Genus-zero two-point hyperplane integrals in the Gromov–Witten theory

Aleksey Zinger
2009 Communications in analysis and geometry  
In this paper, we compute certain two-point integrals over a moduli space of stable maps into projective space. Computation of onepoint analogs of these integrals constitutes a proof of mirror symmetry for genus-zero one-point Gromov-Witten (GW) invariants of projective hypersurfaces. The integrals computed in this paper constitute a significant portion in the proof of mirror symmetry for genus-one GW-invariants completed in a separate paper. These integrals also provide explicit mirror
more » ... icit mirror formulas for genus-zero twopoint GW-invariants of projective hypersurfaces. The approach described in this paper leads to a reconstruction algorithm for all genus-zero GW-invariants of projective hypersurfaces. Aleksey Zinger Proof of Lemma 1.1 986 3.3 Proof of Lemma 1.2 991 3.4 Proof of Lemma 3.1 994 Acknowledgment 998 References 998 1 The fiber of V 0 over a point [C, f] ∈ M 0,m (P n−1 , d) is H 0 (C; f * O P n−1 (a))/ Aut(C, f). 2 In Chapters 29 and 30 of [11], the roles of the marked points 1 and 2 in (1.2) are switched; the analogs of V 0 and V 0 over M 0,2 (P n , d) are denoted by E 0,d and E 0,d , respectively. Aleksey Zinger equivariant cohomology. In particular, V 0 , V 0 and V 0 have well-defined equivariant Euler classes e(V 0 ), e(V 0 ), e(V 0 ) ∈ H * T M 0,m (P n −1 , d) . These classes are related by where x ∈ H * T (P n−1 ) is the equivariant hyperplane class. For each i = 1, 2, . . . , m, there is also a well-defined equivariant ψ-class, the first chern of the vertical cotangent line bundle of U pull-backed to M 0,m (P n−1 , d) by the section Since M 0,m (P n−1 , d) is a smooth stack (orbifold), there is an integrationalong-the-fiber homomorphism M0,m(P n−1 ,d)
doi:10.4310/cag.2009.v17.n5.a4 fatcat:k5euvw5ksjbd7c2dunuwfvffgi