Abstract
Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kahler base B. Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let La be convex line bundles over B, Aa smooth divisors of B arising as the zero loci of generic sections of La , and a particular fixed-point section of E. Further assume the {Aa} to be mutually disjoint. The manifold is a new manifold with tautological line bundles over new projective spaces in the geometry, where previously there was a simpler vector bundle in the given local geometry (Section 1.5). Thus, we compute genus-0 Gromov-Witten invariants of in terms of genus-0 Gromov-Witten invariants of B and of {Aa}, the matrix used for the symplectic reduction description of the fiber of the toric fibration E→B, and the restriction maps . The proofs utilize the fixed-point localization technique describing the geometry of and its genus-0 Gromov-Witten theory, as well as the Quantum Lefschetz theorem relating the genus-0 Gromov-Witten theory of A with that of B.
Highlights
Let La be convex line bundles over B, Aa smooth divisors of B arising as the zero loci of generic sections of La, and α : B → E a particular fixed-point section of
Points F ( z) on the overruled Lagrangian cone of the genus-0 T-equivariant Gromov-Witten theory of E α ( A) are certain H-valued formal functions, which we study in terms of
Let La be convex line bundles over B, and Aa smooth divisors of B arising as the zero loci of generic sections of La
Summary
The spaces M0,n,D are moduli spaces of (equivalence classes of) degree-D stable maps into M of genus-0 (possibly nodal) compact connected holomorphic curves with n marked points. Take the ring of coefficients for Novikov’s variables to be the (super-commutative) power series ring (with coefficients in the field of fractions (λ ) := (λ1, , λN ) , in all of our applications) in the formal coordinates along H * (M , ) , and require the variables t0 ,t1, to vanish when Novikov’s variables and formal coordinates along H * (M ) are all set to zero This gives a Novikov ring that is consistent with the formula for IE α( A) in our Main Theorem. The Lagrangian cone of the T-equivariant genus-0 Gromov-Witten theory of E α ( A) lies in the corresponding symplectic loop space ( , Ω) as above. The graded homogeneity, defined by degrees of formal variables, makes the quantum cup product a degree 0 operation, the J-function graded homogeneous of degree 1, and z of degree 1
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