On The Action Of Symplectic Automorphisms Of Fibred Surfaces On CH₀

Let S be a complex smooth projective algebraic surface.Bloch’s conjecture predicts that the symplectic automorphism group Auts(S)acts trivially on the Albanese kernel CH0(S)alb of the 0-th Chow group CH0(S).In the paper,we prove that the conjecture holds for some fibred surfaces of lower genus.Suppose that S is of Kodaira dimension one.We show that Auts(S)acts trivially on CH0(S)alb,unless possibly the geometric genus and the irregularity satisfy pg(S)=q(S)∈{1,2}.In the exceptional cases,the image of the homomorphism Auts(S)→Aut(CH0(S)alb)has order at most 3.Our arguments actually take care of the group Autf(S)of fibration-preserving automorphisms of elliptic surfaces f:S → B.We also prove that,if σ∈ Autf(S)induces the trivial action on Hi,0(S)for i>0,then it induces the trivial action on CH0(S)alb.As a by-product we obtain that if S is an elliptic K3 surface,then Autf(S)Auts(S)acts trivially on CH0(S)alb.Let f:S→C be a fibration by genus 2 curves,where C is a curve of genus g(C)≥2.It is well-known that the relative irregularity qf:=q(S)-g(C)satisfies 0≤qf≤2.If qf>0,we prove that any symplectic automorphism of finite order of S acts trivially on CH0(S)alb.In the qf=0 case,assuming that f*(ωS/C)is an ample and unstable vector bundle and deg(f*(ωS/C))≥3,we show that any fibrationpreserving symplectic automorphism of finite order of S acts trivially on CH0(S)alb except for some limited cases.In the exceptional cases,the image of the homomorphism Autf(S)∩Auts(S)→ Aut(CH0(S)alb)has order at most 3.Let f:S→C be a smooth hyperelliptic fibration,we show that any symplectic automorphism of finite order of S acts trivially on CH0(S)alb.Finally,let S be a surface of general type with pg≥3.Suppose that the canonical linear system of S is composite with a pencil and S has a fibration of genus 2.Then we can prove that,any symplectic automorphism of S acts trivially on CH0(S)alb.