2016 Theses Doctoral

# Kuranishi atlases and genus zero Gromov-Witten invariants

Kuranishi atlases were introduced by McDuff and Wehrheim as a means to build a virtual fundamental cycle on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This thesis considers genus zero Gromov-Witten invariants on a general closed symplectic manifold. We complete the construction of these invariants using Kuranishi atlases. To do so, we show that Gromov-Witten moduli spaces admit a smooth enough Kuranishi atlas to define a virtual fundamental class in any virtual dimension. In the process, we prove a stronger gluing theorem. Once we have defined genus zero Gromov-Witten invariants, we show that they satisfy the Gromov-Witten axioms of Kontsevich and Manin, a series of main properties that these invariants are expected to satisfy. A key component of this is the introduction of the notion of a transverse subatlas, a useful tool for working with Kuranishi atlases.

## Files

- Castellano_columbia_0054D_13218.pdf binary/octet-stream 1.01 MB Download File

## More About This Work

- Academic Units
- Mathematics
- Thesis Advisors
- McDuff, Dusa
- Degree
- Ph.D., Columbia University
- Published Here
- April 11, 2016