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Perturbations by analytic discs along generating CR-submanifolds of C^{n}are considered. In the case all partial indices of a closed path$p$in a generating CR-fibration {$M$(ξ)}_{ξ∈∂$D$}are greater or equal to −1 we can completely parametrize all small holomorphic perturbations of the path$p$along the fibration {$M$(ξ)}_{ξ∈∂$D$}. In this case we also study the geometry of perturbations by analytic discs and their relation to the conormal bundle of the fibration.

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We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to$S$^{1}-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small$J$-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small$J$-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.