Reduced genus-two Gromov-Witten Invariants for complex manifolds

In this article, we construct the reduced genus-two Gromov-Witten invariants for certain almost K"{a}hler manifold $(X,\omega ,J)$ such that $J$ is integrable and satisfies some regularity conditions. In particular, the standard projective space $({\mathrm{\P }}^n,{\omega }_0,J_0)$ of dimension $n\le 7$ satisfies these conditions. This invariant counts the number of simple genus-two $J$-holomorphic curves that satisfy appropriate number of constraints.