Français
Indices of vector fields on singular varieties and the Milnor number
Post-edited
Authors : Seade, José (Author of the conference)
CIRM (Publisher )
14B05 - Singularities
32S65 - Singularities of holomorphic vector fields and foliations
57R20 - Characteristic classes and numbers
CIRM (Publisher )
Loading the player...
|
| Milnor fibration indices of vector fields Schwartz index GSV index Lê numbers and cycles Chern classes Fulton-Johnson classes Schwartz classes Milnor classes Questions |
Abstract :
Let $(V,p)$ be a complex isolated complete intersection singularity germ (an ICIS). It is well-known that its Milnor number $\mu$ can be expressed as the difference:
$$\mu = (-1)^n ({\rm Ind}_{GSV}(v;V) - {\rm Ind}_{rad}(v;V)) \;,$$
where $v$ is a continuous vector field on $V$ with an isolated singularity at $p$, the first of these indices is the GSV index and the latter is the Schwartz (or radial) index. This is independent of the choice of $v$.
In this talk we will review how this formula extends to compact varieties with non-isolated singularities. This depends on two different ways of extending the notion of Chern classes to singular varieties. On elf these are the Fulton-Johnson classes, whose 0-degree term coincides with the total GSV-Index, while the others are the Schwartz-McPherson classes, whose 0-degree term is the total radial index, and it coincides with the Euler characteristic. This yields to the well known notion of Milnor classes, which extend the Milnor number. We will discuss some geometric facts about the Milnor classes.
MSC Codes : $$\mu = (-1)^n ({\rm Ind}_{GSV}(v;V) - {\rm Ind}_{rad}(v;V)) \;,$$
where $v$ is a continuous vector field on $V$ with an isolated singularity at $p$, the first of these indices is the GSV index and the latter is the Schwartz (or radial) index. This is independent of the choice of $v$.
In this talk we will review how this formula extends to compact varieties with non-isolated singularities. This depends on two different ways of extending the notion of Chern classes to singular varieties. On elf these are the Fulton-Johnson classes, whose 0-degree term coincides with the total GSV-Index, while the others are the Schwartz-McPherson classes, whose 0-degree term is the total radial index, and it coincides with the Euler characteristic. This yields to the well known notion of Milnor classes, which extend the Milnor number. We will discuss some geometric facts about the Milnor classes.
14B05 - Singularities
32S65 - Singularities of holomorphic vector fields and foliations
57R20 - Characteristic classes and numbers
|
Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 11/03/15 Conference Date : 26/02/15 Subseries : Research talks arXiv category : Algebraic Geometry ; Complex Variables Mathematical Area(s) : Algebraic & Complex Geometry Format : QuickTime (.mov) Video Time : 00:51:45 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2015-02-26_Seade.mp4 
|
Information on the Event
Event Title : Local and global invariants of singularities / Invariants locaux et globaux des singularitésEvent Organizers : Dutertre, Nicolas ; Pichon, Anne Dates : 23/02/15 - 27/02/15 Event Year : 2015 Event URL : http://chairejeanmorlet-1stsemester2015....
Citation Data
DOI : 10.24350/CIRM.V.18707003Cite this video as: Seade, José (2015). Indices of vector fields on singular varieties and the Milnor number. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18707003 URI : http://dx.doi.org/10.24350/CIRM.V.18707003 |
Bibliography
- Aluffi, P. (1999). Chern classes for singular hypersurfaces. Transactions of the American Mathematical Society, 351(10), 3989-4026 - http://dx.doi.org/10.1090/s0002-9947-99-02256-4
- Brasselet, J.-P., & Schwartz, M.-H. (1981). Sur les classes de Chern d'un ensemble analytique complexe. Astérisque 82-83, 93-147 - https://www.zbmath.org/?q=an:0471.57006
- Brasselet, J.-P., Lehmann, D., Seade, J., & Suwa, T. (2002). Milnor classes of local complete intersections. Transactions of the American Mathematical Society, 354(4), 1351-1371 - http://dx.doi.org/10.1090/S0002-9947-01-02846-X
- Brasselet, J.-P., Seade, J., & Suwa, T. (2009). Vector fields on singular varieties. Berlin: Springer. (Lecture Notes in Mathematics, 1987) - http://dx.doi.org/10.1007/978-3-642-05205-7
- Callejas-Bedregal, R., Morgado, M.F.Z., & Seade, J. (2014). Lê cycles and Milnor classes. Inventiones Mathematicae, 197(2), 453-482; erratum ibid. 197(2), 483-489 - http://dx.doi.org/10.1007/s00222-013-0450-7
- Parusinski, A., & Pragacz, P. (2001). Characteristic classes of hypersurfaces and characteristic cycles. Journal of Algebraic Geometry, 10(1), 63-79 - https://www.zbmath.org/?q=an:1072.14505
