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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities

Author: Alan Landman
Journal: Trans. Amer. Math. Soc. 181 (1973), 89-126
MSC: Primary 14D05
MathSciNet review: 0344248
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Abstract: We consider a holomorphic family $ {\{ {V_t}\} _{t \in D}}$ of projective algebraic varieties $ {V_t}$ parametrized by the unit disc $ D = \{ t \in {\mathbf{C}}:\vert t\vert < 1\} $ and where $ {V_t}$ is smooth for $ t \ne 0$ but $ {V_0}$ may have arbitrary singularities. Displacement of cycles around a path $ t = {t_0}{e^{i\theta }}(0 \leqslant \theta \leqslant 2\pi )$ leads to the Picard-Lefschetz transformation $ T:{H_\ast }({V_{{t_0}}},{\mathbf{Z}}) \to {H_\ast }({V_{{t_0}}},{\mathbf{Z}})$ on the homology of a smooth $ {V_{t0}}$. We prove that the eigenvalues of $ T$ are roots of unity and obtain an estimate on the elementary divisors of $ T$. Moreover, we give a global inductive procedure for calculating $ T$ in specific examples, several of which are worked out to illustrate the method.

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Keywords: Picard-Lefschetz transformation ( $ {\mathbf{P}} - {\mathbf{L}}$ transformation), vanishing cycle, vanishing cone, general pencil of hyperplane sections, branch function, branch curve, normal crossings, monodromy theorem, Gauss-Manin connection, Hodge decomposition
Article copyright: © Copyright 1973 American Mathematical Society

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