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Transactions of the American Mathematical Society

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On a simplicial complex associated to the monodromy


Author: Gerald Leonard Gordon
Journal: Trans. Amer. Math. Soc. 261 (1980), 93-101
MSC: Primary 32C40; Secondary 14D05, 32G13
DOI: https://doi.org/10.1090/S0002-9947-1980-0576865-1
MathSciNet review: 576865
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Abstract: Suppose we have a complex analytic family, $ {V_t}$, $ \left\vert t \right\vert\, \leqslant \,1$, such that the generic fibre is a nonsingular complex manifold of complex dimension n. Let T denote the monodromy induced from going once around the singular fibre and let I denote the identity map. We shall associate to the singular fibre a simplicial complex $ \Gamma $, which is at most n-dimensional. Then under certain conditions on the family $ {V_t}$ (which are satisfied for the Milnor fibration of an isolated singularity or if the $ {V_t}$ are compact Kähler), there is an integer $ N\, > \,0$ such that $ {({T^N}\, - \,I)^k}{H_k}({V_t})\, = \,0$ if and only if $ {H_k}(\Gamma )\, = \,0$.


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  • [1] V. I. Arnol'd, Critical points of smooth function and their normal forms, Russian Math. Surveys 30 (1975), 1-75. MR 0420689 (54:8701)
  • [2] A. Blanchard, Sur les variétés analytiques complexes, Ann. Ecole Norm. Sup. (3) 73 (1956), 157-202. MR 0087184 (19:316e)
  • [3] C. H. Clemens, Picard-Lefshetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities, Trans. Amer. Math. Soc. 136 (1969), 93-108. MR 0233814 (38:2135)
  • [4] -, Degeneration of Kähler manifolds, Duke Math. J. 44 (2) (1977), 215-290. MR 0444662 (56:3012)
  • [5] G. L. Gordon, A geometric study of the monodromy of complex analytic surfaces, Invent. Math. 1 (1977), 11-35. MR 0450267 (56:8563)
  • [6] -, On the degeneracy of a spectral sequence associated to normal crossings, Pacific J. Math. (to appear). MR 600638 (83f:32011)
  • [7] P. A. Griffiths and W. Schmid, Recent developments in Hodge theory: A discussion of techniques and results, Discrete Subgroups of Lie Groups and Application to Moduli (Internat. Colloq. Bombay, 1973), Oxford Univ. Press, Bombay, 1975, pp. 31-137. MR 0419850 (54:7868)
  • [8] H. Hironaka, Bimeromorphic maps, mimeographed notes, Warwick, 1971.
  • [9] G. Kempf, F. Knudson, D. Mumford and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math., vol. 339, Springer-Verlag, New York, 1973. MR 0335518 (49:299)
  • [10] K. Kodaira, On the structures of compact complex analytic surfaces. III, Amer. J. Math. 90 (1968), 55-83. MR 0228019 (37:3603)
  • [11] B. Malgrange, Letter to the editors, Invent Math. 20 (1973), 171-172. MR 0330502 (48:8839)
  • [12] M. Sebastiani and R. Thom, Un résultat sur la monodromie, Invent Math. 13 (1971), 90-96. MR 0293122 (45:2201)
  • [13] J. Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1976), 229-257. MR 0429885 (55:2894)
  • [14] -, Mixed Hodge structure on the vanishing cohomology, Nordic Summer School Sympos. in Math., Oslo, 1976, pp. 525-563. MR 0485870 (58:5670)

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DOI: https://doi.org/10.1090/S0002-9947-1980-0576865-1
Article copyright: © Copyright 1980 American Mathematical Society

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