Real gromov-witten invariants on the moduli space of genus 0 stable maps to a smooth rational projective space

We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a rational projective surface. If a target space is equipped with a real structure, , anti-holomorphic involution, then the results have real enumerative applications. | Turk J Math 32 (2008) , 155 – 186. ¨ ITAK ˙ c TUB Real Gromov-Witten Invariants on the Moduli Space of Genus 0 Stable Maps to a Smooth Rational Projective Space∗ Seongchun Kwon Abstract We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a rational projective surface. If a target space is equipped with a real structure, , anti-holomorphic involution, then the results have real enumerative applications. Firstly, we can define a real version of Gromov-Witten invariants. Secondly, we can prove the invariance of Welschinger’s invariant in algebraic geometric category. Key Words: Gromov-Witten invariant, enumerative invariant, transversality, intersection multiplicity, real structure. 1. Introduction Let M k (X, β) be the moduli space of stable maps from a k-pointed arithmetic genus 0 curve to X, representing a 2nd homology class β. Let [Υ1 ], . . . , [Υk ] be Poincar´e duals to the homology classes represented by Υ1 , . . . , Υk , where Υ1 , . . . , Υk are pure dimensional varieties in the target space X. The Gromov-Witten invariant on M k (X, β) is defined as: Iβ ([Υ1 ], . . . , [Υk ]) := ev1∗ ([Υ1 ]) ∪ · · · ∪ evk∗ ([Υk ]), M k (X,β) MSC 2000 Mathematics Subject Classification: Primary: 14C17 Secondary: 14C25. ∗ Dedicated to the originator Gang Tian 155 KWON where evi is an i-th evaluation map. The Gromov-Witten invariant Iβ ([Υ1 ], . . . , [Υk ]) may be non-trivial only when codim(Υi ) = dimM k (X, β). We say that the GromovWitten invariant has an enumerative meaning if Iβ ([Υ1 ], . . . , [Υk ]) equals to the actual number of points in ev1−1 (Γ1 )∩· · ·∩evk−1 (Γk ), where Γ1 , . . . , Γk are any pure dimensional varieties in a general position such that [Γi ] = [Υi ], i = 1, . . . , k. So, the GromovWitten invariant counts the number of stable maps whose i-th marked point maps into Γi if it has an enumerative implication. Note that the number of intersection points in ev1−1 (Γ1 ) ∩ · · · ∩ .