Some applications of localization to enumerative problems. (Q5954581)

The author surveys applications of the localization technique to enumerative geometry, focussing on the cases of Schubert calculus on the flag manifold and Gromov-Witten invariants of rational curves. NEWLINENEWLINEFirst, equivariant cohomology theory and the Atiyah-Bott localization theorem are reviewed. The next section gives \textit{J. Kong}'s [Schubert calculus on flag manifolds via localization, Ph.D. dissertation, Univ. of Utah, Salt Lake City, 2000] results on the Schubert calculus of the partial flag manifolds. Finally, it is described how formulae for genus zero Gromov-Witten invariants of a smooth projective variety \(X\) can be obtainedNEWLINEby localization even if there is no group action on \(X\). By applying this method, the quantum Lefschetz hyperplane theorem and the genus zero reconstruction theorem are derived.