More about vanishing cycles and mutation

By P. Seidel. 2000

167 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/371dcdee66b0dc041761ef1bfeb10f28e91233a4

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In Geometry and Topology

Citations in the corpus

Global matrix factorizations (2011, background): “Moreover when Y has a complex structure and (X, W ) has a symplectic structure, the derived category of Y should be equivalent to the Fukaya–Seidel category [Sei] of (X, W ).”
Symplectic homology as Hochschild homology (2006, background): “Donaldson’s conjecture, in a form closely related to the one in [16], is:”
The periodic Floer homology of a Dehn twist. (2004, methodology): “The symplectic Floer homology has been computed for a Dehn twist by Seidel [21], more examples have been computed by Eftekhary [4] and Gautschi [8], and a conjecture for a product of positive Dehn twists on a surface with boundary is given in [22].”
Symplectic Floer homology of area-preserving surface diffeomorphisms (2008, background): “Seidel has a conjecture for the symplectic Floer homology of a general composition of positive Dehn twists on a surface with boundary in terms of Hochschild Homology coming from his work on vanishing cycles and mutation [Se3].”
Topological conformal field theories and Calabi–Yau categories (2004, background): “Consider the dg category Perf(X), whose objects are bounded complexes of holomorphic vector bundles on X, and whose morphisms are HomPerf(X)(E,F ) = Ω 0,−∗(E∨ ⊗ F ) (we reverse the grading, as all our differentials are homological).”
Mirror symmetry for toric Fano manifolds via SYZ transformations (2008, unknown): “A version of the Homological Mirror Symmetry Conjecture has also been formulated by Kontsevich [28], which again has been checked in many cases [39], [44], [7], [8], [1], [2].”
Effect of Legendrian surgery (2009, result): “In this sense our results can be viewed as generalizations of P Seidel’s exact triangles for symplectic Dehn twists; see [38; 37; 39].”
Microlocal branes are constructible sheaves (2006, background): “In technical terms, the construction is a close relative of the category of vanishing cycles proposed by Kontsevich [22] and Hori-Iqbal-Vafa [13], and developed by Seidel [30], [31], [33].”
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves (2005, methodology): “[17] or [3]).”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, methodology): “We write the outcome as HF ∗(φ, ), where denotes the amount of perturbation (for small | |, this appears in [37, 29]; for the general case, see [51] or the exposition in the body of the paper).”
Homological Mirror Symmetry for Toric del Pezzo Surfaces (2004, unknown): “The above Theorem 1.1 extends the work of Seidel [ 16 ] and Auroux, Katzarkov and Orlov [1], where the cases of P 2 and P 1 × P 1 , and P 2 blown-up at one point is treated.”
Suspending Lefschetz Fibrations, with an Application to Local Mirror Symmetry (2009, background): “Homological mirror symmetry for such varieties, first considered by Kontsevich [11], is now proved (see [16] for Y = CP (2), [2] for several more cases, and [1, 20] for all remaining ones).”