Aleksey Zinger

On the geometry of genus 1 Gromov-Witten invariants

The mirror symmetry principle of string theory has led to astounding predictions for counts of holomorpic curves, especially for a quintic threefold (a degree 5 hypersurface in P^4). There has been much success in verifying these predictions in genus 0, in part due to a good undertanding of the geometry of genus 0 GW-invariants. In this talk, I will give an overview of geometric properties of genus 1 GW-invariants, including a relation between GW-invariants of a hypersurface and of the ambient projective space. These properties mimic well-known genus 0 properties. Taken together, they provide a method for computing genus 1 GW-invariants of all complete intersections and have led to the verification of the 1993 BCOV mirror symmetry prediction for genus 1 GW-invariants of a quintic threefold.