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Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic Connections Symplectic Properties of the Space of JonathanAidan Fuchsian Equations in the Moduli Space Introduction of Logarithmic Connections StartingPointL0 Deformation SpaceofL0as Operator Jonathan Aidan FromL0to∇0 Deformation Spaceof∇0as Logarithmic InstitutdeMathématiquesdeJussieu,Paris,France Connectionandits Symplectic Structure AMS Special Session on Differential Algebra LocalEmbedding April 15, 2007 ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0 ofordern ofrankn withsingularlocus⊂S withsingularlocus⊂S Fuchsian Logarithmic withnon-resonantexponents withnon-resonantresidues Atanirreduciblepoint, Atanirreduciblepoint, dimension=g dimension=2g whereg issomeintegerdependingonnand|S|. Question Can one find some explanation of this doubling? We give a “symplectic explanation” of it. Introduction: Two Classical Worlds Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic LinearDifferentialOperators LinearDifferentialSystems Connections overP1 overP1 JonathanAidan C C Introduction StartingPointL0 Deformation SpaceofL0as Operator FromL0to∇0 Deformation Spaceof∇0as Logarithmic Connectionandits Symplectic Structure LocalEmbedding ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0 withsingularlocus⊂S withsingularlocus⊂S Fuchsian Logarithmic withnon-resonantexponents withnon-resonantresidues Atanirreduciblepoint, Atanirreduciblepoint, dimension=g dimension=2g whereg issomeintegerdependingonnand|S|. Question Can one find some explanation of this doubling? We give a “symplectic explanation” of it. Introduction: Two Classical Worlds Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic LinearDifferentialOperators LinearDifferentialSystems Connections overP1 overP1 JonathanAidan C C ofordern ofrankn Introduction StartingPointL0 Deformation SpaceofL0as Operator FromL0to∇0 Deformation Spaceof∇0as Logarithmic Connectionandits Symplectic Structure LocalEmbedding ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0 Fuchsian Logarithmic withnon-resonantexponents withnon-resonantresidues Atanirreduciblepoint, Atanirreduciblepoint, dimension=g dimension=2g whereg issomeintegerdependingonnand|S|. Question Can one find some explanation of this doubling? We give a “symplectic explanation” of it. Introduction: Two Classical Worlds Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic LinearDifferentialOperators LinearDifferentialSystems Connections overP1 overP1 JonathanAidan C C ofordern ofrankn Introduction withsingularlocus⊂S withsingularlocus⊂S StartingPointL0 Deformation SpaceofL0as Operator FromL0to∇0 Deformation Spaceof∇0as Logarithmic Connectionandits Symplectic Structure LocalEmbedding ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0 withnon-resonantexponents withnon-resonantresidues Atanirreduciblepoint, Atanirreduciblepoint, dimension=g dimension=2g whereg issomeintegerdependingonnand|S|. Question Can one find some explanation of this doubling? We give a “symplectic explanation” of it. Introduction: Two Classical Worlds Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic LinearDifferentialOperators LinearDifferentialSystems Connections overP1 overP1 JonathanAidan C C ofordern ofrankn Introduction withsingularlocus⊂S withsingularlocus⊂S StartingPointL0 Fuchsian Logarithmic Deformation SpaceofL0as Operator FromL0to∇0 Deformation Spaceof∇0as Logarithmic Connectionandits Symplectic Structure LocalEmbedding ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0 Atanirreduciblepoint, Atanirreduciblepoint, dimension=g dimension=2g whereg issomeintegerdependingonnand|S|. Question Can one find some explanation of this doubling? We give a “symplectic explanation” of it. Introduction: Two Classical Worlds Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic LinearDifferentialOperators LinearDifferentialSystems Connections overP1 overP1 JonathanAidan C C ofordern ofrankn Introduction withsingularlocus⊂S withsingularlocus⊂S StartingPointL0 Fuchsian Logarithmic Deformation withnon-resonantexponents withnon-resonantresidues SpaceofL0as Operator FromL0to∇0 Deformation Spaceof∇0as Logarithmic Connectionandits Symplectic Structure LocalEmbedding ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0 Question Can one find some explanation of this doubling? We give a “symplectic explanation” of it. Introduction: Two Classical Worlds Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic LinearDifferentialOperators LinearDifferentialSystems Connections overP1 overP1 JonathanAidan C C ofordern ofrankn Introduction withsingularlocus⊂S withsingularlocus⊂S StartingPointL0 Fuchsian Logarithmic Deformation withnon-resonantexponents withnon-resonantresidues SpaceofL0as Operator Atanirreduciblepoint, Atanirreduciblepoint, FromL0to∇0 dimension=g dimension=2g Deformation Spaceof∇0as Logarithmic whereg issomeintegerdependingonnand|S|. Connectionandits Symplectic Structure LocalEmbedding ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0 Introduction: Two Classical Worlds Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic LinearDifferentialOperators LinearDifferentialSystems Connections overP1 overP1 JonathanAidan C C ofordern ofrankn Introduction withsingularlocus⊂S withsingularlocus⊂S StartingPointL0 Fuchsian Logarithmic Deformation withnon-resonantexponents withnon-resonantresidues SpaceofL0as Operator Atanirreduciblepoint, Atanirreduciblepoint, FromL0to∇0 dimension=g dimension=2g Deformation Spaceof∇0as Logarithmic whereg issomeintegerdependingonnand|S|. Connectionandits Symplectic Structure Question LocalEmbedding Can one find some explanation of this doubling? ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 We give a “symplectic explanation” of it. Symplectic propertyofE0into M0 Symplectic Propertiesofthe SpaceofFuchsian Equationsinthe ModuliSpaceof Logarithmic Connections Goal JonathanAidan Introduction StartingPointL0 Deformation SpaceofL0as Operator FromL0to∇0 M y<< 0 Deformation y yy Spaceof∇0as -(cid:13) yyyy LCoognanreitchtmioincandits Symplectic E0 Structure LocalEmbedding ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0 Definition of E E = {L ∈ W = C(z)[d/dz], L is a monic Fuchsian operator of order n with singular locus in S}. These L’s are exactly those that can be written as: L = ( d )n + an−1( d )n−1+···+ a0 witha ∈ C[z] dz P dz Pn i ≤i·m Definition of L 0 We fix L ∈ E. Suppose L is irreducible and each of its 0 0 local monodromies has distinct eigenvalues. The Starting Point L Symplectic 0 Propertiesofthe SpaceofFuchsian Equationsinthe Basic Setting ModuliSpaceof Logarithmic Fix n ≥ 1. Fix m ≥ 1, and fix Connections JonathanAidan S := {s0,...,sm,sm+1 = ∞} ⊂ P1C suchthatsj 6= sj0 Introduction StartingPointL0 Then we set P = Qm (z −s) and P = Qm (z −s). j=0 j 0 j=1 j Deformation SpaceofL0as Operator FromL0to∇0 Deformation Spaceof∇0as Logarithmic Connectionandits Symplectic Structure LocalEmbedding ofE0inM0 LocallyfromE0intoN0 FromN0intoM0 Symplectic propertyofE0into M0

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Moduli Space of Logarithmic Connections Jonathan Aidan Introduction Starting Point L0 Deformation Space of L0 as Operator From L0 to

Symplectic Properties of the Space of Fuchsian Equations in the PDF

37 Pages·2007·0.33 MB·English

by Jonathan Aidan

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Moduli Space of Logarithmic Connections Jonathan Aidan Introduction Starting Point L0 Deformation Space of L0 as Operator From L0 to

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