Question

After studying some foundation of Gromow-Witten invariants, I now would like to see an explicit computation. I heard that one should first take a look at the total space of $\mathcal{O}(-1)^{\oplus2}$ over $\mathbb{P}^1$ or the total space of the canonical bundle of Fano surface (local Calabi-Yau). They can be worked out very explicitly via equivariant cohomology and localization. (Or there may be more tractable examples)

Could someone kindly suggest a paper or lecture note where I can start learning these examples and technique? Any suggestion is welcome.

Answer 1

Gromov-Witten classes, quantum cohomology, and enumerative geometry (by Kontsevich & Manin)

J-holomorphic curves and symplectic topology (by McDuff & Salamon)

A tutorial on quantum cohomology (by Givental)

Answer 2

Hope this is among the lines of what you're looking for.

http://mat.uab.es/~kock/invitation.html

The exercises are a very valuable part of this book, for it contains a handful of nice exercises.

As for lecture notes, Renzo Cavalieri has some nice notes you might find helpful. He knows a good deal localization.

http://www.math.colostate.edu/~renzo/