In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degener...In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degeneration of 2-dimensional(perturbed)J-holomorphic maps to 1-dimensional gradient segments.We consider the case when the Floer equations are S^(1)-invariant on parts of their domains whose adiabatic limit has positive length as ε→0,which we call thimble-flow-thimble configurations.The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration.In particular,our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of L by Fukaya-Oh-Ohta-Ono and the pearly complex of L Lalonde and Biran-Cornea.It also provides another proof of the present authors’earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.展开更多
We use the technique of Ruan(1999)and Li and Ruan(2001)to construct the virtual neighborhoods and show that the Gromov-Witten invariants can be defined as integrals over the top strata of the virtual neighborhoods.We ...We use the technique of Ruan(1999)and Li and Ruan(2001)to construct the virtual neighborhoods and show that the Gromov-Witten invariants can be defined as integrals over the top strata of the virtual neighborhoods.We prove that the invariants defined in this way satisfy all the axioms of Gromov-Witten invariants summarized by Kontsevich and Manin(1994).展开更多
文摘In the present paper,we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing,as the prototype of the partial collapsing degeneration of 2-dimensional(perturbed)J-holomorphic maps to 1-dimensional gradient segments.We consider the case when the Floer equations are S^(1)-invariant on parts of their domains whose adiabatic limit has positive length as ε→0,which we call thimble-flow-thimble configurations.The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration.In particular,our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of L by Fukaya-Oh-Ohta-Ono and the pearly complex of L Lalonde and Biran-Cornea.It also provides another proof of the present authors’earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.
基金supported by National Natural Science Foundation of China(Grant Nos.11890660,11821001,11890663,11871352 and 1196131001)。
文摘We use the technique of Ruan(1999)and Li and Ruan(2001)to construct the virtual neighborhoods and show that the Gromov-Witten invariants can be defined as integrals over the top strata of the virtual neighborhoods.We prove that the invariants defined in this way satisfy all the axioms of Gromov-Witten invariants summarized by Kontsevich and Manin(1994).