Stratified Morse Theory: Past and Present

Massey, David B.

doi:10.4310/pamq.2006.v2.n4.a6

Cited by 9 publications

(5 citation statements)

References 35 publications

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Section: Singular Support and The Category Shmentioning

confidence: 99%

“…In the stratified Morse theory [23] for a smooth function f : M × R → R, a non-degenerate critical point q of f (in the stratified sense) is a non-degenerate critical point of f | s 0 (in the usual Morse sense) for some stratum s 0 ∈ S, such that for all strata s 0 ≤ s α there exists v ∈ T q (M × R) ∩ T s α with df q (v) = 0. Fix a metric g on J 0 M and let B δ (x) ⊂ J 0 M denote the open ball of radius δ centered at x.…”

Section: Singular Support and The Category Shmentioning

confidence: 99%

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Generating families and augmentations for Legendrian surfaces

Rutherford

Sullivan

2018

Algebr. Geom. Topol.

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Given an augmentation for a Legendrian surface in a 1-jet space, Λ ⊂ J 1 (M ), we explicitly construct an object, F ∈ Sh • Λ (M × R, K), of the (derived) category from [30] of constructible sheaves on M × R with singular support determined by Λ. In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in 1-jet spaces that, based on [25,26,27], is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of [30] for 1dimensional Legendrian knots to obtain a combinatorial model for sheaves in Sh • Λ (M × R, K) in the 2-dimensional case.

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Section: Singular Support and The Category Shmentioning

confidence: 99%

Section: Singular Support and The Category Shmentioning

confidence: 99%

Generating families and augmentations for Legendrian surfaces

Rutherford

Sullivan

2018

Algebr. Geom. Topol.

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“…In this section, we review the basic facts about stratified Morse theory. While the standard reference for this part is [15], the reader may also consult [16,25,26] for a brief introduction to the theory and its applications.…”

Section: Stratified Morse Theory: Overviewmentioning

confidence: 99%

On the signed Euler characteristic property for subvarieties of abelian varieties

Elduque

Geske

Maxim

2018

jsing

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We give an elementary proof of the fact that a pure-dimensional closed subvariety of a complex abelian variety has a signed intersection homology Euler characteristic. We also show that such subvarieties which, moreover, are local complete intersections, have a signed Euler-Poincaré characteristic. Our arguments rely on the construction of circle-valued Morse functions on such spaces, and use in an essential way the stratified Morse theory of Goresky-MacPherson. Our approach also applies (with only minor modifications) for proving similar statements in the analytic context, i.e., for subvarieties of compact complex tori. Alternative proofs of our results can be given by using the general theory of perverse sheaves.Classically, much of the manifold theory, e.g., Morse theory, Lefschetz theorems, Hodge decompositions, and especially Poincaré Duality, is recovered in the singular stratified context if, instead of the usual (co)homology, one uses Goresky-MacPherson's intersection homology groups [12,13]. We recall here the definition of intersection homology of complex analytic (or algebraic) varieties, and discuss some preliminary results concerning the corresponding intersection homology Euler characteristic. For more details on intersection homology, the reader may consult, e.g., [9] and the references therein.Let X be a purely n-dimensional complex analytic (or algebraic) variety with a fixed Whitney stratification. All strata of X are of even real (co)dimension. By [11], X admits a triangulation which is compatible with the stratification, so X can also be viewed as a PL stratified pseudomanifold. Let (C * (X), ∂) denote the complex of finite PL chains on X, with Z-coefficients. The intersection homology groups of X, denoted IH i (X), are the homology groups of a complex of "allowable chains", defined by imposing restrictions on how chains meet the singular strata. Specifically, the chain complex (IC * (X), ∂) of allowable finite PL chains is defined as follows: if ξ ∈ C i (X) has support |ξ|, then ξ ∈ IC i (X) if, and only if, dim(|ξ| ∩ S) < i − s and dim(|∂ξ| ∩ S) < i − s − 1, for each stratum S of complex codimension s > 0. The boundary operator on allowable chains is induced from the usual boundary operator on chains of X. (The second condition above ensures that ∂ restricts to the complex of allowable chains.)

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“…), i ∈ Z are independent of the choices of g and x for (x, dg x ) staying in a fixed connected component of T * Sα X − D * Sα X. For more details, see [Mas06] Theorems 2.29, 2.31 and the references therein. In the complex setting, T * Sα X −D * Sα X is always connected, so M x,F (F) are quasi-isomorphic for different choices of (x, ξ) in it (but not in a canonical way since there may be monodromies).…”

Section: Local Morse Group Functor Mmentioning

confidence: 99%

Holomorphic Lagrangian branes correspond to perverse sheaves

Jin¹

2015

Geom. Topol.

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Let X be a compact complex manifold, $D_c^b(X)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk(T^*X)$ be the Fukaya category of $T^*X$. A Lagrangian brane in $Fuk(T^*X)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^*X_{\mathbb{C}}$, the holomorphic cotangent bundle of $X$. We prove that under the quasi-equivalence between $D^b_c(X)$ and $DFuk(T^*X)$ established in [NaZa09] and [Nad09], holomorphic Lagrangian branes with appropriate grading correspond to perverse sheaves.Comment: Significant improvement of the previous version. Comments are welcome

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Stratified Morse Theory: Past and Present

Cited by 9 publications

References 35 publications

Generating families and augmentations for Legendrian surfaces

Generating families and augmentations for Legendrian surfaces

On the signed Euler characteristic property for subvarieties of abelian varieties

Holomorphic Lagrangian branes correspond to perverse sheaves

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