# zbMATH — the first resource for mathematics

Singularities of homogeneous quadratic mappings. (English) Zbl 1342.14116
The author considers the affine real variety $$V$$ in $$\mathbb R^n$$ defined as a zero level set of the mapping $$F$$ into $$\mathbb{R}^2$$, build of two quadratic forms. A complete topological description of $$V$$ in all generic cases and the topology of the intersection of $$V$$ with a half-space and the topology of various deformations of $$V_t$$ are found. For the intersection of $$V$$ and the half-space $$Z$$, the author provides a complete description of its topological type in all but three isolated cases in dimension 4, and also in the diagonal case for generic $$V_t$$.

##### MSC:
 14P05 Real algebraic sets 58K05 Critical points of functions and mappings on manifolds 14J17 Singularities of surfaces or higher-dimensional varieties
Full Text:
##### References:
 [1] Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces. Adv. Math. Vol. 225, 1634–1668 (2010) · Zbl 1197.13021 · doi:10.1016/j.aim.2010.03.026 [2] Baskakov, I.V.: Massey triple products in the cohomology of moment-angle complexes. Russ. Math. Surv. 58(5), 1039–1041 (2003) · Zbl 1054.55501 · doi:10.1070/RM2003v058n05ABEH000670 [3] Bazaikin, Y.V., Matvienko, I.V.: On the moment-angle manifolds with positive Ricci curvature, arXiv:1011.5371 (2010) [4] Bosio, F., Meersseman, L.: Real quadrics in $$C\^n$$ , complex manifolds and convex polytopes. Acta Math. 197(1), 53–127 (2006) · Zbl 1157.14313 · doi:10.1007/s11511-006-0008-2 [5] Buchstaber, V.M., Panov, T.E.: Torus Actions and Their Applications in Topology and Combinatorics, University Lecture Seriers. AMS, Providence (2002) · Zbl 1012.52021 [6] Buchstaber, V.M., Ray, N.: Tangential structures on toric manifolds, and connected sums of polytopes. Int. Math. Res. Notes Vol. 4, 193–219 (2001) · Zbl 0996.52013 · doi:10.1155/S1073792801000125 [7] Camacho, C., Kuiper, N., Palis, J.: The topology of holomorphic flows with singularities. Publications Mathématiques. I.H.E.S 48, 5–38 (1978) · Zbl 0411.58018 [8] Chaperon, M.: Géométrie différentielle et singularités de systèmes dynamiques. Astérisque 138–139, 434 (1986) [9] Chaperon, M., López de Medrano, S.: Birth of attracting compact invariant submanifolds diffeomorphic to moment-angle manifolds in generic families of dynamics. C. R. Acad. Sci. Paris, Ser. I 346, 1099–1102 (2008) · Zbl 1152.37014 [10] Denham, G., Suciu, A.: Moment-angle complexes, monomial ideals, and Massey products. Pure Appl. Math. Quart. 3(1), 25–60 (2007) · Zbl 1169.13013 · doi:10.4310/PAMQ.2007.v3.n1.a2 [11] Fernández de Bobadilla, J.: On homotopy types of complements of analytic sets and Milnor Fibers, arXiv:0907.2176 (2009) · Zbl 1214.14005 [12] Franz, M., Puppe, V.: Freeness of equivariant cohomology and mutants of compactified representations, arXiv:0710.2302 (2007) · Zbl 1154.55005 [13] Gitler, S., López de Medrano, S.: Intersections of Quadrics, Moment-Angle Manifolds and Connected Sums, arXiv:0901.2580v4 (2012) · Zbl 1276.14087 [14] Hirzebruch, F.: Arrangements of Lines and Algebraic Surfaces, in Arithmetic and Geometry, Vol. II (= Progr. Math. 36), 113–140, Boston: Birkhauser (1983) · Zbl 0527.14033 [15] Kettner, M.: Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomials, arXiv:0709.3283v (2007) [16] López de Medrano, S.: Topology of the Intersection of Quadrics in $$R\^n$$ , in Algebraic Topology (Arcata Ca, 1986), Springer Verlag LNM 1370 (1989), pp. 280–292. Springer, Berlin [17] López de Medrano, S., Verjovsky, A.: A new family of complex, compact non-symplectic manifolds. Bol. Soc. Brasil. Mat 28, 253–269 (1997) · Zbl 0901.53021 · doi:10.1007/BF01233394 [18] Meersseman, L.: A new geometric construction of compact complex manifolds in any dimension. Math. Ann. 317, 79–115 (2000) · Zbl 0958.32013 · doi:10.1007/s002080050360 [19] Meersseman, L., Verjovsky, A.: Holomorphic principal bundles over projective toric varieties. J. Reine Angew. Math. 572, 57–96 (2004) · Zbl 1070.14047 [20] Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton (1968) · Zbl 0184.48405 [21] Mironov, A., Panov, T.: Intersections of quadrics, moment-angle manifolds and Hamiltonian-minimal Lagrangian embeddings, arXiv:1103.4970, 2011. · Zbl 1282.53066 [22] Wall, C.T.C.: Stability, pencils and polytopes. Bull. London Math. Soc. 12, 401–421 (1980) · Zbl 0433.58006 · doi:10.1112/blms/12.6.401
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.