Mixed Bruce-Roberts numbers

[EN] We extend the notions of mu*- sequences and Tjurina numbers of functions to the framework of Bruce-Roberts numbers, that is, to pairs formed by the germ at 0 of a complex analytic variety X. Cn and a finitely R( X)-determined analytic function germ f : (Cn, 0). (C, 0). We analyze some fundamental properties of these numbers.Part of this work was developed during the stay of the first author at the Departamento de Matematica of ICMC, Sao Carlos, Universidade de Sao Paulo (Brazil), in February and July 2018. The first author wishes to thank this institution for their hospitality and working conditions and to FAPESP for financial support. The first author was partially supported by MICINN Grant PGC2018-094889-B-I00 and FAPESP Grant 2014/00304-2. The second author was partially supported by CNPq Grant 306306/2015-8 and FAPESP Grant 2014/00304-2.Bivià-Ausina, C.; Ruas, M. (2020). Mixed Bruce-Roberts numbers. Proceedings of the Edinburgh Mathematical Society. 63(2):456-474. https://doi.org/10.1017/S0013091519000543S456474632Damon, J. (1996). Higher multiplicities and almost free divisors and complete intersections. Memoirs of the American Mathematical Society, 123(589), 0-0. doi:10.1090/memo/0589Wahl, J. M. (1983). Derivations, automorphisms and deformations of quasihomogeneous singularities. Proceedings of Symposia in Pure Mathematics, 613-624. doi:10.1090/pspum/040.2/713285De Goes Grulha, N. (2008). THE EULER OBSTRUCTION AND BRUCE-ROBERTS’ MILNOR NUMBER. The Quarterly Journal of Mathematics, 60(3), 291-302. doi:10.1093/qmath/han011Greuel, G.-M. (1975). Der Gau�-Manin-Zusammenhang isolierter Singularit�ten von vollst�ndigen Durchschnitten. Mathematische Annalen, 214(3), 235-266. doi:10.1007/bf01352108Gaffney, T. (1996). Multiplicities and equisingularity of ICIS germs. Inventiones Mathematicae, 123(1), 209-220. doi:10.1007/bf01232372Damon, J. (2002). On the freeness of equisingular deformations of plane curve singularities. Topology and its Applications, 118(1-2), 31-43. doi:10.1016/s0166-8641(01)00040-2Bruce, J. W., & Roberts, R. M. (1988). Critical points of functions on analytic varieties. Topology, 27(1), 57-90. doi:10.1016/0040-9383(88)90007-9Decker, W. , Greuel, G.-M. , Pfister, G. and Schönemann, H. , Singular 4-0-2. A computer algebra system for polynomial computations. Available at http://www.singular.uni-kl.de (2015).Looijenga, E. J. N. (1984). Isolated Singular Points on Complete Intersections. doi:10.1017/cbo9780511662720AHMED, I., RUAS, M. A. S., & TOMAZELLA, J. N. (2013). Invariants of topological relative right equivalences. Mathematical Proceedings of the Cambridge Philosophical Society, 155(2), 307-315. doi:10.1017/s0305004113000297Aleksandrov, A. G. (1986). COHOMOLOGY OF A QUASIHOMOGENEOUS COMPLETE INTERSECTION. Mathematics of the USSR-Izvestiya, 26(3), 437-477. doi:10.1070/im1986v026n03abeh001155Briançon, J., & Maynadier-Gervais, H. (2002). Sur le nombre de Milnor d’une singularité semi-quasi-homogène. Comptes Rendus Mathematique, 334(4), 317-320. doi:10.1016/s1631-073x(02)02256-2Giusti, M., & Henry, J.-P.-G. (1980). Minorations de nombres de Milnor. Bulletin de la Société mathématique de France, 79, 17-45. doi:10.24033/bsmf.1907Hauser, H., & Müller, G. (1993). Affine varieties and lie algebras of vector fields. Manuscripta Mathematica, 80(1), 309-337. doi:10.1007/bf03026556Liu, Y. (2018). Milnor and Tjurina numbers for a hypersurface germ with isolated singularity. Comptes Rendus Mathematique, 356(9), 963-966. doi:10.1016/j.crma.2018.07.004Nuno-Ballesteros, J. J., Orefice, B., & Tomazella, J. N. (2011). THE BRUCE-ROBERTS NUMBER OF A FUNCTION ON A WEIGHTED HOMOGENEOUS HYPERSURFACE. The Quarterly Journal of Mathematics, 64(1), 269-280. doi:10.1093/qmath/har032Ohmoto, T., Suwa, T., & Yokura, S. (1997). A remark on the Chern classes of local complete intersections. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 73(5), 93-95. doi:10.3792/pjaa.73.93Lê Tráng, D. (1974). Calculation of Milnor number of isolated singularity of complete intersection. Functional Analysis and Its Applications, 8(2), 127-131. doi:10.1007/bf0107859

Real map germs and higher open books

We present a general criterion for the existence of open book structures defined by real map germs (\bR^m, 0) \to (\bR^p, 0), where $m> p \ge 2$, with isolated critical point. We show that this is satisfied by weighted-homogeneous maps. We also derive sufficient conditions in case of map germs with isolated critical value.Comment: 12 page

Singular open book structures from real mappings

We prove extensions of Milnor's theorem for germs with nonisolated singularity and use them to find new classes of genuine real analytic mappings $\psi$ with positive dimensional singular locus \Sing \psi \subset \psi^{-1}(0), for which the Milnor fibration exists and yields an open book structure with singular binding.Comment: more remark

Maps of manifolds into the plane which lift to standard embeddings in codimension two

AbstractLet f:M→R2 be a smooth map of a closed n-dimensional manifold (n⩾2) into the plane and let πn+22 :Rn+2→R2 be an orthogonal projection. We say that f has the standard lifting property, if every embedding f̃ :M→Rn+2 with πn+22∘f̃=f is standard in a certain sense. In this paper we give some sufficient conditions for a generic smooth map f to have the standard lifting property when M is a closed surface or an n-dimensional homotopy sphere