10] U. Karras, Deformations of cusp singularities, Proc. Sympos. Pure Math., vol. 30, Amer. Math. Soc., 1977, 37–40.
• [11] N. Kasuya, The canonical contact structure on the link of a cusp singularity, Tokyo J. Math. 37(1) (2014), 1–20.
• [12] F. Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Teubner, 1884, Dover, 1956.
• [13] H. B. Laufer, Normal Two Dimensional Singularities, Ann. of Math. Stud., vol. 71, 1971.
• [14] H. B. Laufer, Two dimensional taut singularities, Math. Ann. 205 (1973), 131–164.
• [15] H. B. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), 1257–1295.
• [16] Y. Lekili, B. Ozbagci, Milnor fillable contact structures are universally tight, Math. Res. Lett. 17(6) (2010), 1055–1063.
• [17] E. Looijenga, Isolated Singular Points on Complete Intersections, London Math. Soc. Lecture Note Ser., vol. 77, 1984.
• [18] J. McKay, Graphs, singularities and finite groups, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., 1980, 183–186.
• [19] J. W. Milnor, On the 3-dimensional Brieskorn manifolds M(p, q, r), Ann. of Math. Stud. 84 (1975), 175–225.
• [20] Y. Mitsumatsu, Anosov flows and non-Stein symplectic manifolds, Ann. Inst. Fourier 45 (1995), 1407–1421.
• [21] A. Mori, Reeb foliations on S5 and contact 5-manifolds violating the Thuston-Bennequin inequality, preprint, 2009. arXiv:0906.3237v2 [math.GT]
• [22] A. Mori, The Reeb foliation arises as a family of Legendrian submanifolds at the end of a deformation of the standard S3 in S5, C. R. Math. Acad. Sci. Paris, Ser. I 350 (2012), 67–70.
• [23] A. Nemethi, The signature of f(x, y) + zn = 0, London Math. Soc. Lecture Note Ser. 263 (1999), 131–149.
• [24] W. D. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299–344.
• [25] W. D. Neumann, Geometry of quasihomogeneous surface singularities, Proc. Sympos. Pure Math., vol. 40, part 2, Amer. Math. Soc., 1983, 245–258.
• [26] P. Orlik, P. Wagreich, Isolated singularities of algebraic surfaces with C* action, Ann. of Math. 93 (1971), 205–228.
• [27] H. Pinkham, Normal surface singularities with C*-action, Math. Ann. 227 (1977), 183–193.
• [28] K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142.
• [29] K. Saito, Einfach elliptische Singularitäten, Invent. Math. 23 (1974), 289–325.
• [30] J. Seade, On the Topology of Isolated Singularities in Analytic Spaces, Progress in Mathematics, vol. 241, Birkhäuser, Basel, 2006.
• [31] P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Math., vol. 815, Springer, 1980.
• [32] P. Wagreich, The structure of quasihomogeneous singularities, Proc. Sympos. Pure Math., vol. 40, part 2, Amer. Math. Soc., 1983, 593–611.