On the Kernel of the Gysin Homomorphism on Chow Groups of Zero Cycles

Given a smooth projective connected surface over C embedded into a projective space P d and a smooth projective curve C embedded into the surface we study the kernel of the Gysin homomorphism between the Chow groups of 0-cycles of degree zero of the curve and the surface induced by the closed embedding. Following the approach of Bannerjee and Guletskii we prove that the kernel of the Gysin homomorphism is a countable union of translates of an abelian subvariety A inside the Jacobian J of the curve C . We also prove that there is a c -open subset U 0 contained in the set U ⊂ ( P d ) * parametrizing the smooth projective curves such that A = 0 or A = B for all curves parametrized by U 0 , where B is the abelian subvariety of J corresponding to the vanishing cohomology H 1 ( C , Q ) van of C . We give a background of algebraic cycles, Chow groups, Hodge structures, the Abel–Jacobi map, Lefschetz pencils and the irreducibility of the monodromy representation.