## On a simplicial complex associated to the monodromy

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- by Gerald Leonard Gordon
- Trans. Amer. Math. Soc.
**261**(1980), 93-101 - DOI: https://doi.org/10.1090/S0002-9947-1980-0576865-1
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## Abstract:

Suppose we have a complex analytic family, ${V_t}$, $\left | t \right | \leqslant 1$, 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 ${H_k}(\Gamma ) = 0$.

## References

- V. I. Arnol′d,
*Critical points of smooth functions, and their normal forms*, Uspehi Mat. Nauk**30**(1975), no. 5(185), 3–65 (Russian). MR**0420689** - André Blanchard,
*Sur les variétés analytiques complexes*, Ann. Sci. École Norm. Sup. (3)**73**(1956), 157–202 (French). MR**0087184** - C. H. Clemens Jr.,
*Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities*, Trans. Amer. Math. Soc.**136**(1969), 93–108. MR**233814**, DOI 10.1090/S0002-9947-1969-0233814-9 - C. H. Clemens,
*Degeneration of Kähler manifolds*, Duke Math. J.**44**(1977), no. 2, 215–290. MR**444662** - Gerald Leonard Gordon,
*A geometric study of the monodromy of complex analytic surfaces*, Invent. Math.**40**(1977), no. 1, 11–35. MR**450267**, DOI 10.1007/BF01389859 - Gerald Leonard Gordon,
*On the degeneracy of a spectral sequence associated to normal crossings*, Pacific J. Math.**90**(1980), no. 2, 389–396. MR**600638** - Phillip Griffiths and Wilfried Schmid,
*Recent developments in Hodge theory: a discussion of techniques and results*, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 31–127. MR**0419850**
H. Hironaka, - G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat,
*Toroidal embeddings. I*, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR**0335518** - K. Kodaira,
*On the structure of compact complex analytic surfaces. III*, Amer. J. Math.**90**(1968), 55–83. MR**228019**, DOI 10.2307/2373426 - B. Malgrange,
*Letter to the editors*, Invent. Math.**20**(1973), 171–172. MR**330502**, DOI 10.1007/BF01404064 - M. Sebastiani and R. Thom,
*Un résultat sur la monodromie*, Invent. Math.**13**(1971), 90–96 (French). MR**293122**, DOI 10.1007/BF01390095 - Joseph Steenbrink,
*Limits of Hodge structures*, Invent. Math.**31**(1975/76), no. 3, 229–257. MR**429885**, DOI 10.1007/BF01403146 - J. H. M. Steenbrink,
*Mixed Hodge structure on the vanishing cohomology*, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525–563. MR**0485870**

*Bimeromorphic maps*, mimeographed notes, Warwick, 1971.

## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- 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