New PDF release: Hodge Theory and Complex Algebraic Geometry II

By Claire Voisin, Leila Schneps

ISBN-10: 0521802830

ISBN-13: 9780521802833

The second one quantity of this contemporary account of Kaehlerian geometry and Hodge idea starts off with the topology of households of algebraic kinds. the most effects are the generalized Noether-Lefschetz theorems, the well-known triviality of the Abel-Jacobi maps, and most significantly, Nori's connectivity theorem, which generalizes the above. The final half bargains with the relationships among Hodge thought and algebraic cycles. The textual content is complemented through routines providing worthy leads to advanced algebraic geometry. additionally on hand: quantity I 0-521-80260-1 Hardback $60.00 C

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This theorem is better than the proposition because the value of the X s are more elementary than the value of the Y s. Next we will see that in the above situation that the bilinear relations R( fi', fi') generate all homogeneous forms vanishing on X when we embed X in projective space via fi'. Let Jt be an ample sheaf on X. If Jt is an m-power with m ~ 3. 9 we have a surjection SymdT(X,Jt)] ~ ED T(X,Jt0 n ) . n~O Let I be the kernel of this surjection. 13. a) If m = 3 then I is generated as an ideal by its forms of degree 2 and 3.

9 =I o. Assume that H is degenerate we need to see that X(f£) = O. 9a) and b). 2'») = 0 as the cohomology of the structure sheaf is an exterior algebra. If H is nondegenerate then z = 0 and Hi(X,f£) is non-zero when i = n and its dimension is ±JdetLE. Thus we need only check that ± = (_l)n. This is a question in linear alg;ebra which we don't do. (Hint: if H' is a pOf>itive definite form on V when (ImH,)g is positive). 11. 'Jx(D)) = the intersection number ~Dg and g. 'Jx(D)) = 0 ifi > O. 12. 'Jx, then Hi(X,f£) all i.

H. data. The sheaf f£ = f£( a, H) is called excellent with respect to the decomposition A EB B = L, or just excellent if there is no confusion. The whole theory of theta function revolves around the special properties of excellent sheaves. Consider L~ = {v E VIE(v,l) E 71 for all 1 in L}. We have L~ = A ~ EB B~ where A = L ~ n A 0 1R and B = L ~ n B 0 IR. Thus K( f£) = L ~ f L is the direct sum A ~ fA EB B~ f B. Let A(f£) = A ~ fA and B(f£) = B~ f B. One may check that A(f£) and B(f£) are maximal isotropic subgroups of K(f£).

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Hodge Theory and Complex Algebraic Geometry II by Claire Voisin, Leila Schneps

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