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Thom-Porteous formulas in algebraic cobordism
This PhD thesis grew out of the attempt to understand the forms in which the Thom-Porteous formula for the Chow ring could be extended to other oriented cohomology theories and in particular to algebraic cobordism, the universal such theory.
For a suitably generic morphism of vector bundles f:E->F over a scheme X of pure dimension, the Thom-Porteous formula expresses the fundamental class of a degeneracy locus as a determinant in the Chern classes of E and F. The core of the proof consists in dealing with the universal case of morphisms of bundles over the flag bundle Fl(V). In this setting, the degeneracy loci are the so-called Schubert varieties, whose classes in the Chow ring are given by the double Schubert polynomials. The same approach led Buch to prove similar results for the Grothendieck ring of locally coherent sheaves which, once it is made into a graded ring, represents another example of oriented cohomology theory. In this case the fundamental classes of Schubert varieties are described by means of the double Grothendieck polynomials.
In order to generalise these results, I used the work of Hornbostel and Kiritchenko on the push-forward morphism for P^1-bundles in algebraic cobordism to give an explicit description of the push-forward cobordism classes of the Bott-Samelson resolutions in the algebraic cobordism of Fl(V). These schemes over Fl(V) play an imporant role in the classical theory since every Schubert variety is birationally isomorphic to, at least, one member of the family. It then follows that knowing these classes in CH(Fl(V)) and K^0(Fl(V)) automatically gives the fundamental classes of all Schubert varieties.
In the attempt to obtain an extension of this result for a more general oriented cohomology theory, I have specialized the expression obtained in the algebraic cobordism ring off Fl(V) to connected K-theory, the universal theory satisfying birational invariance for smooth schemes. In this context I have been able to give a geometrical interpretation to the double beta-polynomials of Fomin and Kirillov: the push-forward classes of the Bott-Samelson resolutions can be obtained by evaluating the double beta-polynomials at the Chern roots of the bundles used to describe the corresponding Schubert variety, exactly as it was happening for the Chow ring and for the Grothendieck ring.