×
Bosio, Frédéric; Meersseman, Laurent

Real quadrics in \(\mathbb C^n\), complex manifolds and convex polytopes. (English) Zbl 1157.14313

Acta Math. 197, No. 1, 53-127 (2006).
Summary: We investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics \(\mathbb{C}^n\) which are invariant with respect to the natural action of the real torus \((S^1)^n\) onto \(\mathbb{C}^n\). The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.

Cited in 5 Reviews
Cited in 66 Documents

MSC:

14P05 Real algebraic sets
32V40 Real submanifolds in complex manifolds

Keywords:

Affine complex manifolds; Combinatorics of convex polytopes; Real quadrics; Equivariant surgery; Subspace arrangements
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Aleksandrov, P.S.: Combinatorial Topology, vol. 1–3. Graylock Press, Rochester, NY (1956)
[2] Baskakov, I.V.: Cohomology of K-powers of spaces and the combinatorics of simplicial divisions. Uspekhi Mat. Nauk 57(5), 147–148 (2002) (Russian). English translation in Russian Math. Surveys 57, 989–990 (2002) · Zbl 1044.55004
[3] Baskakov, I.V., Buchstaber, V.M., Panov, T.E.: Algebras of cellular cochains, and torus actions. Uspekhi Mat. Nauk 59(3), 159–160 (2004) (Russian). English translation in Russian Math. Surveys 59, 562–563 (2004)
[4] Bayer, M.M., Lee, C.W.: Combinatorial aspects of convex polytopes, in Handbook of Convex Geometry, vol. A, pp. 485–534. North-Holland, Amsterdam (1993) · Zbl 0789.52014
[5] Borcea, C.: Some remarks on deformations of Hopf manifolds. Rev. Roumaine Math. Pures Appl. 26, 1287–1294 (1981) · Zbl 0543.32010
[6] Bosio, F.: Variétés complexes compactes: une généralisation de la construction de Meersseman et Löpez de Medrano-Verjovsky. Ann. Inst. Fourier (Grenoble) 51, 1259–1297 (2001) · Zbl 0994.32018
[7] Bredon, G.E.: Introduction to Compact Transformation Groups. Academic Press, New York (1972) · Zbl 0246.57017
[8] Brieskorn, E., van de Ven, A.: Some complex structures on products of homotopy spheres. Topology 7, 389–393 (1968) · Zbl 0174.54901 · doi:10.1016/0040-9383(68)90014-1
[9] Buchstaber, V.M., Panov, T.E.: Torus actions and their applications in topology and combinatorics. University lecture series, 24. Amer. Math. Soc., Providence, RI (2002) · Zbl 1012.52021
[10] Calabi, E., Eckmann, B.: A class of compact, complex manifolds which are not algebraic. Ann. of Math. 58, 494–500 (1953) · Zbl 0051.40304 · doi:10.2307/1969750
[11] Camacho, C., Kuiper, N.H., Palis, J.: The topology of holomorphic flows with singularity. Inst. Hautes Études Sci. Publ. Math. 48, 5–38 (1978) · Zbl 0411.58018 · doi:10.1007/BF02684312
[12] Davis, M.W., Januszkiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62, 417–451 (1991) · Zbl 0733.52006 · doi:10.1215/S0012-7094-91-06217-4
[13] Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29, 245–274 (1975) · Zbl 0312.55011 · doi:10.1007/BF01389853
[14] Denham, G., Suciu, A.I.: Moment-angle complexes, monomial ideals, and Massey products. To appear in Pure Appl. Math. Q. arXiv:math.AT/0512497. · Zbl 1169.13013
[15] Girbau, J., Haefliger, A., Sundararaman, D.: On deformations of transversely holomorphic foliations. J. Reine Angew. Math. 345, 122–147 (1983) · Zbl 0538.32015 · doi:10.1515/crll.1983.345.122
[16] Goresky, M., MacPherson, R.: Stratified Morse Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 14. Springer, Berlin (1988)
[17] Grünbaum, B.: Convex Polytopes. Pure and Applied Mathematics, 16. Wiley, New York (1967) · Zbl 0163.16603
[18] Haefliger, A.: Deformations of transversely holomorphic flows on spheres and deformations of Hopf manifolds. Compositio Math. 55, 241–251 (1985) · Zbl 0582.32026
[19] Hirsch, M.W.: Differential Topology. Springer-Verlag, New York (1976) · Zbl 0356.57001
[20] Hopf, H.: Zur Topologie der komplexen Mannigfaltigkeiten, in Studies and Essays, pp. 167–185. Interscience Publishers, New York (1948) · Zbl 0033.02501
[21] Jewell, K.: Complements of sphere and subspace arrangements. Topology Appl. 56, 199–214 (1994) · Zbl 0797.55015 · doi:10.1016/0166-8641(94)90075-2
[22] Kobayashi, S., Horst, C.: Topics in complex differential geometry, in Complex Differential Geometry, DMV Sem., 3, pp. 4–66. Birkhäuser, Basel (1983) · Zbl 0506.53029
[23] Loeb, J.J., Nicolau, M.: Holomorphic flows and complex structures on products of odd-dimensional spheres. Math. Ann. 306, 781–817 (1996) · Zbl 0860.32001 · doi:10.1007/BF01445277
[24] de Longueville, M.: The ring structure on the cohomology of coordinate subspace arrangements. Math. Z. 233, 553–577 (2000) · Zbl 0956.52020 · doi:10.1007/s002090050487
[25] López de Medrano, S.: The space of Siegel leaves of a holomorphic vector field, in Holomorphic Dynamics (Mexico, 1986), Lecture Notes in Math., 1345, pp. 233–245. Springer, Berlin (1988)
[26] López de Medrano, S.: Topology of the intersection of quadrics in \(\mathbf{R}^n\) , in Algebraic Topology (Arcata, CA, 1986), Lecture Notes in Math., 1370, pp. 280–292. Springer, Berlin (1989)
[27] 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
[28] McGavran, D.: Adjacent connected sums and torus actions. Trans. Amer. Math. Soc. 251, 235–254 (1979) · Zbl 0428.57013 · doi:10.1090/S0002-9947-1979-0531977-5
[29] McMullen, P.: On simple polytopes. Invent. Math. 113, 419–444 (1993) · Zbl 0803.52007 · doi:10.1007/BF01244313
[30] 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
[31] Meersseman, L., Verjovsky, A.: Holomorphic principal bundles over projective toric varieties. J. Reine Angew. Math. 572, 57–96 (2004) · Zbl 1070.14047 · doi:10.1515/crll.2004.054
[32] Milnor, J.W.: On the 3-dimensional Brieskorn manifolds \(M(p,q,r)\) , in Knots, Groups, and 3-Manifolds, Ann. of Math. Studies, 84, pp. 175–225. Princeton University Press, Princeton, NJ (1975) · Zbl 0305.57003
[33] Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton, NJ (1974) · Zbl 0298.57008
[34] Morrow, J., Kodaira, K.: Complex Manifolds. Holt, Rinehart and Winston, New York (1971) · Zbl 0325.32001
[35] Panov, T.E.: Cohomology of face rings and torus actions. To appear in London Math. Soc. Lecture Note Ser. arXiv:math.AT/0506526.
[36] Stanley, R.P.: Combinatorics and Commutative Algebra. Progress in Mathematics, 41. Birkhäuser, Boston, MA (1983) · Zbl 0537.13009
[37] Taubes, C.H.: The existence of anti-self-dual conformal structures. J. Differential Geom. 36, 163–253 (1992) · Zbl 0822.53006
[38] Timorin, V.A.: An analogue of the Hodge–Riemann relations for simple convex polyhedra. Uspekhi Mat. Nauk 54(2), 113–162 (1999) (Russian). English translation in Russian Math. Surveys 54, 381–426 (1999) · Zbl 0944.52005
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.