scholarly journals Gauss-Manin Connections for Boundary Singularities and Isochore Deformations

2015 ◽  
Vol 48 (2) ◽  
Author(s):  
Konstantinos Kourliouros
Keyword(s):  
Deformation Theory ◽  
Normal Forms ◽  
Hyperplane Section ◽  
Isolated Singularity ◽  
Boundary Singularity ◽  
Relative Cohomology ◽  
Manifold With Boundary ◽  
Vanishing Cycles ◽  
Volume Forms ◽  
Singularity Of A Function

AbstractWe study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary singularities with respect to a volume form. In particular, we prove the relative analog of J. Vey's isochore Morse lemma, J .-P. Fran~oise's generalisation on the local normal forms of volume forms with respect to the boundary singularity-preserving diffeomorphisms, as well as M. D. Garay's theorem on the isochore version of Mather's versa! unfolding theorem.

scholarly journals Generic sections of essentially isolated determinantal singularities

2017 ◽  
Vol 28 (11) ◽  
pp. 1750083 ◽  
Author(s):  
Jean-Paul Brasselet ◽  
Nancy Chachapoyas ◽  
Maria A. S. Ruas
Keyword(s):  
Hyperplane Section ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Dimension Formula ◽  
Determinantal Singularities ◽  
Hyperplane Sections

We study the essentially isolated determinantal singularities (EIDS), defined by Ebeling and Gusein-Zade [S. M. Guseĭn-Zade and W. Èbeling, On the indices of 1-forms on determinantal singularities, Tr. Mat. Inst. Steklova 267 (2009) 119–131], as a generalization of isolated singularity. We prove in dimension [Formula: see text], a minimality theorem for the Milnor number of a generic hyperplane section of an EIDS, generalizing the previous results by Snoussi in dimension [Formula: see text]. We define strongly generic hyperplane sections of an EIDS and show that they are still EIDS. Using strongly general hyperplanes, we extend a result of Lê concerning the constancy of the Milnor number.

scholarly journals Singularities of Legendre varieties, of evolvents and of fronts at an obstacle

1982 ◽  
Vol 2 (3-4) ◽  
pp. 301-309 ◽  
Author(s):  
V. I. Arnol'd
Keyword(s):  
Shortest Path ◽  
Normal Forms ◽  
Contact Manifold ◽  
Shortest Path Problem ◽  
Binary Forms ◽  
Manifold With Boundary ◽  
Generic Singularities

AbstractThe Legendre manifolds are the maximal integral submanifolds of a contact manifold. The shortest path problem on a manifold with boundary leads to Legendre varieties. We find normal forms of their generic singularities in terms of binary forms invariants theory.

scholarly journals Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

2015 ◽  
Vol 218 ◽  
pp. 125-173
Author(s):  
Tadashi Ochiai ◽  
Kazuma Shimomoto
Keyword(s):  
System Theory ◽  
Deformation Theory ◽  
Galois Representations ◽  
Hyperplane Section ◽  
Euler System ◽  
Local Rings ◽  
Local Domain ◽  
Strong Version ◽  
Mixed Characteristic ◽  
Torsion Modules

AbstractIn this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

EXTENSIONS OF VECTOR BUNDLES WITH APPLICATION TO NOETHER–LEFSCHETZ THEOREMS

2013 ◽  
Vol 15 (05) ◽  
pp. 1350003 ◽  
Author(s):  
G. V. RAVINDRA ◽  
AMIT TRIPATHI
Keyword(s):  
Characteristic Zero ◽  
Deformation Theory ◽  
Vector Bundles ◽  
Projective Variety ◽  
Hyperplane Section ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Open Set ◽  
Higher Rank ◽  
Lefschetz Theorems

Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle [Formula: see text] on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck–Lefschetz theory. As a consequence, we prove a Noether–Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether–Lefschetz theorems of Joshi and Ravindra–Srinivas.

scholarly journals Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

2015 ◽  
Vol 218 ◽  
pp. 125-173 ◽  
Author(s):  
Tadashi Ochiai ◽  
Kazuma Shimomoto
Keyword(s):  
System Theory ◽  
Deformation Theory ◽  
Galois Representations ◽  
Hyperplane Section ◽  
Euler System ◽  
Local Rings ◽  
Local Domain ◽  
Strong Version ◽  
Mixed Characteristic ◽  
Torsion Modules

AbstractIn this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

Deformations of Poisson structures on fibered manifolds and adiabatic slow–fast systems

2017 ◽  
Vol 14 (06) ◽  
pp. 1750086 ◽  
Author(s):  
Misael Avendaño-Camacho ◽  
Yury Vorobiev
Keyword(s):  
Perturbation Theory ◽  
Hamiltonian Systems ◽  
Deformation Theory ◽  
Normal Forms ◽  
Geometric Approach ◽  
Fiber Bundles ◽  
Poisson Structures ◽  
Averaging Theorem ◽  
Fibered Manifolds ◽  
Adiabatic Perturbation Theory

In the context of normal forms, we study a class of slow–fast Hamiltonian systems on general Poisson fiber bundles with symmetry. Our geometric approach is motivated by a link between the deformation theory for Poisson structures on fibered manifolds and the adiabatic perturbation theory. We present some normalization results which are based on the averaging theorem for horizontal 2-cocycles on Poisson fiber bundles.

scholarly journals Extended Hodge theory for fibred cusp manifolds

2018 ◽  
Vol 10 (03) ◽  
pp. 531-562 ◽  
Author(s):  
E. Hunsicker
Keyword(s):  
Exact Sequence ◽  
Hodge Theory ◽  
Intersection Cohomology ◽  
Boundary Values ◽  
Cohomology Groups ◽  
Harmonic Forms ◽  
Relative Cohomology ◽  
Manifold With Boundary ◽  
The Given ◽  
Boundary Space

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted [Formula: see text] harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted [Formula: see text] harmonic forms are harmonic forms that are almost in the given weighted [Formula: see text] space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. In analogy with that setting, in the unweighted [Formula: see text] case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.

scholarly journals Relative cohomology and volume forms

1988 ◽  
Vol 20 (1) ◽  
pp. 207-222 ◽  
Author(s):  
J.-P. Francoise
Keyword(s):  
Relative Cohomology ◽  
Volume Forms
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