Vector fields on manifolds play a major role in mathematics and other
sciences. In particular, the Poincaré-Hopf index theorem gives rise to the theory of
Chern classes, key manifold-invariants in geometry and topology.
It is natural to ask what is the ‘good’ notion of the index of a vector
field, and of Chern classes, if the underlying space becomes singular. The question has
been explored by several authors resulting in various answers, starting with the
pioneering work of M.-H. Schwartz and R. MacPherson.
We present these notions in the framework of the obstruction theory and the
Chern-Weil theory. The interplay between these two methods is one of the main features of
the monograph.
Table of Contents:
1 The Case of Manifolds
1.1 Poincar´e–Hopf Index Theorem
1.1.1 Poincar´e–Hopf Index at Isolated Points
1.1.2 Poincar´e–Hopf Index at Nonisolated Points
1.2 Poincar´e and Alexander Dualities
1.3 Chern Classes via Obstruction Theory
1.3.1 Chern Classes of Almost Complex Manifolds
1.3.2 Relative Chern Classes
1.4 Chern–Weil Theory of Characteristic Classes
1.5 ˇCech-de Rham Cohomology
1.5.1 Integration on the ˇCech-de Rham Cohomology .
1.5.2 Relative ˇCech-de Rham Cohomology – Alexander Duality
1.6 Localization of Chern Classes
1.6.1 Characteristic Classes in the ˇCech-de Rham Cohomology
1.6.2 Localization of Characteristic Classes of Complex Vector Bundles
1.6.3 Localization of the Top Chern Class
1.6.4 Hyperplane Bundle
1.6.5 Grothendieck Residues
1.6.6 Residues at an Isolated Zero
1.6.7 Examples
2 The Schwartz Index
2.1 Isolated Singularity Case
2.2 Whitney Stratifications
2.3 Radial Extension of Vector Fields
2.4 The Schwartz Index on a Stratified Variety
2.4.1 Case of Vector Fields with an Isolated Singularity
2.4.2 Case of Vector Fields with Nonisolated Singularity
3 The GSV Index
3.1 Vector Fields Tangent to a Hypersurface
3.2 The Index for Vector Fields on ICIS
3.3 Some Applications and Examples
3.4 The Case of Isolated Smoothable Singularities
3.5 Nonisolated Singularities
3.5.1 The Strict Thom Condition for Complex Analytic Maps
3.5.2 The Hypersurface Case
3.5.3 The Complete Intersection Case
3.6 The Proportionality Theorem
3.7 Geometric Applications
3.7.1 Topological Invariance of the Milnor Number
3.7.2 The Canonical Contact Structure on the Link
3.7.3 On the Normal Bundle of Holomorphic Singular Foliations
4 Indices of Vector Fields on Real Analytic Varieties
4.1 The Schwartz Index on Real Analytic Varieties
4.2 The GSV Index on Real Analytic Varieties.
4.3 A Geometric Interpretation of the GSV Index
4.4 Topological Invariants and Curvatura Integra
4.5 Relation with the Milnor Number for Real Singularities
5 The Virtual Index
5.1 The Virtual Tangent Bundle of a Local Complete Intersection
5.2 Chern–Weil Theory for Virtual Bundles
5.3 Characteristic Numbers on Singular Varieties
5.4 The Virtual Index
5.5 Identification with GSV Index When Singularities are Isolated
5.6 A Generalization of the Adjunction Formula
5.7 An Integral Formula for the Virtual Index
6 The Case of Holomorphic Vector Fields
6.1 Baum–Bott Residues of Holomorphic Vector Fields
6.2 One-Dimensional Singular Foliations
6.3 Residues of Holomorphic Vector Fields on Singular Varieties
6.3.1 Grothendieck Residues Relative to a Subvariety
6.3.2 Residues for the Ambient Tangent Bundle (Generalized Variation)
6.3.3 Residues for the Normal Bundle (Residues of Type Camacho–Sad)
6.3.4 Residues for the Virtual Tangent Bundle (Singular Baum–Bott)
7 The Homological Index and Algebraic Formulas
7.1 The Homological Index
7.2 The Hypersurface Case
7.3 The Index of Real Analytic Vector Fields
7.3.1 The Signature Formula of Eisenbud– Levine–Khimshiashvili
7.3.2 The Index on Real Hypersurface Singularities
8 The Local Euler Obstruction
8.1 Definition of the Euler Obstruction. The Nash Blow Up .
8.1.1 Proportionality Theorem for Vector Fields
8.2 Euler Obstruction and Hyperplane Sections
8.3 The Local Euler Obstruction of a Function
8.4 The Euler Obstruction and the Euler Defect
8.5 The Euler Defect at General Points
8.6 The Euler Obstruction via Morse Theory
9 Indices for 1-Forms
9.1 Some Basic Facts About 1-Forms
9.2 Radial Extension and the Schwartz Index
9.3 Local Euler Obstruction of a 1-Form and the Proportionality Theorem
9.4 The Radial Index
9.5 The GSV Index
9.5.1 Isolated Singularity Case
9.5.2 Nonisolated Singularity Case
9.6 The Homological Index
9.7 On the Milnor Number of an Isolated Singularity
9.8 Indices for Collections of 1-Forms
9.8.1 The GSV Index for Collections of 1-Forms
9.8.2 Local Chern Obstructions
10 The Schwartz Classes
10.1 The Local Schwartz Index of a Frame
10.2 Proportionality Theorem.
10.3 The Schwartz Classes
10.4 Alexander and Other Homomorphisms
10.5 Localization of the Schwartz Classes
10.5.1 The Topological Viewpoint
10.5.2 The Differential Geometric Viewpoint
10.6 MacPherson and Mather Classes
11 The Virtual Classes
11.1 Virtual Classes
11.2 Lifting a Frame to the Milnor Fiber
11.3 The Fulton–Johnson Classes
11.4 Localization of the Virtual Classes
12 Milnor Number and Milnor Classes
12.1 Milnor Classes
12.2 Localization of Milnor Classes
12.3 Differential Geometric Point of View
12.4 Generalized Milnor Number
13 Characteristic Classes of Coherent Sheaves on Singular Varieties
13.1 Local Chern Classes and Characters in the ˇCech-de Rham Cohomology
13.2 Thom Class
13.3 Riemann-Roch Theorem for Embeddings
13.4 Homology Chern Characters and Classes
13.5 Characteristic Classes of the Tangent Sheaf
References
Index
254 pages, Paperback
