Vanishing Cycles and Mutation

By P. Seidel. 2000

156 citations recorded by the source corpus.

Using Floer cohomology, we establish a connection between PicardLefschetz theory and the notion of mutation of exceptional collections in homological algebra.

View Paper: https://www.semanticscholar.org/paper/271444ec09f659fd2fe6b2a7bf970f4052eb6b9d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

D branes in Landau-Ginzburg models and algebraic geometry (2002, background): “[12, 17 ]. In particular, the mirrors of \exceptional" bundles on CPn have been identifled, and in the case of CP1 and CP2 it has been checked that morphisms between these bundles in the derived category of coherent sheaves on CPn agree with the Floer homology between their mirror”
Brane tilings and their applications (2008, background): “Although we are choosing a specific choice of the distinguished basis of vanishing cycles, you can verify that the final categoryDb(FukW) is independent of such a choice [221,241].”
Homological Mirror Symmetry for Toric del Pezzo Surfaces (2004, methodology): “Roughly speaking, the Floer complex between two vanishing cycles are the vector space spanned by intersection points between them, and the compositions of morphisms are given by “counting polygons.” By Seidel [ 15 ], the derived category D b Fuk → W of Fuk → W is independent of the”
Mirror symmetry for weighted projective planes and their noncommutative deformations (2004, methodology): “Although the category Lagvc(f, {γi}) depends on the chosen ordered collection of arcs {γi}, Seidel has obtained the following result [34]:”
A long exact sequence for symplectic Floer cohomology (2001, background): “The Floer cohomology of two such submanifolds is comparatively easy to define, since the corresponding action functional has no periods, so that bubbling is impossible.”
Fukaya categories of symmetric products and bordered Heegaard-Floer homology (2010, methodology): “Following Abouzaid and Seidel [2], we actually define hom(L1, L2) to be the infinitely generated complex ⊕∞ w=1 CF ∗(φwHρ(L1), L2)[q], or rather the quasi-isomorphic trunca- tion ⊕∞ w=m CF ∗(φwHρ(L1), L2)[q] for some m ≥ 1 (see Definition A.1 below), where the formal variable q has degree −1 and sat”
Homological mirror symmetry for the quartic surface (2003, background): “ the vanishing cycle, up to isotopy inside the group of exact symplectic automorphisms: (6.12) hγ≃τVc ∈Aut(Mc(0)). To be precise, Vc carries a small additional piece of structure, called a framing in [53, 55], and the deﬁnition of Dehn twist uses that too. However, framings contain no information if”
Special Lagrangians, stable bundles and mean curvature flow (2001, methodology): “proceed more directly by applying Seidel’s exact sequence [ S3 ].”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “It is well-known that there is a version of the Fukaya category tailored to a Lefschetz fibration (the idea is originally due to Kontsevich, and a first rigorous definition appears in [36]).”
Legendrian fronts for affine varieties (2016, background): “Seidel [67, 68], including the study of Fukaya categories [65, 66] and homological mirror symmetry [69], but they have never been treated before from the contact topology viewpoint.”
Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves (2005, background): “Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of Xk, and give an explicit correspondence between the deformation parameters for Xk and the cohomology class [B+ iω] ∈ H 2(Mk,C).”
Microlocal branes are constructible sheaves (2006, background): “[33] P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, preprint of book in progress available at math.uchicago.edu/∼seidel.”
Exact Lagrangian submanifolds in simply-connected cotangent bundles (2007, background): “By definition (see [19] or [22, Section 5m]) this is a strictly unital A∞-category, with a finite ordered set of objects ObA = {Y1, .”