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As i learn the local index theory , Chern roots appears and i cannot understand what it is and i can't find any references about it .Can anyone tell me something about it and give me some references,also tell me how to compute it ?

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Say you want to do some computation of Chern classes of a finite "combination" (e.g. tensor product) of vector bundles. Then by the splitting principle you can pretend that the bundles are direct sums of line bundles. The Chern classes of these hypothetical line bundles are the Chern roots. The Chern classes of the original bundles are symmetric polynomials of the roots. It's hard to say more for a nonspecific question like this. –  Donu Arapura Dec 14 '10 at 14:07
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Two books come to mind: $$ $$ Milnor and Stasheff "Characteristic classes" $$ $$ and $$ $$ Hirzebruch's "Topological methods in algebraic geometry". –  José Figueroa-O'Farrill Dec 14 '10 at 15:27
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They are used to state Hirzebruch's "Riemann Roch" theorem. E.g. the arithmetic genus chi(O) is a polynomial in the chern classes, but how does one write these polynomials for manifolds of all dimensions? Writing it symmetrically in terms of the chern roots, and using Donu's remark, allows it to be written in terms of the chern classes. There is a brief discussion in the last part of my notes on RRT at math.uga.edu/~roy. See also Hartshorne p.431ff, Macdonald: Alg. geom. - intro to schemes, p.102 ff, and of course Hirzebruch, as Jose said. –  roy smith Dec 14 '10 at 16:01
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Chern root algorithm and local index theorems:

http://books.google.com/books?id=sqwa47R6Ds4C&pg=PA324&lpg=PA324

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