Vanishing cycles over a base of dimension ≥1

1983 ◽  
pp. 143-150 ◽  
Author(s):  
G. Laumon
Keyword(s):  
Vanishing Cycles

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

scholarly journals Vanishing cycles and Thom's a f condition

2007 ◽  
Vol 39 (4) ◽  
pp. 591-602 ◽  
Author(s):  
David B. Massey
Keyword(s):  
Vanishing Cycles

scholarly journals Wild ramification and a vanishing cycles formula

2004 ◽  
Vol 273 (1) ◽  
pp. 108-128 ◽  
Author(s):  
Mohamed Saı̈di
Keyword(s):  
Wild Ramification ◽  
Vanishing Cycles

SPIN STRUCTURES ON LEFSCHETZ FIBRATIONS

2001 ◽  
Vol 33 (4) ◽  
pp. 466-472 ◽  
Author(s):  
ANDRÁS I. STIPSICZ
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lefschetz Fibrations ◽  
Spin Structures ◽  
Vanishing Cycles ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lefschetz fibrations over D2 and S2 to carry spin structures are presented. The conditions are given in terms of vanishing cycles and homology sections.

scholarly journals EXTENSION OF FUNCTORS FOR ALGEBRAS OF FORMAL DEFORMATION

2013 ◽  
Vol 56 (1) ◽  
pp. 103-141
Author(s):  
ANA RITA MARTINS ◽  
TERESA MONTEIRO FERNANDES ◽  
DAVID RAIMUNDO
Keyword(s):  
Fourier Transform ◽  
Open Subset ◽  
Inverse Image ◽  
Explicit Construction ◽  
Comparison Theorems ◽  
Complex Manifolds ◽  
Formal Parameter ◽  
Formal Deformation ◽  
Characteristic Case ◽  
Vanishing Cycles

AbstractSuppose we are given complex manifoldsXandYtogether with substacks$\mathcal{S}$and$\mathcal{S}'$of modules over algebras of formal deformation$\mathcal{A}$onXand$\mathcal{A}'$onY, respectively. Also, suppose we are given a functor Φ from the category of open subsets ofXto the category of open subsets ofYtogether with a functorFof prestacks from$\mathcal{S}$to$\mathcal{S}'\circ\Phi$. Then we give conditions for the existence of a canonical functor, extension ofFto the category of coherent$\mathcal{A}$-modules such that the cohomology associated to the action of the formal parameter$\hbar$takes values in$\mathcal{S}$. We give an explicit construction and prove that when the initial functorFis exact on each open subset, so is its extension. Our construction permits to extend the functors of inverse image, Fourier transform, specialisation and micro-localisation, nearby and vanishing cycles in the framework of$\mathcal{D}[[\hbar]]$-modules. We also obtain the Cauchy–Kowalewskaia–Kashiwara theorem in the non-characteristic case as well as comparison theorems for regular holonomic$\mathcal{D}[[\hbar]]$-modules and a coherency criterion for proper direct images of good$\mathcal{D}[[\hbar]]$-modules.

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