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arxiv: math/0010032 · v1 · submitted 2000-10-03 · 🧮 math.SG · math.AG

More about vanishing cycles and mutation

classification 🧮 math.SG math.AG
keywords cyclestheoryvanishingcategorycohomologydiscussionmutationpicard-lefschetz
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The paper continues the discussion of symplectic aspects of Picard-Lefschetz theory begun in "Vanishing cycles and mutation" (this archive). There we explained how to associate to a suitable fibration over a two-dimensional disc a triangulated category, the "derived directed Fukaya category" which describes the structure of the vanishing cycles. The present second part serves two purposes. Firstly, it contains various kinds of algebro-geometric examples, including the "mirror manifold" of the projective plane. Secondly there is a (largely conjectural) discussion of more advanced topics, such as (i) Hochschild cohomology, (ii) relations between Picard-Lefschetz theory and Morse theory, (iii) a proposed "dimensional reduction" algorithm for doing certain Floer cohomology computations.

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