I wanted to know if there is something analogous to Kontsevich's recursion formula for enumeration of genus zero curves in $\mathbb{C}\mathbb{P}^2$, for higher genus curves. There is a similar formula for genus one curves. See the book "Mirror Symmetry and Algebraic Geometry" by Katz, Page 211.

Any partial results known for g>1? That is, maybe its not known for all d, but for some small values of d?