If only the cohomology of projective non-singular varieties is considered, one speaks of pure motives. Grothendieck proposed a definition of a category of pure motives, and showed that if the category defined had a number of properties, modelled on those of Hodge structures, the Weil conjectures ...
Grothendieck不是为了推广而推广,一个原因是为了更好地理解已有的对象。 His answer opened my eyes! Grothendieck did not develop the theory of schemes for the sake of generalization, but he did see that in order to understand algebraic varieties you do need schemes, and this was the reason (or a...
If X and Y are nonsingular varieties and f:X→Y is a projective morphism (or, more generally, a proper morphism), then there is a well-defined push-forward f* on Chow groups. There is also a kind of push-forward for vector bundles. The Grothendieck group of vector bundles on X, ...
[Edin] A. GROTHENDIECK, The cohomology theory of abstract algebraic varieties , 1960 Proc. Internat. Congress Math. (Edinburgh, 1958), pp. 103–118, Cambri dge Univ. Press, New York. [FAC] J.-P. SERRE, Faisceaux algébriques cohérents, Ann. of Math. 61 (1955) , 197–278. [Fest...
For this, we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs. Equivalently, the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of Schubert varieties...
Let K 0 (Var k ) be the Grothendieck ring of algebraic varieties over a field k . Let X , Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X 1 , ..., X n , Y 1 , ..., Y n into locally closed subvarieties such that...
We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$, assuming $k$ has characteristic 0, and for any ...
In each dimension n3, there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally ...
The theory of motives was created by Grothendieck in the 1960s as he searched for a universal cohomology theory for algebraic varieties. The theory of pure motives is well established as far as the construction is concerned. Pure motives... JP Murre,J Nagel,CAM Peters - Lectures on the Theo...
The power structure over the Grothendieck (semi)ring of complex quasi-projective varieties constructed by the authors is used to express the generating series of classes of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety as an exponent of that for the complex aff...