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Hodge Style Chern Classes for Vector Bundles on Schemes
| Content Provider | Semantic Scholar |
|---|---|
| Author | Kumar, Nishanth Rao, A. Prabhakar Ravindra, G. V. |
| Copyright Year | 2005 |
| Abstract | In this note, we develop the formalism of Hodge style chern classes of vector bundles over arbitrary quasi-projective schemes defined over K, a field of characteristic zero. The theory of Chern classes is well known by now and without any restriction on the characteristic, can be defined in many theories with rational coefficients, like for example the Chow ring. Atiyah [1] developed the theory of Chern classes of vector bundles with values in the Hodge ring ⊕H(X,ΩpX) for smooth complex varieties. Grothendieck [2] remarked that Atiyah’s constructions could be transposed (“sans difficulté”) to the case of any S-scheme X and referred to a future paper where it would appear. To the best of our knowledge, this has not occured. The primary purpose of this note is to satisfy ourselves that indeed the formalism extends to the case of arbitrary schemes and at the same time to fill a gap in the existing literature. Everything in this paper is “known” to experts and has indeed been written down many times in the case of smooth varieties and has been generalised in various directions. Our purpose is to provide a suitable reference for our own use of this theory on schemes as well as to provide a self-contained exposition that would be suitable for a new-comer to the subject. Our personal motivation for making sure that the theory was valid for arbitrary schemes was in understanding intersection theory on nonreduced hypersurfaces in projective spaces (see [7]). Unlike in the theory of Chern classes with values in the Chow ring, there is a closer connection between the Hodge cohomology of these hypersurfaces and that of the projective space. The goal of this paper is to verify the following: let X be a quasiprojective scheme over a field K of characteristic zero. For any vector bundle E of rank r onX, there is an element c(E) = 1+ ∑r i=1 ci(E) in the graded commutative K-algebra ⊕Hi(X,ΩX) satisfying the following two properties: |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.cs.umsl.edu/~girivaru/chern.pdf |
| Alternate Webpage(s) | http://www.cs.umsl.edu/~rao/papersdir/chern.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
