Normal Surface Singularities

Description

This monograph provides a comprehensive introduction to the theory of complex normal surface singularities, with a special emphasis on connections to low-dimensional topology. In this way, it unites the analytic approach with the more recent topological one, combining their tools and methods. In the first chapters, the book sets out the foundations of the theory of normal surface singularities. This includes a comprehensive presentation of the properties of the link (as an oriented 3-manifold) and of the invariants associated with a resolution, combined with the structure and special properties of the line bundles defined on a resolution. A recurring theme is the comparison of analytic and topological invariants. For example, the Poincaré series of the divisorial filtration is compared to a topological zeta function associated with the resolution graph, and the sheaf cohomologies of the line bundles are compared to the Seiberg–Witten invariants of the link. Equivariant Ehrhart theory is introduced to establish surgery-additivity formulae of these invariants, as well as for the regularization procedures of multivariable series. In addition to recent research, the book also provides expositions of more classical subjects such as the classification of plane and cuspidal curves, Milnor fibrations and smoothing invariants, the local divisor class group, and the Hilbert–Samuel function. It contains a large number of examples of key families of germs: rational, elliptic, weighted homogeneous, superisolated and splice-quotient. It provides concrete computations of the topological invariants of their links (Casson(–Walker) and Seiberg–Witten invariants, Turaev torsion) and of the analytic invariants (geometric genus, Hilbert function of the divisorial filtration, and the analytic semigroup associated with the resolution). The book culminates in a discussion of the topological and analytic lattice cohomologies (as categorifications of the Seiberg–Witten invariant and of the geometric genus respectively) and of the graded roots. Several open problems and conjectures are also formulated. Normal Surface Singularities provides researchers in algebraic and differential geometry, singularity theory, complex analysis, and low-dimensional topology with an invaluable reference on this rich topic, offering a unified presentation of the major results and approaches. Preface Contents 1 Introduction 1.1 General Introduction 1.2 Why a New Book? 1.3 Examples and Constructions 1.4 Prologue of the Chapters 1.5 What Is New? 1.6 Organization of the Sections 1.7 What Is Not Covered 2 Resolution of Surface Singularities 2.1 Modifications and Resolutions 2.2 The Embedded Resolution Graph 2.2.A Continued Fraction and Dedekind Sums 2.3 Example: Hirzebruch–Jung Singularities 2.4 Existence and Uniqueness of the Minimal Resolution 2.5 Analytic Realizations: Theorems of Grauert and Winters 3 The Link 3.1 The Local Conic Structure of Isolated Analytic Germs. The Link 3.2 Embedded Links. Milnor Fibration 3.2.A The Homological Package of the Milnor Fibration 3.3 Plumbed 3-Manifolds: Plumbing Graphs 3.3.A The Plumbing Calculus 3.4 Homological Properties of the Link 3.5 Examples of Special Graphs 3.5.A Star-Shaped Graphs 3.5.B Cyclic Graphs 3.5.C Surgery 3-Manifolds 3.6 Basic Classification Theorems on 3-Manifolds and Links 3.7 The Fundamental Group of the Link 4 Coverings 4.1 Cyclic Coverings 4.1.A Cyclic Coverings of Graphs 4.1.B The Universal Covering of the Embedded Resolution Graph 4.1.C Ramified Cyclic Coverings of Germs 4.1.D Algebraic Monodromy and Cyclic Coverings 4.1.E o–Ramified Cyclic Coverings 4.2 Abelian Coverings 5 Examples 5.1 Weighted Homogeneous Singularities 5.2 Superisolated Singularities 5.3 Splice Diagrams 5.4 Splice Quotient Singularities 5.5 Newton Non-degenerate Singularities 5.5.A The Classical Case of Hypersurfaces in ps: [/EMC pdfmark [/Subtype /Span /ActualText (left parenthesis double struck upper C Superscript n plus 1 Baseline comma 0 right parenthesis) /StPNE pdfmark [/StBMC pdfmark(Cn+1,0)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 5.5.B Weil Divisors in Affine Toric Singularities 6 Invariants Associated with a Resolution 6.1 Local Divisor Class Group 6.1.A ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper Q) /StPNE pdfmark [/StBMC pdfmarkQps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark-Cartier Divisors and Canonical Coverings 6.2 Natural Line Bundles 6.3 The Canonical Cycle 6.3.A ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper Z Subscript upper K Superscript 2 Baseline plus StartAbsoluteValue script upper V EndAbsoluteValue) /StPNE pdfmark [/StBMC pdfmarkZK2+|V|ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Formulae 6.3.B The Gorenstein Property 6.4 Vanishing Theorems 6.4.A The Cohomological Cycle 6.5 Base Point Freeness 6.6 The Monoids ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S) /StPNE pdfmark [/StBMC pdfmarkSps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S prime) /StPNE pdfmark [/StBMC pdfmarkSps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 6.6.A The Representatives rh and sh 6.6.B The `Local' Zariski Decomposition 6.7 The Monoids ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S Subscript a n) /StPNE pdfmark [/StBMC pdfmarkSanps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S prime Subscript a n) /StPNE pdfmark [/StBMC pdfmarkSanps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 6.7.A Kulikov Singularities 6.8 The (Equivariant) Geometric Genus and Laufer's Duality 6.8.A The Linear Subspace Arrangement ps: [/EMC pdfmark [/Subtype /Span /ActualText (left brace normal upper Omega Subscript upper X overTilde Baseline left parenthesis script upper I right parenthesis right brace Subscript script upper I subset of script upper V) /StPNE pdfmark [/StBMC pdfmark{ΩX"0365X(I)}IVps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark of Forms 6.8.B A Topological Upper Bounds for pg. The Invariant Pathi 6.8.C A Topological Lower Bounds for pg 6.8.D Plurigenera 6.9 Relations with Smoothing Invariants 6.9.A The Formulae of Laufer, Durfee and Wahl 6.9.B Thom–Sebastiani Type Results and Suspension-Formulae 6.9.C Some Open Problems for Hypersurface Singularities 6.10 Spin and Spinc Structures 6.10.A Turaev's Euler Structures 7 The Artin–Laufer Program 7.1 Rational Singularities 7.1.A Some Analytic Invariants Described Topologically 7.1.B Some Invariants of the Universal Abelian Covering 7.2 Elliptic Singularities 7.2.A Elliptic Kulikov Singularities 7.2.B Minimally Elliptic Singularities 7.2.C The Elliptic Sequence, General Case 7.2.D The Elliptic Sequence, Numerically Gorenstein Case 7.2.E The `second' Elliptic Sequence, General Case 7.2.F Maximally Elliptic Numerically Gorenstein Germs 7.2.G The Multiplicity and Hilbert–Samuel Function 7.2.H Non-maximally Elliptic Singularities 7.2.I The `generic' Analytic Structure of Elliptic Singularities 7.3 Weighted Cubes and Generalized Computation Sequences 7.3.A The Topology of the Spaces {Sk,n}n. Deformation Retractions 7.3.B Measure of Non-rationality. `Bad' Vertices 7.3.C Lattice Fibrations. Universal Cycles in the Fibers 7.3.D Concatenated Computation Sequences of AR Graphs 8 Multivariable Divisorial Filtration 8.1 Multi-Variable Series 8.1.A Multigradings 8.1.B Poincaré series of Weighted Homogeneous Singularities 8.1.C Filtrations 8.2 Divisorial Filtration and its Multivariable Series 8.2.A The Series H(t) and P(t) 8.3 Linear Subspace Arrangements Associated with the Filtration 8.4 The Topological Series Z(t) 8.4.A The Extension of Z(t) to the Grothendieck Ring 8.4.B Motivic Extension of Z(t) via the Space of Divisors 8.4.C ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper Z left parenthesis double struck upper L comma bold t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkZ(L,t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper P left parenthesis double struck upper L comma bold t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkP(L,t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark for Cusp Singularities 8.5 Singularities Satisfying the `end curve condition' 8.5.A The Identity P(t)=Z(t) and the Monomial Filtration 8.5.B Topological Characterization of ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S prime Subscript a n) /StPNE pdfmark [/StBMC pdfmarkSanps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/Subtype /Span /ActualText (script upper S Subscript a n) /StPNE pdfmark [/StBMC pdfmarkSanps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 8.5.C Base Points of ps: [/EMC pdfmark [/Subtype /Span /ActualText (phi Superscript asterisk Baseline German m Subscript upper X comma o) /StPNE pdfmark [/StBMC pdfmarkϕ*mX,ops: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and Multiplicity 8.5.D The Monomial Condition and End Curve Theorem 8.6 Reductions of Variables in the Series P(t) and Z(t) 8.6.A Examples. Is ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper P Subscript h comma script upper I) /StPNE pdfmark [/StBMC pdfmarkPh, Ips: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Topological/Combinatorial? 8.6.B Example. P0 and Z0 for Superisolated Singularities 8.6.C The Ring ps: [/EMC pdfmark [/Subtype /Span /ActualText (circled plus Subscript n greater than or equals 0 Baseline phi Subscript asterisk Baseline left parenthesis script upper O Subscript upper X overTilde Baseline left parenthesis minus n l right parenthesis right parenthesis Subscript o Baseline) /StPNE pdfmark [/StBMC pdfmarkn≥0 ϕ*(OX"0365X(-nl))ops: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 8.7 The Periodic Constant of One-Variable Series 8.7.A Okuma's Additivity Formula 9 Topological Invariants. The Seiberg–Witten Invariant 9.1 The Casson Invariant 9.2 The Casson Invariant Conjecture of Neumann–Wahl 9.2.A The Proof of CIC for Splice Type Singularities 9.3 The Casson–Walker Invariant 9.3.A Additivity Formulae for λ and ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper K squared plus StartAbsoluteValue script upper V EndAbsoluteValue) /StPNE pdfmark [/StBMC pdfmarkK2+|V|ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 9.4 The Reidemeister–Turaev Torsion. Generalities 9.4.A The Fourier Transform 9.5 The Reidemeister–Turaev Torsion of Graph 3-Manifolds 9.5.A Additivity Formula for the Torsion 9.6 The Seiberg–Witten Invariant 9.6.A The Seiberg–Witten Invariant and the Series Z(t) 9.7 The Seiberg–Witten Invariant Conjecture/Coincidence 9.7.A SWIC for Weighted Homogeneous Singularities 9.7.B SWIC and Superisolated Singularities 9.7.C The Seiberg–Witten Invariant and Abelian Coverings 10 Ehrhart Theory and the Seiberg–Witten Invariant 10.1 Introduction into Ehrhart Theory 10.1.A Equivariant Multivariable Ehrhart Theory 10.2 Multivariable Rational Functions and Their Periodic Constants 10.3 Ehrhart Theory of Z(t) Associated with a Plumbing Graph 10.3.A Reduction of the Variables of Z(t) 10.3.B Technical Lemmas Regarding the Intersection Form 10.4 The Modified Counting Function 10.4.A Convexity Property of the Modified Counting Functions 10.4.B Additivity Formula for the Modified Counting Functions 10.5 General Additivity Formulae 10.6 Duality for the Topological Series 10.6.A The `Polynomial Part' of the Series Z(t) 10.6.B Polytopes, Lattice Points and the Seiberg–Witten Invariant 11 Lattice Cohomology 11.1 The Lattice Cohomology Associated with a System of Weights 11.1.A The Lattice Cohomology Associated with a Plumbing Graph 11.1.B The Lattice Cohomology and the Seiberg–Witten Invariant 11.2 Graded Roots and Their Cohomologies 11.2.A The Graded Root Associated with a Plumbing Graph 11.3 Graded Roots of Almost Rational Graphs 11.3.A Example. Star-Shaped Graphs 11.3.B Example. The Surgery Manifold ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper S Subscript negative d Superscript 3 Baseline left parenthesis upper K right parenthesis) /StPNE pdfmark [/StBMC pdfmarkS3-d(K)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 11.3.C Superisolated Singularities with One Cusp 11.4 The Reduction Theorem 11.5 Application. ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper H Superscript asterisk) /StPNE pdfmark [/StBMC pdfmarkH*ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark of the Surgery Manifold ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper S Subscript negative d Superscript 3 Baseline left parenthesis number sign Subscript i Baseline upper K Subscript i Baseline right parenthesis) /StPNE pdfmark [/StBMC pdfmarkS3-d(#iKi)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 11.5.A ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper H Superscript 0 Baseline left parenthesis upper S Subscript negative d Superscript 3 Baseline left parenthesis number sign Subscript i Baseline upper K Subscript i Baseline right parenthesis right parenthesis) /StPNE pdfmark [/StBMC pdfmarkH0(S3-d(#iKi))ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and the Multiplicity Sequences 11.5.B Superisolated Singularities with More Cusps 11.6 Path Lattice Cohomology 11.6.A Lattice Cohomology of Newton Non-degenerateGerms 11.7 Lattice Cohomology and Heegaard Floer Homology 11.8 Combinatorial Lattice Cohomology with Special Weight Functions 11.9 Analytic Lattice Cohomology of Normal Surface Singularities 11.9.A Analytic Reduction Theorem 11.9.B Examples 11.9.C ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper H Subscript a n comma 0 Superscript asterisk) /StPNE pdfmark [/StBMC pdfmarkH*an,0ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and pg–Constant Deformations 12 Appendix. Complex Analytic Spaces 12.1 Analytic Algebras 12.2 Complex Spaces 12.3 Analytic Coverings 12.4 Complex Reduced Surfaces Bibliography Index