More about vanishing cycles and mutation

Abstract

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.