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I can find no references for the fundamental theorem of Picard-Lefschetz theory (other than a quick mention in the lecture notes of Jonathan Evans: http://www.homepages.ucl.ac.uk/~ucahjde/ST-lectures/lecture10.pdf). Can anybody point me towards anything that gives a precise statement on the existence of vanishing cycles?

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Welcome to MSE! Am I misunderstanding your question? Book. Regards –  Amzoti Feb 16 '13 at 17:18
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