Chern Classes for Singular Hypersurfaces

We prove a simple formula for MacPherson’s Chern class of hypersurfaces in nonsingular varieties. The result highlights the relation between MacPherson’s class and other definitions of homology Chern classes of singular varieties, such as Mather’s Chern class and the class introduced by W. Fulton in [Fulton], 4.2.6. §0. Introduction 2 §1. Statements of the result 4 §1.1. MacPherson’s Chern class and Segre classes 4 §1.2. MacPherson’s Chern class and Fulton’s Chern class 5 §1.3. MacPherson’s Chern class and Mather’s Chern class 6 §1.4. Notational device 8 §1.5. MacPherson’s Chern class and μ-classes 9 §2. The proof: preliminaries and divisors with normal crossings 10 §2.1. c∗(X) = c∗(Xred) 11 §2.2. Divisors with normal crossings: proof 13 §3. The proof: behavior under blow-ups 16 §3.1. (3) in terms of classes in PP M L, PP M̃ L 16 §3.2. (3) in terms of classes in P(P M L ⊕ P M̃ L) over M̃ 18 §3.3. (3) in terms of classes in P(P M L ⊕ P M̃ L) over BL 20 §3.4. The graph construction 21 §3.5. Coordinate set-up in BL 22 §3.6. Three lemmas 26 §3.7. Computing Z∞ 28 §3.8. End of the proof of (3) 31 §4. Remarks and applications 33 §4.1. μ-class and Parusiński’s Milnor number 33 §4.2. Blowing up μ-classes 34 §4.3. Contact of two hypersurfaces 35 §4.4. A geometric application 37 Supported in part by NSF grant DMS-9500843