Algebraic and symplectic Gromov-Witten invariants coincide

Gromov-Witten invariants “count” (pseudo-) holomorphic curves on algebraic or symplectic manifolds. This amounts to intersection theory on moduli spaces of such curves. Because in general these are non-compact, singular and not of “expected dimension”, a rigorous mathematical definition is far from trivial. For a reasonably large class of manifolds including Fano and Calabi-Yau manifolds this has first been done using symplectic techniques by Ruan and Tian [Ru1], [RuTi1], [RuTi2]. The point in this approach is to restrict to sufficiently “generic” almost complex structures (tamed by the symplectic