Thom polynomials of Morin singularities. (English) Zbl 1247.58021
Authors’ abstract: We prove a formula for Thom polynomials of \(A_d\) singularities in any codimension. We use a combination of the test-curve model of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and are computationally effective up to \(d=6\).
Reviewer: Viorel Vâjâitu (Bucureşti)
MSC:
| 58K30 | Global theory of singularities |
| 57R45 | Singularities of differentiable mappings in differential topology |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
References:
| [1] | V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasilliev, ”Singularity theory. I,” in Dynamical Systems. VI, New York: Springer-Verlag, 1993, vol. 6, p. iv. · Zbl 0786.00006 |
| [2] | G. Bérczi, Multidegrees of singularities and nonreductive quotients, 2008. |
| [3] | G. Bérczi, L. M. Fehér, and R. Rimányi, ”Expressions for resultants coming from the global theory of singularities,” in Topics in Algebraic and Noncommutative Geometry, Providence, RI: Amer. Math. Soc., 2003, vol. 324, pp. 63-69. · Zbl 1052.57043 |
| [4] | N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, New York: Springer-Verlag, 1992. · Zbl 0744.58001 |
| [5] | N. Berline and M. Vergne, ”Zéros d’un champ de vecteurs et classes caractéristiques équivariantes,” Duke Math. J., vol. 50, iss. 2, pp. 539-549, 1983. · Zbl 0515.58007 · doi:10.1215/S0012-7094-83-05024-X |
| [6] | J. M. Boardman, ”Singularities of differentiable maps,” Inst. Hautes Études Sci. Publ. Math., iss. 33, pp. 21-57, 1967. · Zbl 0165.56803 · doi:10.1007/BF02684585 |
| [7] | R. Bott, ”Vector fields and characteristic numbers,” Michigan Math. J., vol. 14, pp. 231-244, 1967. · Zbl 0145.43801 · doi:10.1307/mmj/1028999721 |
| [8] | R. Bott and L. W. Tu, Differential forms in algebraic topology, New York: Springer-Verlag, 1982, vol. 82. · Zbl 0496.55001 |
| [9] | J. Damon, Thom polynomials for contact class singularities, 1972. |
| [10] | B. Doran and F. Kirwan, Towards nonreductive geometric invariant theory. · Zbl 1143.14039 · doi:10.4310/PAMQ.2007.v3.n1.a3 |
| [11] | M. Duflo and M. Vergne, ”Orbites coadjointes et cohomologie équivariante,” in The Orbit Method in Representation Theory, Boston, MA: Birkhäuser, 1990, vol. 82, pp. 11-60. · Zbl 0741.22008 |
| [12] | D. Eisenbud and J. Harris, The Geometry of Schemes, New York: Springer-Verlag, 2000, vol. 197. · Zbl 0960.14002 · doi:10.1007/b97680 |
| [13] | D. Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry, New York: Springer-Verlag, 1995, vol. 150. · Zbl 0819.13001 |
| [14] | L. M. Fehér and R. Rimányi, ”Thom polynomial computing strategies. A survey,” in Singularity theory and its applications, Tokyo: Math. Soc. Japan, 2006, pp. 45-53. · Zbl 1135.58303 |
| [15] | L. M. Fehér and R. Rimányi, ”Calculation of Thom polynomials and other cohomological obstructions for group actions,” in Real and Complex Singularities, Providence, RI: Amer. Math. Soc., 2004, vol. 354, pp. 69-93. · Zbl 1074.32008 |
| [16] | L. M. Fehér and R. Rimányi, ”On the structure of Thom polynomials of singularities,” Bull. Lond. Math. Soc., vol. 39, iss. 4, pp. 541-549, 2007. · Zbl 1140.32019 · doi:10.1112/blms/bdm023 |
| [17] | W. Fulton, Intersection Theory, New York: Springer-Verlag, 1984, vol. 2. · Zbl 0541.14005 |
| [18] | W. Fulton, Introduction to Toric Varieties, Princeton, NJ: Princeton Univ. Press, 1993, vol. 131. · Zbl 0813.14039 · doi:10.1515/9781400882526 |
| [19] | W. Fulton, Young Tableaux, Cambridge: Cambridge Univ. Press, 1997, vol. 35. · Zbl 0878.14034 |
| [20] | T. Gaffney, ”The Thom polynomial of \(\overline{\sum ^{1111}}\),” in Singularities, Part 1, Providence, R.I.: Amer. Math. Soc., 1983, vol. 40, pp. 399-408. · Zbl 0531.58012 |
| [21] | V. Gasharov and V. Reiner, ”Cohomology of smooth Schubert varieties in partial flag manifolds,” J. London Math. Soc., vol. 66, iss. 3, pp. 550-562, 2002. · Zbl 1064.14056 · doi:10.1112/S0024610702003605 |
| [22] | V. W. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, New York: Springer-Verlag, 1999. · Zbl 0934.55007 |
| [23] | P. Griffiths and J. Harris, Principles of Algebraic Geometry, New York: Wiley-Interscience [John Wiley & Sons], 1978. · Zbl 0408.14001 |
| [24] | A. Haefliger and A. Kosiński, ”Un théorème de Thom sur les singularités des applications différentiables,” Séminaire Henri Cartan, vol. 9, 1956-57. |
| [25] | R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52. · Zbl 0367.14001 |
| [26] | A. Joseph, ”On the variety of a highest weight module,” J. Algebra, vol. 88, iss. 1, pp. 238-278, 1984. · Zbl 0539.17006 · doi:10.1016/0021-8693(84)90100-5 |
| [27] | M. Kazarian, Thom polynomials. · Zbl 1149.58013 |
| [28] | B. KHomHuves, Thom polynomials via restriction equations, 2003. |
| [29] | V. Lakshmibai and B. Sandhya, ”Criterion for smoothness of Schubert varieties in \({ Sl}(n)/B\),” Proc. Indian Acad. Sci. Math. Sci., vol. 100, iss. 1, pp. 45-52, 1990. · Zbl 0714.14033 · doi:10.1007/BF02881113 |
| [30] | J. M. Lee, Introduction to Smooth Manifolds, New York: Springer-Verlag, 2003, vol. 218. · Zbl 1030.53001 |
| [31] | J. Martinet, ”Déploiements versels des applications différentiables et classification des applications stables,” in Singularités d’Applications Différentiables, New York: Springer-Verlag, 1976, vol. 535, pp. 1-44. · Zbl 0362.58004 · doi:10.1007/BFb0080494 |
| [32] | V. Mathai and D. Quillen, ”Superconnections, Thom classes, and equivariant differential forms,” Topology, vol. 25, iss. 1, pp. 85-110, 1986. · Zbl 0592.55015 · doi:10.1016/0040-9383(86)90007-8 |
| [33] | J. N. Mather, ”On Thom-Boardman singularities,” in Dynamical Systems, New York: Academic Press, 1973, pp. 233-248. · Zbl 0292.58004 |
| [34] | J. N. Mather, ”Stability of \(C^{\infty }\) mappings. VI: The nice dimensions,” in Proceedings of Liverpool Singularities-Symposium, I (1969/70), New York, 1971, pp. 207-253. · Zbl 0211.56105 |
| [35] | B. Morin, ”Formes canoniques des singularités d’une application différentiable,” C.R. Acad. Sci. Paris, vol. 260, pp. 5662-5665; 6503, 1965. · Zbl 0178.26801 |
| [36] | D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Third ed., New York: Springer-Verlag, 1994, vol. 34. · Zbl 0797.14004 · doi:10.1007/978-3-642-57916-5 |
| [37] | E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, New York: Springer-Verlag, 2005, vol. 227. · Zbl 1090.13001 |
| [38] | P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Bombay: Tata Institute of Fundamental Research, 1978, vol. 51. · Zbl 0411.14003 |
| [39] | I. R. Porteous, ”Simple singularities of maps,” in Proceedings of Liverpool Singularities Symposium, I (1969/70), New York, 1971, pp. 286-307. · Zbl 0221.57016 · doi:10.1007/BFb0066829 |
| [40] | I. R. Porteous, ”Probing singularities,” in Singularities, Part 2, Providence, R.I.: Amer. Math. Soc., 1983, vol. 40, pp. 395-406. · Zbl 0523.58010 |
| [41] | P. Pragacz, Thom polynomials and Schur-functions I.. · Zbl 1157.58015 |
| [42] | F. Ronga, ”Le calcul des classes duales aux singularites de Boardman d’ordre 2,” C. R. Acad. Sci. Paris Sér. A-B, vol. 270, p. a582-a584, 1970. · Zbl 0191.54202 |
| [43] | F. Ronga, ”A new look at Faà de Bruno’s formula for higher derivatives of composite functions and the expression of some intrinsic derivatives,” in Singularities, Part 2, Providence, RI: Amer. Math. Soc., 1983, vol. 40, pp. 423-431. · Zbl 0541.58010 |
| [44] | R. Rimányi, ”Thom polynomials, symmetries and incidences of singularities,” Invent. Math., vol. 143, iss. 3, pp. 499-521, 2001. · Zbl 0985.32012 · doi:10.1007/s002220000113 |
| [45] | W. Rossmann, ”Equivariant multiplicities on complex varieties,” in Orbites Unipotentes et Représentations, III, , 1989, vol. 173-174, pp. 313-330. · Zbl 0691.32004 |
| [46] | A. Szenes, ”Iterated residues and multiple Bernoulli polynomials,” Internat. Math. Res. Notices, iss. 18, pp. 937-956, 1998. · Zbl 0968.11015 · doi:10.1155/S1073792898000567 |
| [47] | R. Thom, ”Les singularités des applications différentiables,” Ann. Inst. Fourier, Grenoble, vol. 6, pp. 43-87, 1955-1956. · Zbl 0075.32104 · doi:10.5802/aif.60 |
| [48] | M. Vergne, ”Polynômes de Joseph et représentation de Springer,” Ann. Sci. École Norm. Sup., vol. 23, iss. 4, pp. 543-562, 1990. · Zbl 0718.22009 |
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.
