Degeneracy Loci and Grassmannians

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 2)


Let σ: EF be a homomorphism of vector bundles of ranks e and f on a variety X, and let k ≦ min (e, f). The degeneracy locus
$$D_{k}\left ( \sigma \right)=\left \{ x\in X|\,\mathrm{rank}\left ( \sigma \left ( x \right) \right)\leqq k\right \}$$
has codimension at most (ek) (fk) in X, if it is not empty. We construct a class
$$\mathbb{D} _{k}\left ( \sigma \right)\in A_{m}\left ( D_{k} \left ( \sigma \right)\right)$$
, m = dim (X) − (ek) (fk), whose image in A m (X) is given by the Thom-Porteous formula:
$$\mathbb{D} _{k}=\Delta _{f-k}^{\left ( e-k \right)}\left ( c\left ( F-E \right) \right)\cap [X]$$


Vector Bundle Line Bundle Young Diagram Chern Class Schubert Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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