Download Algebraic Geometry: Proceedings of the U. S.-U. S. S. R. by Spencer J. Bloch, I. Dolgachev, W. Fulton PDF

By Spencer J. Bloch, I. Dolgachev, W. Fulton

This quantity includes the lawsuits of a joint USA-USSR symposium on algebraic geometry, held in Chicago, united states, in June-July 1989.

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Additional info for Algebraic Geometry: Proceedings of the U. S.-U. S. S. R. Symposium Held in Chicago, June 20-July 14, 1989

Example text

PROOF. By Riemann-Roch: hO(E)+h°(E*(K)) = 4+h~(E). If h°(E*(K)) = 0, the assertion is obvious. Assume h°(E*(K)) ~ 0. Let (gs(D) be an effective saturated line subbundle o f E*(K). It defines an exact sequence 0 -~ (gs(D)---~ E*(K) --~ 9z(D') - o 0 for some divisor D' and an effective 0-dimensional cycle Z. We have cl(E*(K)) = - A = D + D ' < 0. Dualizing the above exact sequence and twisting it by egs(K), we get an exact sequence 0 --~ (gs(-D'+K)-~ E --~ 9z(-D) -~ 0. It yields h°(E) _>h°(-D'+K) = h°(D+A+K) >_h°(A) = 6.

FPHk(Z,ZA strong validity to bijectivity (X,YUZ) Remark. (Z,ZAY) FPHk(x,Y) that is equivalent is true > this with morphisms prove are bijective. > ~k(x,~zp~(X, ZAy ) ) ,,, > Mk(x,~mpQiX,y )) intersections. and using Remark assertion is the [2]. k >k+l. Suppose that Sk+IXCY and Z=SkX. assumption the Theorem is true for the pair (x,YUZ). Induction step: By inductive Since dimZ

Hodge-De Rham) ~p,an . E~ 'q= Hq(x(c) ,U(X,y)) ===> HP+q(x(c),Y(C) ;C) In g e n e r a l case is d i f f e r e n t ourselves nothing if X is known to only such case. rewrittenas Hq(x,~ interpreted as sequence Our main result First, y)). compact spectral cohomology a) Hodge t h e o r e m Let Theorem. sequence. p+qcx is the following case the c l a s s i c a l complete. 4. about is a c o m p l e t e variety, (X,Y) theorem generalizing to the toroidal (see [2]). e. b) the limit filtration on Hodge filteration.

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