Let S be a complex smooth projective surface with a genus two fibration, and Aut s (S) the group of symplectic automorphisms, fixing every holomorphic 2-forms (if any) on S. Based on the work of Jin-Xing Cai, we show that, if (O S ) 5, then |Aut s (S)| 2. Then we verify, under some conditions, that Aut s (S) acts trivially on the Albanese kernel CH 0 (S) alb of the 0-th Chow group, which is predicted by a conjecture of Bloch and Beilinson.As a consequence, if an automorphism Aut(S) acts trivially on H i,0 (S) for 0 i 2, then it also acts trivially on CH 0 (S) alb . Introduction 259 2. Preliminaries 2632.1.Fibered surfaces and their automorphisms 263 2.2.The induced action on the Albanese variety and the 0-th Chow group 264 2.3.Useful criteria for a symplectic automorphism to act trivially on CH 0 (S) alb 266 3. Symplectic automorphisms of surfaces with a genus two fibration 267
Du et al. (Thu,) studied this question.