Motives and oriented cohomology of generically cellular varieties

摘要

For a cellular variety $X$ over a field $k$ of characteristic 0 and an algebraic oriented cohomology theory $mathtt{h}$ of Levine-Morel we construct a filtration on the cohomology ring $mathtt{h}(X)$ such that the associated graded ring is isomorphic to the Chow ring of $X$. Using this filtration we establish the following comparison result between Chow motives and $mathtt{h}$-motives of generically cellular varieties: any irreducible Chow-motivic decomposition of a generically cellular variety $Y$ gives rise to an $mathtt{h}$-motivic decomposition of $Y$ with the same generating function. Moreover, under some conditions on the coefficient ring of $mathtt{h}$ the obtained $mathtt{h}$-motivic decomposition will be irreducible. We also prove that if the Chow motives of two twisted forms of $Y$ coincide, then their $mathtt{h}$-motives coincide as well.