Algebraic Intersection Theory on Singular Varieties

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In this project, we studied algebraic intersection theory on singular varieties by the following two methods and obtained the following results :1) Bivariant sheaf theory. If the Chow group of a singular algebraic variety X has a ring structure, then X is called an Alexander scheme. We constructed the topos C of algebraic varieties with the Grothendieck topology which is obtained by proper morphisms between algebraic varieties. Using this topos C, we introduced the Bivariant sheaves for algebraic varieties. It is showned that an algebraic variety X is an Alexander scheme if and only if H'(X, A) =0, where A is the Bivariant sheaf on X.In addition, we have started to study the higher cohomologies of Bivarinat sheaves in order to generalize the above result which might concern the problem on finite dimensionality of Motives that is the most important problem in the field of algebraic cycles and introduced the theory of Hyper-Covering to compute the higher cohomologies of Bivariant sheaves concretely.2) Splitting of Vector Bundles. As for the splitting problem for rank two vector bundles on projective spaces which is one of the most important problem in the field of algebraic vector bundles, we obtained the following two results. (1) Let E be a rank two very ample vector bundle on P^n (n*4) and X an determinantal variety defined by global sections of E.Analyzing the structure of the Hilbert scheme of those determinantal varieties, it is shown that E splits into line bundles if and only if H^1 (P, End(E))=0, where P is a 4- or 5- dimensional linear subspace of P^n. (2) E is a direct sum of line bundles if and only if dimH^1(X, O_x(r-Z)) *O(r^1)(r*0) and diinH ^k(X, O_x(-Rz-_sH)) * P_k (s) (r, s>O) (l*k*dimX-l), where Z and H are specific effective divisors on the determinantal variety X and P_k (s) is a polynomial on s which is independent of r.