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2]) Let E and F be elliptic curves (Riemann surfaces of genus 1). We propose to calculate the Chow group of the quotient surface X = (F × E)/{1, σ}, where σ is a fixed-point-free involution on F × E obtained by fixing a point η ∈ E of order 2, η 0, and taking σ( f, e) = (− f, e + η). Let E = E/{1, η}. There is a natural map ρ : X → E with all fibres of ρ F. Notice Γ(X, Ω1X ) Γ(F × E, Ω1F×E ){1,σ} Γ(E, Ω1E ) ⊕ Γ(F, Ω1F ) Γ(E, Ω1E ) {1,σ} C since the automorphism f → − f acts by −1 on Γ(F, Ω1F ).

N. Tyurin, Five lectures on three-dimensional varieties (in Russian), Uspehi Mat. Nauk, 27 (1972), no. 5, (167) 3–50. [Translation: Russian Math. Surveys, 27 (1972), no. ] 20 Lecture 1 Some references for the conjectures at the end of Lecture 1 are: [13] S. Bloch, Some elementary theorems about algebraic cycles on abelian varieties, Invent. , 37 (1976), 215–228. [14] S. Bloch, An example in the theory of algebraic cycles, pp. , no. 551, Springer, Berlin (1976). [15] S. Kleiman, Algebraic cycles and the Weil conjectures, pp.

For further discussion of this sort of conjecture, the reader can see Bloch [14]. References for Lecture 1 For general facts about algebraic surfaces over C, a good reference is [1] A. Beauville, Surfaces alg´ebriques complexes, Ast´erisque, 59 (1978), as well as the references cited there. 2) is done more generally in [2] S. Bloch, A. Kas, and D. , 33 (1976), 135–145, along with other surfaces with Pg = 0 and Kodaira dimension < 2. 5) is due to H. Inose and M. Mizukami. Details and other analogous examples are in [3] H.