In this article, using the WDVV equation, the author first proves that all Gromov-Witten invariants of blowups of surfaces can be computed from the Cromov- Witten invariants of itself by some recursive relations. Furt...In this article, using the WDVV equation, the author first proves that all Gromov-Witten invariants of blowups of surfaces can be computed from the Cromov- Witten invariants of itself by some recursive relations. Furthermore, it may determine the quantum product on blowups. It also proves that there is some degree of functoriality of the big quantum cohomology for a blowup.展开更多
Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kähler base B. Then each torus-fixed point of the toric manifold fiber defines a section of t...Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kähler 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.展开更多
基金Supported in part by NSF of China (1017114, 10231050 and NCET)
文摘In this article, using the WDVV equation, the author first proves that all Gromov-Witten invariants of blowups of surfaces can be computed from the Cromov- Witten invariants of itself by some recursive relations. Furthermore, it may determine the quantum product on blowups. It also proves that there is some degree of functoriality of the big quantum cohomology for a blowup.
文摘Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kähler 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.
