Lagrange-Singularitäten [Elektronische Ressource] / vorgelegt von Christian Sevenheck

Sujets

• Mathematics

Informations

• Publié par: johannes_gutenberg-universitat_mainz
• Publié le: 01 janvier 2003
• Nombre de lectures: 9
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Lagrange-Singularitaten˜
Dissertation zur Erlangung des Grades
Doktor der Naturwissenschaften\
"
am Fachbereich Mathematik
der Johannes-Gutenberg-Universitat in Mainz˜
vorgelegt von
Christian Sevenheck
geboren in Gorlitz.˜Mundliche Prufung: 2003˜ ˜Preface
In this thesis lagrangian singularities are studied. This topic lies on the bor-
derofdifierentbranchesofmathematics,likesingularitytheoryandalgebraic
geometry, symplecticgeometry, mathematicalphysics, algebraic analysis etc.
The main goal of this work is to develop a deformation theory for lagrangian
singularities and to investigate its relationship to D-module theory. Algo-
rithms for computations of deformation spaces are derived and applied to
concrete examples.
ItisgreatpleasureformetoacknowledgethehelpIreceivedfromdifierent
people during the work on this thesis. In the flrst place, I would like to
express my deep gratitude to my advisor D. van Straten for his constant
support which begun a long time before I started this work. His way to do
andexplainmathematicsverymuchimpressedandin uenced mealloverthe
years. I am particular grateful to him for bringing the subject of this thesis
tomyattentionandforstimulatingagoodpartoftheideascontainedherein.
Secondly, I like to thank C. Sabbah who kindly accepted to undertake the
adventure of co-directing my thesis and whose interest in my work as well as
his explanations onD-modules were of great help during my stay in Paris.
It is tempting but hopeless to try to list all people who contributed in
some way to this thesis. To name only a few, I thank Th. Warmt for sharing
myenthusiasmformathematicsoverthelasteightyears. IalsothankCh.van
Enckevort, K. M˜ohring and O. Labs for discussions on various mathematical
subjects. Many tanks goes to C. Hertling for explaining me at difierent
occasions his work on Frobeniusmanifolds andtherelationship to lagrangian
singularities. It is a pleasure to thank M. Garay for his interest in
singularities and for many fruitful discussion on the subject, some of which
are at the origin of results contained in this thesis.
I would like to thank P. Seidel and G.-M. Greuel for having been the
referees of this thesis and for their useful remarks. I thank C. Roger for his
willingness to participate in the jury of the defense.
This work was done while I was enrolled as Ph.D. student both at the
Fachbereich Mathematik of the Johannes-Gutenberg-Universit˜at Mainz andat the Centre de Math¶ematiques, Ecole Polytechnique. I would like to thank
these institutions for the excellent working conditions they provided. In
particular, I thank R. Emerenziani who helped me a lot handling all type of
administrative di–culties.
Mainz, 18th February 2003 Christian SevenheckContents
1 Examplesoflagrangiansingularities 11
1.1 Involutive ideals and generating families . . . . . . . . . . . . 11
1.2 Open Swallowtails . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Conormal cones . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Integrable systems . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5 The „=2-stratum . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.6.1 Spectral covers of Frobenius manifolds . . . . . . . . . 39
1.6.2 Special lagrangian singularities . . . . . . . . . . . . . 41
2 Lagrangiandeformations 45
2.1 Real lagrangian submanifolds . . . . . . . . . . . . . . . . . . 45
2.2 Curve singularities . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 The lagrangian deformation functor . . . . . . . . . . . . . . . 47
3 Lagrangiansubvarieties 53
3.1 Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1 Lie algebroids and difierential operators . . . . . . . . 53
3.1.2 Modules over Lie algebroids . . . . . . . . . . . . . . . 56
3.1.3 The de Rham complex . . . . . . . . . . . . . . . . . . 58
3.2 The lagrangian Lie algebroid . . . . . . . . . . . . . . . . . . . 61
3.3 Applications to deformation theory . . . . . . . . . . . . . . . 63
3.3.1 Inflnitesimal deformations . . . . . . . . . . . . . . . . 64
3.3.2 Obstructions . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.3 Stability of families . . . . . . . . . . . . . . . . . . . . 68
3.3.4 Integrable systems . . . . . . . . . . . . . . . . . . . . 71
3.4 Properties of the lagrangian de Rham complex . . . . . . . . . 74
3.4.1 Constructibility and Coherence . . . . . . . . . . . . . 74
3.4.2 Freeness of the relative cohomology . . . . . . . . . . . 81
3.5 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
v4 IsotropicMappings 95
4.1 Generalities and basic examples . . . . . . . . . . . . . . . . . 95
4.2 Corank 1 mappings . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3 Symplectic and Lagrange stability . . . . . . . . . . . . . . . . 105
4.4 Further computations and conjectures . . . . . . . . . . . . . . 109
A DeformationTheory 119
A.1 Formal deformation theory . . . . . . . . . . . . . . . . . . . . 119
A.1.1 Difierential graded Lie algebras . . . . . . . . . . . . . 120
A.1.2 Categories flbred in groupoids and deformation functors122
A.1.3 Obstruction theory . . . . . . . . . . . . . . . . . . . . 126
A.1.4 The functors MC , G and Def . . . . . . . . . . . . 131L L L
1A.1.5 The T -lifting property . . . . . . . . . . . . . . . . . . 133
A.2 Examples of controlling dg-Lie algebras . . . . . . . . . . . . . 137
A.2.1 The Kodaira-Spencer algebra . . . . . . . . . . . . . . 137
A.2.2 Deformation of associative, commutative and Lie alge-
bras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A.2.3 The cotangent complex . . . . . . . . . . . . . . . . . . 145
B Algebraicanalysis 151
B.1 The characteristic variety. . . . . . . . . . . . . . . . . . . . . 151
B.2 HolonomicD -modules . . . . . . . . . . . . . . . . . . . . . . 155XIntroduction
Lagrangian singularities flrst appeared in the work of Arnold and his school
around 1980. Arnold recognized their importance in relation with problems
from mathematical physics, in particular, variational problems with con-
straints ([Arn82]). Most prominently, the so-called obstacle problem leads to
the open swallowtail, a singular subvariety in a certain space of polynomials
in one variable of flxed degree, which comes equipped with a natural sym-
plectic form. Some years later, Givental studied immersions of lagrangian
surfaces in four space ([Giv86]), also called isotropic mappings and discov-
ered a generic mapping the image of which is called open Whitney umbrella.
More recently, lagrangian subvarieties associated to any Frobenius manifold
have been studied extensively by Hertling [Her02]. Singular subspaces of
symplectic manifolds also arises in algebraic analysis, the characteristic vari-
ety of a holonomicD-module is a lagrangian subvariety. These few examples
show that Lagrangian singularities occur at rather difierent places in math-
ematics, as subspaces of holomorphic symplectic manifolds as well as in the
1C -setting. There are also classes of lagrangian submanifolds involving real
and complex structures, namely the so-called special lagrangians are sub-
spaces of Calabi-Yau manifolds such that the K˜ahler form as well as the
imaginary part of the holomorphic form of maximal degree vanish on them.
Singularities of such special lagrangians play an important role in the (con-
jectural) version of mirror symmetry as developed by Strominger, Yau and
Zaslow (see, e.g., [Joy00]).
The central topic of this thesis is the problem how lagrangian singulari-
ties behave under deformations. Partial aspects of this question can already
be found in the work of Givental ([Giv88]). However, the deformations that
are considered in that paper are only perturbations of the symplectic struc-
ture which flxes the lagrangian subspace. In order to take into account
deformations of the space itself, we are led to use rather sophisticated tools
from abstract deformation theory, which have been developed since the six-
ties (quite independently from classical singularity theory) by Grothendieck,
Schlessinger, Illusie, Artin, Deligne and others. In this approach, the main
12
idea is to associate to any object that one wants to deform a functor on a
certain category (which is the category of base spaces of the families un-
der consideration) and to study its representability, at least in a somewhat
weaker sense (existence of a so-called \hull"). The classical notion of semi-
universal deformations (e.g., for functions with isolated critical points) is a
special case of this more general principle.
To make this deformation theory program work, the flrst step is to deflne
the appropriate functor. Hence we need to know what exactly is meant by
a Lagrangian deformation. We will give in the sequel an informal deflnition,
postponing the exact formulation to the second chapter (deflnition 2.4 on
page 48). Given any germ (L;0)‰ (M;0) of a reduced (complex, say) ana-
lytic subspaceLinsidea(holomorphic)symplecticmanifoldM withdeflning
ideal I ‰ O , the question arises how to detect whether L is lagrangianM;0
only in terms of the ideal I. It turns out that a necessary condition is that
I is stable under the Poisson bracket, i.e.,fI;Ig‰I. Such ideals are called
involutive. In addition, the space L m