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Zbl 0786.14006
Voisin, Claire
On zero cycles of some hypersurfaces with automorphism. (Sur les zéro-cycles de certaines hypersurfaces munies d'un automorphisme.) (French)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 19, No.4, 473-492 (1992). ISSN 0391-173X

This paper gives a simple proof of Bloch's conjecture $$(p\sb g(S)=0\Longrightarrow\text{CH}\sp 0\sb 0(S)\simeq\text{Alb}(S))$$ for any Godeaux surface, quotient of a complete intersection by a finite group. The proof is also applied in higher dimension and gives the following result: Let $X\subset\bbfP\sp 4$ be a quintic hypersurface invariant under an involution acting trivially on $H\sp{3,0}(X)$. Then $\text{CH}\sb 0(X)\sp -=\{0\}$. To get this result one needs to show that $(JX)\sp -$ is parametrized by algebraic cycles, which is done using the Noether-Lefschetz locus for invariant quintic fourfolds containing $X$.
[C.Voisin (Orsay)]

MSC 2000:: *14C25 Algebraic cycles
14C05 Parametrization
14J70 Hypersurfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties

Keywords: Chow group; Bloch's conjecture; Godeaux surface; algebraic cycles

Cited in: Zbl 1191.14011

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