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

Landman, Alan

doi:10.1090/S0002-9947-1973-0344248-1

On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities
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Trans. Amer. Math. Soc. 181 (1973), 89-126 Request permission

Abstract:

We consider a holomorphic family ${\{ {V_t}\} _{t \in D}}$ of projective algebraic varieties ${V_t}$ parametrized by the unit disc $D = \{ t \in {\mathbf {C}}:|t| < 1\}$ 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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